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The author or concept searched is found in the following 53 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Accessibility | Bigelow | I 126 Accessibility relation: can be restricted: for example, by the requirement that a possible world w from the accessible possible world u does not contain any individuals that do not also contain u. That is, that the one world is only a re-structured one of the other. This would e.g. contradict Lewis counterpart theory. >Possible worlds, >Possible worlds/Lewis, >Counterpart theory/Lewis. I 136 Definition weak centering/accessibility/Lewis/Bigelow/Pargetter: we will say that degrees of accessibility are weakly centered if no possible world is more accessible from a given possible world than this possible world itself. This is best satisfied with: d(w, w) = 0. N.b.: this ensures that some additional sentences will be true in all possible worlds, in addition to those guaranteed by the above axioms. These are derivable as theorems if we take the following axioms: A9 (reflexivity) and A16. (B would > would g)> (b> g) Everyday translation: no world can be more accessible to a world than this world is accessible to itself. This leaves open the possibility that some possible worlds have the accessibility "zero-distance" from the world w. Definition strong centering/Lewis/Bigelow/Pargetter: (in the semantics for counterfactual conditionals): no possible world can be accessible from a given world as this world is accessible from itself. This is best satisfied: If w is not equal to u, then either d(w, u) is undefined or d (w, u) > 0. This semantic condition allows a completeness proof for the axiom system which we obtain by adding the axiom of the strong centering to the above axioms: (a ∧ b)> (a would > would b) >Completeness. Counterfactual logic/Lewis/Bigelow/Pargetter: with these axioms, we get Lewis' favored counterfactual logic. BigelowVsStrong centering. Modal logic/Axiom system/Bigelow/Pargetter: our system will be the one Lewis calls VW: V ": "variably strict", "W". "Weakly centered". 139 Accessibility Relation/Bigelow/Pargetter: Problem: we must restrict it, and for a proof of completeness for S5, we must show that it is reflexive, transitive, and symmetric. >Systems S4/S5. S5/Canonical Model/Bigelow/Pargetter: does not only contain the Leibnizian necessity (truth in all worlds). S5: is interesting because it allows a reductionist access to possible worlds. >Reduction. Necessity: in the canonical model a proposition is necessarily true if it is true in all accessible possible worlds. >Necessity. Possible worlds: when they are designed as the maximum consistent extensions of S5, they disintegrate into different equivalence classes. ((s) i.e. for each world there is an additional sentence describing an individual with possibly different descriptions which do not contradict the other sentences). >Possible worlds, >Equivalence classes. Equivalence classes/accessibility/Bigelow/Pargetter: within an equivalence class, all worlds are accessible to one another. But between equivalence classes there is no accessibility from one possible world to the other. ((s), then the maximum consistent extensions must be something other than I suspected, then an extension will modify all existing propositions and makes them incomparable with a subset of the previous consistent set). >Maximum consistent. Accessibility/canonical model/Bigelow/Pargetter: in a canonical model, not all possible worlds are accessible to one another. >Canonicalness. We show it this way: Fa: (spelling: latin a) be an atomic sentence that can be added to the axioms of S5, or its negation, whereby the result being a maximally consistent set or world. With this, we are constructing a world where Fa is true. If it were accessible from all other worlds, MFa would be true in all possible worlds. But a proposition which is true in all worlds must be a theorem. But we know that Fa is not Problem: R2 (universal substitution) would ensure that Mα would be true for every α, even if α = (b u ~ b). Interpretation/Bigelow/Pargetter: if the intended interpretation of S5 is Leibnizean, as we hope ((s) necessity = truth in all worlds) then it follows that this intended interpretation of S5 is not captured by the canonical model. Possible world/Bigelow/Pargetter: that supports what we want to show, namely that possible worlds are not sets of sentences. Accessibility/Bigelow/Pargetter: ...and it also shows that the accessibility relation... I 140 ... which is relevant to alethic modal logic, is not an equivalence relation. Logical truth/Bigelow/Pargetter: is truth in all possible worlds (pro Leibniz!) not merely truth in all accessible worlds? >Logical truth. I 242 Accessibility Relation/Accessibility/Bigelow/Pargetter: nevertheless, we do not believe that the accessibility relation supervenes to properties and relations of the first level of the possible worlds, but on higher level universes! >Universals, >Supervenience. Two worlds can be perfectly similar in terms of universals of the first level and still have different accessibility relations! Humean World/Bigelow/Pargetter: is an example for the failure of the supervenience of the 1st level of the accessibility relation. >Humean world. For example, "all Fs are Gs", whereby F and G are universals of the 1st level, and higher-level universals that supervene on them. I 243 Counterfactual conditional: then also counterfactual conditionals should be valid like: "If this thing had been an F, it would have been a G". We would never be sure if it was a law, even if there were no exceptions. This uncertainty is reflected in uncertainty as to whether the counterfactual conditional is true. >Counterfactuals, >Counterfactual conditionals. Even if we live in a world with laws, we allow the possibility that this world is a Humean world. It might be that the generalization is correct, but without necessity. The world would look the same in both cases. Humean World/Bigelow/Pargetter: is, with respect to the actual world, precisely a world, which is the same, without laws. For other worlds there would be other Humean worlds. I 245 Accessibility/Bigelow/Pargetter: nevertheless, there are strong reasons to believe in a supervenience of the accessibility relation on the contents of the world. This allows us to assume that the contents of the 1st level do not exhaust all the contents of the world. Combinatorial theories. Therefore, must accept higher-level universals, and hence the property theory of the world's properties. Universals/Natural Law/Bigelow/Pargetter: Higher-level universals are the key to laws. >Levels/order, >Description levels. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Addition | Hilbert | Berka I 122 Addition/disjunction/union/Hilbert: the addition of the numbers can be traced back to the disjunction of predicates. If F and G are incompatible predicates, and if the number m is assigned to the predicate F, and number n to G, the predicate F v G corresponds to the number m + n. I 122 Extended function calculus: with the extended function calculus, numerical equations such as 1 + 1 = 2 become purely logical, provable sentences. E.g. 1 + 1 = 2, logical form: (F)(G)([Unv (F,G) & 1(F) & 1(G)] > 2(F v G)).(1) >Natural deduction, >G. Gentzen, >Derivation, >Derivability, >Axioms, >Axiom systems, >Calculus, >Logic. 1. D. Hilbert & W. Ackermann: Grundzüge der Theoretischen Logik, Berlin, 6. Aufl. Berlin/Göttingen/Heidelberg 1972, §§ 1,2. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Axioms | Bigelow | I 119 Axioms/Intuition/Bigelow/Pargetter: nevertheless, intuitions should not be allowed to throw over entire axiom systems. E.g. the principle of distribution of the disjunction can be explained as follows: Suppose that in natural languages a conditional "If A, then B" is equivalent to a quantification over situations: "In all situations where A applies, B also applies." Then you could read the distribution of the disjunction like this: Logical form: (x)((Ax v Bx) would > would Cx) (x) (Ax would > would Cx) u (x)(Bx would > would Cx)). This is indisputably logical! >Distribution, >Disjunction, >Counterfactual conditional. Bigelow/Pargetter: therefore the quantified form seems to capture the everyday language better than the unquantified. E.g. "In any situation where you would eat..." This is then a logical truth. I 120 This again shows the interplay of language and ontology. Axioms/Realism/Bigelow/Pargetter: our axioms are strengthened by a robust realistic correspondence theory. And this is an argument for a conservative, classic logic. >Correspondence theory. I 133 Theorems/Bigelow/Pargetter: Need a semantic justification because they are derived. This is the foundation (soundness). >Foundation. Question: Will the theorems also be provable? Then it is about completeness. >Proofs, >Provability, >Completeness. Axioms/Axiom/Axiom system/Axiomatic/Bigelow/Pargetter: can be understood as a method of presenting an interpretation of the logical symbols without using a meta-language (MS). >Metalanguage. That is, we have here implicit definitions of the logical symbols. This means that the truth of the axioms can be seen directly. And everyone who understands it can manifest it by simply repeating it without paraphrasing it. >Definition, >Definability. 134 Language/Bigelow/Pargetter: ultimately we need a language which we speak and understand without first establishing semantic rules. In this language, however, we can later formulate axioms for a theory: that is what we call Definition "extroverted axiomatics"/terminology/Bigelow/Pargetter: an axiomatics that is developed in an already existing language. Definition introverted axiomatics/terminology/Bigelow/Pargetter: an axiomatics with which the work begins. Extrovert Axiomatics/Bigelow/Pargetter: has no problems with "metatheorems" and no problems with the mathematical properties of the symbols used. We already know what they mean. Understanding and accepting the axioms is one thing here. That is, the implicit definition precedes the explicit definition. We must understand what we are working with. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Axioms | Cresswell | Hughes I 120 Axiomatization/propositional calculus/Hughes/Cresswell: done in other way than with the propositional calculus. Instead of axioms we use axiom schemes and parallel theorem schemes, i.e. general principles which determine that any well-formed formula (wff) of a certain shape is a theorem. >Theorems, >Propositional calculus, >Predicate calculus, >Predicate logic, >Propositional logic, >Axiom systems. |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 Hughes I G.E. Hughes Maxwell J. Cresswell Einführung in die Modallogik Berlin New York 1978 |
| Axioms | Dedekind | Thiel I 208 Axioms/Dedekind/Thiel: From axioms, evidence, i.e. a brief insight into their truth, is required. Euclid's axioms are manageable, today's axiom systems can grow rapidly and can become unclear. From the axioms, every theorem should be derivable. This derivability, however, exists separately for each sentence. >Derivation, >Derivability, >Axiom systems. The plural of "geometries" shows a change in the concept of geometry itself. >Geometry. I 209 Dedekind was the first to try to axiomatize the calculating discipline of arithmetic (not Peano). Definition "basic properties"/Dedekind: are those which cannot be derived from each other. Cf. >Properties. Dedekind Peano Axioms: (1) 1 ε Z (2) (m)((m ε Z) > (m' ε Z)) (3) (m ε Z)(n ε Z)((m' = n') > (m = n)) (4) (m ε Z) ~(m' = 1) (5) (m ε Z)((E(m) > E(m')) >(E(1) > (n ε Z)((E)(n)) I 210 Dedekind and Peano use in the 5th axiom instead of "ε" "m in the set M". Thiel: that is not necessary. We convince ourselfs that the natural numbers satisfy the axiom system by inserting. The five axioms are then transformed into true sentences, for which we also say that the natural numbers with the properties and relations mentioned form a model of the axiom system. >Models. I 211 The constructive arithmetic with the calculus N and the construction equality of counting signs provides an operative model of the axioms. Mathematicians do not work like this either in practice or in books. The practice is not complete. I 213 Insisting on "clean" solutions only arises with metamathematical needs. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Axioms | d’Abro | A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 35 Axiomatics/d'Abro: This new science was developed mainly by the formalists Hilbert and Peano. >Formalism. 37 Hilbert/d'Abro: Examples of Hilbert's typical claims: 1. Two different points, A and B, always form a straight line. 2. Three different points, A, B, and C, which do not lie on a straight line, always form a plane. 3. Of three points lying on a straight line, there is one and only one between the other two. 4. If the segment AB is equal to the segments A'B 'and A''B'', then A'B' is equal to A''B''. The N.B. of Hilbert's postulates: points, lines, and planes are not the only quantities which satisfy these relations: with some imagination others can be found. E.g It originally refers to plane geometry and can be given a different meaning: circles as new lines, with angles as distances. All relations are fulfilled, so the new model and the old (Euclidean) model can be regarded as different models or so-called "concrete representations", both corresponding to the postulates. >Models. 38 It may seem absurd, but Hilbert warns against assigning a priori certain characteristics to the points and lines which he mentions in his postulates. We can replace the words point, straight, plane, in all postulates by letters a, b, c. If we then employ points, lines, and planes, we obtain the Euclidean geometry, if we employ others, whose relations, however, must be the same, we have a new model between point, lines and planes. They are isomorphic. >Isomorphism. For example, the new elements are expressed by a group of three of numbers and by algebraic terms which relate these numbers to one another. He had this idea when he chose cartesian coordinates instead of points, lines and planes. The fact that the new elements, here numerical, satisfied Hilbert's postulates, proves only that the simple geometrical ways of concluding and the Cartesian method are equivalent to analytical geometry. 39 This proves the logical equivalence of the geometric and arithmetic continuum. Long before Hilbert, mathematicians had realized that mathematics has to do with relationships, and not with content. With Hilbert's postulates, we can create the Euclidean geometry, even without knowing what is meant by point, line and plane. 49 The achievements of axiomatics: 1. They are of invaluable value, from the analytical as well as from the constructive point of view. 2. It has shown that mathematics is about relationships and not about content. 3. It has shown that logic itself cannot confirm the consistency. 4. It has also shown that we have to go beyond axiomatics and have to show their origin. >Ultimate justification, >Foundation, >Axiom systems. |
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| Axioms | Hilbert | Berka I 294 Definition/Axiom/Hilbert: the established axioms are at the same time the definitions of the elementary concepts whose relations they regulate. ((s) Hilbert speaks of relationships, not of the use of concepts). >Definitions, >Definability, >Basic concepts. Independence/Axiom/Hilbert: the question is whether certain statements of individual axioms are mutually dependent, and whether the axioms do not contain common components which must be removed so that the axioms are independent of each other(1). >Independence. 1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. S 299-301). --- Thiel I 262 We consider the first three axioms of Hilbert: 1. There are exactly two straight lines at each of two distinct points P, Q, which indicate(2) with P and Q. 2. For every line g and to any point P, which does not indicate with it, there is exactly one line that is indicated with P, but with no point of g. 3. There are three points which do not indicate with one and the same straight line. In Hilbert's original text, instead of points one speaks of "objects of the first kind" instead of straight lines of "objects of the second kind" and instead of the incidence of "basic relation". Thus, the first axiom is now: For each of two different objects of the first kind, there is precisely one object of the second kind, which is in a basic relation with the first two. Thiel I 263 If the axioms are transformed quantifier-logically, then only the schematic sign "π" (for the basic relation) is free for substitutions, the others are bound by quantifiers, and can no longer be replaced by individual names of points or lines. >Quantification, >Quantifiers. They are thus "forms of statements" with "π" as an empty space. >Propositional functions. They are not statements like those before Hilbert's axioms, whose truth or falsehood is fixed by the meanings of their constituents. >Truth values. In the Hilbert axiom concept (usually used today), axioms are forms of statements or propositional schemata, the components of which must be given a meaning only by interpretation by specifying the variability domains and the basic relation. The fact that this can happen in various ways, shows that the axioms cannot determine the meaning of their components (not their characteristics, as Hilbert sometimes says) themselves by their co-operation in an axiom system. Thiel I 264 Multiple interpretations are possible: e.g. points lying on a straight line, e.g. the occurrence of characters in character strings, e.g. numbers. Thiel I 265 All three interpretations are true statements. The formed triples of education regulations are models of our axiom system. The first is an infinite, the two other finite models. >Models, >Infinity. Thiel I 266 The axioms can be combined by conjunction to form an axiom system. >Conjunction. Through the relationships, the objects lying in the subject areas are interwoven with each other in the manner determined by the combined axioms. The regions V .. are thereby "structured" (concrete and abstract structures). >Domains, >Structures (Mathematics). One and the same structure can be described by different axiom systems. Not only are logically equivalent axiom systems used, but also those whose basic concepts and relations differ, but which can be defined on the basis of two systems of explicit definitions. Thiel I 267 Already the two original axiom systems are equivalent without the assumption of reciprocal definitions, i.e. they are logically equivalent. This equivalence relation allows an abstraction step to the fine structures. In the previous sense the same structures, are now differentiated: the axiom systems describing them are not immediately logically equivalent, but their concepts prove to be mutually definable. For example, "vector space" "group" and "body" are designations not for fine structures, but for general abstract structures. However, we cannot say now that an axiom system makes a structure unambiguous. A structure has several structures, not anymore "the" structure. Thiel I 268 E.g. body: the structure Q has a body structure described by axioms in terms of addition and multiplication. E.g. group: the previous statement also implies that Q is also e.g. a group with respect to the addition. Because the group axioms for addition form part of the body axioms. Modern mathematics is more interested in the statements about structures than in their carriers. From this point of view, structures which are of the same structure are completely equivalent. >Indistinguishability. Thiel: in algebra it is probably the most common to talk of structures. Here, there is often a single set of carriers with several links, which can be regarded as a relation. Thiel I 269 E.g. relation: sum formation: x + y = z relation: s (x, y, z). In addition to link structures, the subject areas often still carry order structures or topological structures. Thiel I 270 Bourbaki speaks of a reordering of the total area of mathematics according to "mother structures". In modern mathematics, abstractions, especially structures, are understood as equivalence classes and thus as sets. >N. Bourbaki, >Equivalence classes. 2. Indicate = belong together, i.e. intersect, pass through the point, lie on it. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Axioms | Tarski | Berka I 400 Axiom System/Tarski: problem: the choice of axioms is arbitrary and depends on the level of knowledge.(1) Berka I 530 Axiom System/Tarski: methodological problem: to simply assume that an axiom system is complete and therefore can solve every problem in its domain.(2) >Arbitrariness, >Truth theory, >Method, >Truth definition. 1. A.Tarski, Grundlegung der wissenschaftlichen Semantik, in: Actes du Congrès International de Philosophie Scientifique, Paris 1935, Bd. III, ASI 390, Paris 1936, S. 1-8 2. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Axioms | Zermelo | Thiel I 341 Axioms/Zermelo/Thiel: Zermelo himself has emphasized that the question of the origin and validity domain of his set theoretical axioms remains philosophically excluded. It will be at most possible to say that logicism has survived in the axiomatic set theory insofar as cardinal numbers are defined there as definite sets and a cardinal number arithmetic is created with them. >Axioms, >Axiom systems, >Set theory. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Canonicalness | Bigelow | I 137 Canonical models/Bigelow/Pargetter: deal with maximally consistent sets of sentences to provide completeness proofs. >Models, >Completeness, >Proofs, >Provability. Canonical models were discovered only after Hughes/Cresswell 1968(1), they were described in the later work (Hughes/Cresswell 1984)(2). Definition completeness theorem/Bigelow/Pargetter: is a theorem that proves that if a proposition in a certain semantics is guaranteed true this proposition can be proved as a theorem. How can we prove this? How can we prove that each such proposition is a theorem? Solution: we prove the contraposition of the theorem: Instead: If a is assuredly true in semantics, a is a theorem We prove If a is not a theorem, it is not assuredly true in semantics. >Semantics. Then we prove this by finding an interpretation according to which it is false. >Interpretation, >Valuation. Def canonical model/Bigelow/Pargetter: provides an interpretation which guarantees that every non-theorem is made wrong in at least one possible world. >Possible worlds. I 138 We begin that there will be a sentence a, for which either a or ~a is a theorem. This can be added to the axioms to give another consistent set of sentences. Maximum consistent set of sentences/Bigelow/Pargetter: it can be proved that for the axiom systems which we deal with, there is always a maximally consistent set of sentences. >Maximum consistent. That is, a consistent set of sentences to which no further sentence can be added without making the set inconsistent. That is, for each sentence g is either γ in the set or ~ γ. W: be the set of all maximally consistent extensions of the axiom system with which we have begun. >Expansion. 1. Hughes, G. E. and Cresswell, M.C. (1968) An introduction to modal logic. London: Methuen. 2. Hughes, G. E. and Cresswell, M.C. (1984) A companion to modal logic. London: Methuen. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Completeness | Bigelow | I 134 Completeness/Bigelow/Pargetter: completeness occurs when our explicit semantics guarantees all and only the extroverted asserted theorems. That is, our semantics does not read anything into our language, which is not already there. >Semantics/Bigelow. Def "extroverted axiomatics"/Terminology/Bigelow/Pargetter: an axiomatics that is developed in an already existing language. >Axioms, >Axiom systems. I 135 Completeness/correspondence theory/Bigelow/Pargetter: the existence of completeness proofs provides a kind of correspondence theory. >Correspondence theory, >Proofs, >Provability. Completeness: for us, we can show that all the propositions that are true to our semantics in all possible worlds can be derived. >Derivation, >Derivability, >Possible worlds. I 137 Def completeness theorem/Bigelow/Pargetter: is a theorem that proves that if a proposition in a certain semantics is assuredly true, this proposition can be proved as a theorem. How can we prove this? How can we prove that each such proposition is a theorem? Solution: we prove the contraposition of the theorem: Instead: If a is assuredly true in semantics, a is a theorem. We prove: If a is not a theorem, it is not assuredly true in semantics. We prove this by finding an interpretation according to which it is false. >Falsification, >Verification, >Verifiability. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Consistency | Gödel | F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann I 72 ff Consistency/Gödel/Waismann: proof that a system is consistent cannot be provided by means of this system. Gödel: if one adds to the Peanoic axioms that of the logic calculus and calls the resulting system P, then no proof for the consistency of P can be given, which could be formulated in P, provided that P is consistent. >Proofs, >Provability, >Axioms, >Axiom systems, >Contradictions. (If P were contradictory, any statement could be proven, e.g. also that P is consistent). I 73 Gödel: every arithmetic is incomplete, in each of the formal systems mentioned above there are undecidable arithmetic sentences and for each of these systems arithmetic terms can be specified which cannot be defined in this system. >Arithmetics, >Completeness, >Incompleteness. Example: a real number that cannot be defined in S can be constructed for each formal system S. This should not be interpreted as proof that there are unsolvable mathematical problems. Rather, the term "solvable" or "decisionable" always refers to a certain formal system only. If a sentence is undecidable in this system, there is still the possibility to construct a richer system in which the sentence can be decided. But there is no system in which all arithmetic sentences can be decided or all terms can be defined. This is the deeper meaning of Brouwer: all mathematics is essentially intellectual action: a series of construction steps, and not a rigid system of formulas that is ready or could even exist. Mathematics is incomplete. The statement that System S is consistent cannot be made in S. I 74 Waismann: can arithmetic be justified at all by such investigations? And geometry: If there are several geometries, how can they be applied to our experience? Reasons for geometry/Waismann: a) select a group of sentences that demonstrate independence, completeness and consistency and b) ensure applicability. >Independence, >Completeness, >Geometry. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
| Consistency | Hilbert | Berka I 413 Hilbert/Lecture: "Mathematical Problems" (1900)(1): the second problem of the mathematical problems is to prove the consistency of the arithmetic axioms. Consistency/arithmetics/problem/Schröter: at first, there is no way to see, since a proof by specifying a model is self-banning, since arithmetic is the simplest area on whose consistency all consistency proofs should be returned in other areas. So a new path must be taken. >Proofs, >Provability, >Ultimate justification, >Models, >Model theory. Consistency proof/Schröter: for the arithmetic axioms: the consistency requires the proof that an arithmetical statement cannot also be used to derive the contradictory negation of this statement from the axioms. >Axioms, >Axiom systems, >Derivation, >Derivability. To do this, it suffices to prove the non-derivability of any statement e.g. 0 unequal 0. If this is to be successful, it must be shown that all the deductions from the arithmetic axioms have a certain property which come off the statement that states 0 unequal 0. I 414 Problem: the amount of the consequences is completely unpredictable. Solution/Hilbert: the process of infering (logical inference) has to be formalized itself. With this however, the concluding/infering is deprived of all content. >Conditional, >Implication. Problem: now, one can no longer say that a theory, e.g. is about the natural numbers. Formalism/Schröter: according to formalism, mathematics is no longer concerned with objects which refer to a real or an ideal world, but only by certain signs, or their transformations, which are made according to certain rules. >Formalism. WeylVsHilbert: that would require a reinterpretation of all the mathematics so far. 1. David Hilbert: Mathematische Probleme, in: Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, mathematisch-physikalische Klasse, issue 3, 1900, pp. 253–297. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Consistency | Mates | I 234 Consistency/Mates: can be doubly checked: a) semantically: by specifying an interpretation in which all axioms are true b) syntactically: by showing without referring to an interpretation that there is no statement j such that both j and ~j can be derived from the axioms. >Derivation, >Derivability, >Consequence, >Inference, >Conclusion, >Conditional, >Axioms, >Axiom systems, >Semantics, >Syntax. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Consistency | Thiel | I 223 Consistency/Maximum Consistency/Thiel: In 1920 Post proved that if one adds to the axiom system examined (in his case Principia Mathematica(1)) a proposition that cannot be derived from them, the extended system is contradictory. I 224 ("completeness as maximum consistency") > maximum consistency. >Completeness, >Derivation, >Derivability. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Correctness | Tarski | Berka I 489 Correctness/domain/Tarski: according to the sentences 14-16 (or Lemma I) there is for each natural number k such a statement that is true in any area with k elements and in any area of the other thickness. In contrast: Every statementthat is true in an infinite range is also true in any other infinite domain. Properties/classes: so we conclude that the object language allows us to express such a property of classes of individuals, such as the existence of exactly k elements. There is no means for designating any specific type of infinity (e.g. countability) and we cannot distinguish by means of a single or a finite number of statements... I 490 ...two such properties of classes such as finiteness, infinity from each other. >Infinity. I 491 Truth (in the domain): depends on the scope in the finite case, not in the infinite. I 491 Correctness in the doamin/provability/Tarski: if we add the statement a (every nonempty class contains a singleton class as a part) to the axiom system correctness and provability will be coextensive terms. >Provability. N.B.: this does not work in the logical algebra, because here a is not satisfied in all interpretations. I 516 "In every correct domain"/Tarski: this term stands according to the extent in the middle between the provable sentence and the true statement, but is narrower than the class of all true statements generally. It does not contain statements whose validity depends on how big the total number of individuals is.(1) 1. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Derivability | Hilbert | Thiel I 97 Derivability/Hilbert/Thiel: the methods used for the proof of the non-derivability of a formula from others by means of given derivation rules have been given for the first time by Bernays in the Hilbert school. They were first published by Bernays in his postdoctoral thesis on the proof of the independence of axiom systems of classical propositional logic. Neither of these axioms is to be derived from the others. Classic: ~~p > p effective: p > ~~p I 102 Axiomatic derivations of logical sentences were unrivaled up to the twenties in this form, then alternative procedure calculus of the "natural concluding" were developed, whose rule usually bring exactly one logical symbol into a conclusion chain or eliminate. The actual kind of mathematical approach is closer than the axiomatic approach. >Natural deduction, >G. Gentzen, >Derivation, >Axioms, >Axiom systems, >Calculus, >Logic. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Derivation | Hilbert | Berka I 113 Derivation/insertion/"evidence threads"/Hilbert: any derivation can be dissolved into evidence threads, that is, we start with the final formula by applying the schemes (α), (β), (...). I 114 N.B.: then by the dissolution of a derivative into evidence threads, one can put back the insertions into the initial formulas. >Proofs, >Provability, >Derivability. Inserting/insertion rules/variables/evidence threads/Hilbert: we can do without rules of insertion by putting back the insertions (by means of evidence threads). From the derivation of formulas which contain no formula variable, we can eliminate the formula variables altogether, so that the formally deductive treatment of axiomatic theories can take place without any formula variables. >Inserting. Hilbert: the rule that identical formulas of the propositional calculus are permitted as initial formulas is modified in such a way that each formula which results from an identical formula of the propositional calculus by insertion is permitted as the initial formula. Evidence(s): the rule of insertion is also superfluous by the fact that one can study the practical application in the course of time. That is, each case is documented, so you do not need a rule for non-current cases. Hilbert: Instead of the basic formula (x)A(x) > A(a) is now: (x)A(x) > A(t) And in place of (Ex) A (x) is now: A(t) > (Ex)A(x) t: term. Formulas are replaced by formula schemes. Axioms are replaced with axiom schemata. In the axiom schemata, the previously free individual variables are replaced by designations of arbitrary terms, and in the formula schemes, the preceding formula variables are replaced by arbitrary formulas(1). >Schemes, >Axioms, >Axiom systems. 1. D. Hilbert & P. Bernays: Grundlagen der Mathematik, I, II, Berlin 1934-1939 (2. Aufl. 1968-1970). |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Derivation | Mates | I 158 Derivability/Derivative/Mates: "Fa" can not be derived from "(Ex) Fx" as "Fa" is no implication - but you can introduce "Fa" as a premise. >Premises, >Introduction, >Introduction rules, >Quantification, >Existential quantification, >Consequence, >Inference, >Conclusion, >Derivability, >Axioms, >Axiom systems. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Dialogical Logic | Lorenzen | Wessel I 260 Dialogical Logic/Paul Lorenzen/Wessel: P: Proponent, O: Opponent. 1. The dialogue begins with the setting of a formula by the proponent. 2. The proponent may attack only one of the opponent's formulas claimed by the the latter, or the proponent may defend itself against the opponent's last attack. 3. The opponent may only attack the formula set in the preceding proponent's move, or defend itself against the attack in the previous move. Operational rules: Assertion attack defense ~A A? not possible! A u B ?L A A u B ?R B A v B ? A A v B ? B A > B A? B AiA ?(j) A{i/j} EiA ? A{i/j} --- I 261 Rule of winning: the proponent has won if he has to defend a statement variable or predicate formula c that has already been asserted by the opponent. Definition Dialogical Tautology: a dialogical tautology is present, precisely when it wins against any opponent's strategy. A dialogue always leads from complicated to simpler formulas and finally to statement variables or predicate formulas. --- Berka I 206 Dialogical Logic/Lorenzen/Berka: Dialogical logic suppresses in the recent discussion the conception of a symmetry of rule and general formula in favor of the rule-logical representation. If the logic is realized in sensible speech actions, a process schema constructed as a set of rules is the adequate description of the logical as the regulation of the generation of actions from given actions. >Operationalism, >pragmatism. This is a context of actions which is itself a kind of action. --- Thiel I 103 Logic/Dialogical Logic/Lorenzen: It was only in the sixties that a construction of logic was developed, which can also be described as a justification in the scientific theory and philosophical sense. It provides a possibility, not yet seen, for the foundation of both the classical and the constructive concept of the "validity" of logical propositions. (Lorenzens' "dialogical logic" with proponent and opponent, also "argumentation-theoretic structure of logic"). >Validity, >Justification, >Ultimate justification. Dialogical logic should show that the axiomatic derivation does not constitute the whole meaning of the proof, but that a proof should provide reasons for the truth or validity of the proved proposition. .. + .. I 105. >Proofs, >Provability, >Axioms, >Axiom systems. |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Wessel I H. Wessel Logik Berlin 1999 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Dimensions | Thiel | I 278 Space/Dimensions/Axioms/Thiel: The 19th century has shown that the replacement of the Euclidean axiom system of geometry by a contradictory one is also possible without contradiction for the assumption of a space with more than three dimensions. >Space, >Minkowski-Space, >Geometry. I 279 The derivation of the geometric sentences is entirely formal, without reference to the life world. The dispute over the "justification" of non-Euclidean geometry actually only concerned the hasty claim that it was now the "right" instead of the "wrong" Euclidean. Rather important is the change of meaning of the terms "axiom", "definition", "space", and "geometry" connected with this development. >Axioms, >Derivation, >Derivability. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Finiteness | Hilbert | Thiel I 245 Finite/Hilbert: in the sense of Hilbert, it is only a question of how statements about infinite objects can be justified by means of "finite" methods. >Infinity, >Circularity, cf. >Recursion, >Recursivity. Hilbert found the finiteness in the "operational" method, especially of the combinatorics, arithmetics, and elemental algebra already exemplarily realized. They were "genetically" (constructively) built up into the second third of the 19th century, while the construction of geometry was a prime example for the axiomatic structure of a discipline. >Constructivism, >Geometry, >Number theory, >Arithmetics, >Axioms, >Axiom systems. I 246 Each finite operation is an area that is manageable for the person who is acting. This area can change during the process. I 247 The fact that the arithmetic functions required for Goedel's proof are even primitively recursive is remarkable in that not all effectively computable functions are primitively recursive, and the primitive recursive functions are a true subclass of the computable functions. >K. Gödel, >Completeness/Gödel, >Incompleteness/Gödel. I 248 An effectively computable, but not primitive, recursive function is e.g. explained by the following scheme for the calculation of their values (not proved) (x 'is the successor of x): ψ(0,n) = n' ψ(m',0) = ψ(m,1) ψ(m',n')= ψ(m,ψ(m',n)). (I 247) If one wants to approach the general concept of comprehensibility, one has to accept the so-called μ operator as a new means of expression. Thiel I 249 Computability/Church/Thiel: how close is this to a concept of "general computability"? There is the concept of "Turing computability", the concept of the "l definability" in Church and the "canonical systems" in Post. >Calculability, >A. Turing, >E. Post. Each function, which is in one of these classes, is also demonstrable in the others. Church has then uttered the presumption that with this an adequate clarification of the general concept of computability is achieved. >Church Thesis. But it means that this is a "non-mathematical" presumption, and is not capable of any mathematical proof. It is an intuitive term: whether such a specification is "adequate" cannot be answered with mathematical means. >Proofs, >Provability, >Adequacy. I 250 Apart from finiteness and constructivity, there remain other questions: none of the definitions for the offered functional classes is finite: e.g. μ-recursive functions. The attempt to describe effective executability with classical means remains questionable, but if we interpret the existence quantifier constructively, we have already assumed the concept of constructivity. >Quantification, >Quantifiers, >Existential quantification. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Goedel | Mates | I 289 Goedel/Mates: main result: Goedel showed by the incompleteness theorem that one can not identify mathematical truth with derivability from a particular system of axioms. >K. Gödel, >Incompleteness/Gödel, >Mathematical truth, >Validity, >Derivation, >Derivability, >Axioms, >Axiom systems. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Goedel | Quine | XIII 82 Goedel/Goedel Theorem/Quine: Evidence/Self-Evidence/Quine: it is too much to ask that a proof should be self-evident. E.g. Euclid's parallel axiom is not self-evident. E.g. set theory is also not self-evident because it is shaken by paradoxes. Self-Evidence/Quine: we find it in a small number of axioms of number theory. They are the axioms of Dedekind, which are called the axioms of Peano. Elementary Number Theory/Quine: there was always the question whether there were still valid laws that could not be derived from the axioms. They existed! That was a question of adequacy. Laws/Quine: the question of further, still undiscovered laws seemed to be a problem of all branches of mathematics. By supplementing the axioms, perhaps this could be remedied? But Goedel proved in 1931 that this cannot be done! Goedel/Quine: proved that there can be no complete deductive system for even the smallest fragment of mathematics, such as Elementary Number Theory. XIII 82 Tendency: Goedel/Quine: proved that there can be no complete deductive system for even the smallest fragment of mathematics, such as the elementary number theory. Def Elementary Number Theory/Quine: includes digits, notation for plus, times, power and equality. >Numbers/Quine. XIII 83 Sentence operators: for "not", "and" and "or" and the quantifiers "Each number x is such that..." and "there is a number x so that...". The numbers are the positive integers and the zero. With this you can express e.g. Fermat's last theorem. Goedel/Quine: Thesis: No axiom system or other deductive apparatus can cover all truths that can be expressed even in this most moderate notation. Any valid proof procedure will disregard some true sentences, even infinitely many of them. Self-Evidence/Mathematics/Goedel/Quine: therefore we must drop the requirement of self-evidence. Wrong solution/Quine: could one not simply take all discovered truths as axioms? Vs: this is not impossible because there could be no axiom system with infinitely many axioms - which exist. Rather, it is the case that a proof must be able to be examined in finite time. Goedel/Goedel's Theorem/Quine: is related to the reflexive paradoxes. The point is that the notation of the elementary number theory must be able to speak about itself. ((s) Self-Reference). Goedel Numbering/Goedel Number/Quine: ...+... XIII 84 Mention/Use/Goedel/Quine: Goedel's evidence also requires this distinction. For example, the digit "6" names the number 6 and has the Goedel number 47. We can say that the Goedel number 47 names the number 6. Syntax/Arithmetic/Goedel/Quine: after all expressions have their naming by Goedel numbers, the syntactic operations can be mirrored by expressions, by arithmetic operations via numbers. Quote/Goedel/Quine: Problem: the corresponding notation is not part of symbolic logic and arithmetic. Quotation marks cannot be simply named by Goedel numbers. Quote/Quine: of an expression: names this expression. Goedel Numbers/Goedel number/Quine: 47 names 6, furthermore 5361 names 47 if 53 and 61 are randomly the Goedel numbers of the digits "4" and "7". ((s) Quotation marks sic). Quote/Goedel/Quine: the quote relation is represented as by the arithmetic relation that has 5361 to 47 and 47 to 6. The general relation can be expressed in the notation of the elementary number theory, though not easily. The arithmetic reconstruction of syntactic concepts like this was a substantial part of Goedel's work. Liar/Liar's Paradox/Goedel/Quine: is useful in one of the two parts where Goedel's proof can be split. The bomb explodes when the two parts are put together. The liar can be completely XIII 85 expressed by Goedel numbering with the exception of a single expression: "truth". If that could be done, we would have solved the paradox, but discredited the elementary number theory. Truth/Goedel Number/Goedel Number/Quine: truth is not definable by Goedel numbers, within the elementary number theory. >Goedel Numbers/Quine. Goedel's Theorem/Quine: formal: no formula in the notation of the elementary number theory is true of all and only the Goedel numbers of truths of the elementary number theory. (This is the one part). Other part/Quine: deals with every real evidence procedure, here it is about that every evidence must be testable. Formal: a given formula in the notation of the elementary number theory is true of all and only the Goedel numbers of provable formulas. Church/Quine: here I skip his thesis (Church-Thesis), (see recursion below). Goedel/Quine: the two parts together say that the provable formulas do not coincide with the truths of the elementary number theory. Either they contain some falsehoods, or they do not cover some truths. God forbids that. Goedel/Quine: his own proof was more direct. He showed that a given sentence, expressed in Goedel numbers, cannot be proved. Either it is false or provable, or true and not provable. Probably the latter. Wrong solution/Quine: one could add this lost truth as an axiom, but then again others remain unprovable. Goedel/N.B./Quine: ironically, it was implausible that there could be a proof procedure for all truths of the elementary number theory. This would clarify Fermat's theorem, and much more. XIII 86 On the other hand, Goedel's result hit him like a bomb. N.B.: these two shortcomings turned out to be equivalent! Because: Kleene/Quine: showed that if there is a complete evidence procedure, any statement could be tested as true or false as follows: a computer would have to be programmed to rewind any statement, in alphabetical order, the shortest first, then always longer. In the end, because of the completeness of the procedure, he will have proved or refuted every single sentence. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Hilbert | d’Abro | A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 36 Hilbert's postulate system has 21 postulates that should define relationships between points, lines, and planes. E.g. Continuity had been assumed tacitly by Euclid, and was explicitly demanded by Hilbert. ("Archimedean Postulate") Euclid was unconsciously guided by the idea of solid bodies. Def "Archimedean Postulate"/Hilbert: Assumption of Continuity (was assumed tacitly in Euclid). >Axioms/Hilbert, >D. Hilbert as author, >Axioms, >Axiom systems. |
|
| Incompleteness | Gödel | Thiel I 227 ff Incompleteness Theorem/Goedel/Thiel: ... this metamathematical statement corresponds in F to a one-digit statement form G(x) which then must occur somewhere in the counting sequence. If G(x) takes the h'th place, it is therefore identical with the propositional form called Ah(x) there. Goedel's result will be, that in F neither the proposition G(h) arising from G(x) by the insertion of h nor its negative ~G(h) is derivable. "Undecidable in F". Suppose G(h) is derivable in F, then only the derivation of true statements would be allowed, so G(h) would also be true. Thus, since G(x) was introduced as an image of $Ax(x) in F, $Ah(h) would be valid. But that would mean, since Ah(x) is identical with G(x), $G(h). G(h) would therefore be non-derivable in F - this is a contradiction. >Derivation, >Derivability. This derivation first only proves the validity of the "if-then-statement" S G(h)>$ G(h). This must now be inserted: (S G(h)>$ G(h))> $ G(h). This follows from the general scheme (A>~A)>~A. On the other hand, if we then assume that the negative ~G(h) is derivable, then ~G(h) would also be true. This would be equivalent to the validity of ~$ Ah(h) thus with S Ah(h). Thiel I 228 This in turn agrees with S G(h), so that both assertion and negative would be derivable, and we would have a formal contradiction. If F is contradiction-free at all, our second assumption S ~G(h) is not valid either. This is an undecidable assertion. Cf. >Decidability, >Indecidability. Thiel I 228 This proof sketch establishes a program. Important roles in the execution of this program are played by the "Goedelization" and the so-called "negative representability" of certain relations in F. Def Goedelization: Goedelization is first of all only a reversibly definite assignment of basic numbers to character sequences. We want to put the expressions of F into bracket-free form. >Goedel numbers. For this we write the logical connective signs not between, but in front of the expressions. We write the logical operators as "indices" to the order functor G. Terminology order functor G. Quantifiers: we treat quantifiers as two-digit functors whose first argument is the index, the second the quantified propositional form. >Quantifiers, >Quantification. Thiel I 229 Then the statement (x)(y)(z) ((x=y)>(zx = zy) gets the form (x)(y)(z)G > G = xyG = G times zxG times zy. We can represent the members of the infinite variable sequences in each case by a standard letter signaling the sort and e.g. prefixed points: thus for instance x,y,z,...by x,°x,°°x,...As counting character we take instead of |,||,|||,... zeros with a corresponding number of preceding dashes 0,'0,''0,... >Sequences. With this convention, each character in F is either a 0 or one of the one-digit functors G1 (the first order functor!), ', ~. Two-digit is G2, three-digit is G4, etc. Thiel I 229 E.g. Goedelization, Goedel number, Goedel number: Prime numbers are assigned in each case:.... Primes. Thiel I 230 In this way, each character string of F can be uniquely assigned a Goedel number and told how to compute it. Since every basic number has a unique representation as a product of prime numbers, it can be said of any given number whether it is a Goedel number of a character string of F at all. Metamathematical and arithmetical relations correspond to each other: example: Thiel I 230 We replace the x by 0 in ~G=x'x and obtain ~G = 0'0. The Goedel number of the first row is: 223 x 313 x 537 x 729 x 1137, the Goedel number of the second row of characters is: 223 x 313 x 531 x 729 x 1131. The transition from the Goedel number of the first row to that of the second row is made by division by 56 x 116 and this relation (of product and factor) is the arithmetic relation between their Goedel numbers corresponding to the metamathematical relation of the character rows. Thiel I 231 These relations are even effective, since one can effectively (Goedel says "recursively") compute the Goedel number of each member of the relation from those of its remaining members. >Recursion. The most important case is of course the relation Bxy between the Goedel number x, a proof figure Gz1...zk and the Goedel number y of its final sequence... Thiel I 233 "Negation-faithful representability": Goedel shows that for every recursive k-digit relation R there exists a k-digit propositional form A in F of the kind that A is derivable if R is valid, and ~A if R does not (..+..). We say that the propositional form A represents the relation R in F negation-faithfully. Thiel I 234 After all this, it follows that if F is ω-contradiction-free, then neither G nor ~G is derivable in F. G is an "undecidable statement in F". The occurrence of undecidable statements in this sense is not the same as the undecidability of F in the sense that there is no, as it were, mechanical procedure. >Decidability. Thiel I 236 It is true that there is no such decision procedure for F, but this is not the same as the shown "incompleteness", which can be seen from the fact that in 1930 Goedel had proved the classical quantifier logic as complete, but there is no decision procedure here, too. Def Incomplete/Thiel: a theory would only be incomplete if a true proposition about objects of the theory could be stated, which demonstrably could not be derived from the axiom system underlying the theory. ((s) Then the system would not be maximally consistent.) Whether this was done in the case of arithmetic by the construction of Goedel's statement G was for a long time answered in the negative, on the grounds that G was not a "true" arithmetic statement. This was settled about 20 years ago by the fact that combinatorial propositions were found, which are also not derivable in the full formalism. Goedel/Thiel: thus incompleteness can no longer be doubted. This is not a proof of the limits of human cognition, but only a proof of an intrinsic limit of the axiomatic method. Thiel I 238 ff One of the points of the proof of Goedel's "Underivability Theorem" was that the effectiveness of the metamathematical derivability relation corresponding to the self-evident effectiveness of all proofs in the full formalism F, has its exact counterpart in the recursivity of the arithmetic relations between the Goedel numbers of the proof figures and final formulas, and that this parallelism can be secured for all effectively decidable metamathematical relations and their arithmetic counterparts at all. >Derivation, >Derivability. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Incompleteness | Thiel | I 222 Incompleteness/Thiel: Incompleteness keeps reappearing. The logical rules do not contain all the operations actually performed when closing, nor are the prerequisites for these operations formalized. For example, the order of the premises of a rule is regarded as insignificant. For example, the separation rule is also formulated with reversed premises. >modus ponens. I 223 The usual axiom systems of the logical connectives are complete since Frege. >Completeness, >Axiom systems, >G. Frege, >Junctions. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Independence | Cresswell | II 176 Independence/Logic/Cresswell: misunderstanding: independence of an axiom does not mean that you can discard it at will. >Axioms, >Axiom systems. E.g. an independence proof within the axiomatic propositional calculus, for example, the independence of (p v q)> (q v p). Such proof indicates that one can give a semantic definition of an operator that meets all other axioms of disjunction, but is not commutative. But it does not show that disjunction itself is not commutative, and it also does not show that (p v q)> (q v p) is not a logical truth about classic disjunction. >Disjunction, >Logical truth, >Operators. |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 |
| Infinity Axiom | Hilbert | Berka I 122 Definition Number/logical form/extended function calculus/Hilbert: the general number concept can also be formulated logically: if a predicate-predicate φ (F) should be a number, then φ must satisfy the following conditions: 1. For two equal predicates F and G, φ must be true for both or none of them. 2. If two predicates F and G are not equal in number, φ can only be true for one of the two predicates F and G. Logical form: (F)(G){(φ(F) & φ(G) > Glz (F,G) & [φ(F) & Glz (F,G) > φ(G)]}. The entire expression represents a property of φ. If we designate it with Z (φ), then we can say: A number is a predicate-predicate φ that has the property Z (φ). >Numbers, >Definitions, >Definability, >Infinity, >Axioms, >Axiom systems, >Predicates, >Properties. Problem/infinity axiom/Hilbert: a problem occurs when we ask for the conditions under which two predicate-predicates φ and ψ define the same number with the properties Z (φ) and Z (ψ). Infinity Axiom/equal numbers/Hilbert: the condition for equal numbers or for the fact that two predicate-predicates φ and ψ define the same number is that, that φ(P) and ψ(P) are true for the same predicates P and false for the same predicates. So that the relationship arises: (P)(φ(P) ↔ ψ(P)) I 122 Problem: when the object area is finite, all the numbers are made equal which are higher than the number of objects in the individual domain. >Finiteness/Hilbert, >Finitism, >Finiteness. For example, if a number is e.g. smaller than 10 to the power of 60 and if we take φ and ψ the predicates which define the numbers 10 to the power of 60+1 and 10 high 60 + 1, then both φ and ψ do not apply to any predicate P. The relation (P)(φ(P) ↔ ψ(P)) Is thus satisfied for φ and ψ, that is, φ and ψ would represent the same number. Solution/Hilbert: infinity axiom: one must presuppose the individual domain as infinite. A logical proof of the existence of an infinite totality is, of course, dispensed with(1). 1. D. Hilbert & W. Ackermann: Grundzüge der theoretischen Logik, Berlin, 6. Aufl. Berlin/Göttingen/Heidelberg 1972, §§ 1, 2. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Inserting | Hilbert | Berka I 113 Derivation/inserting/"evidence threads"/Hilbert: every derivation can be dissolved into evidence threads, that is, we start with the final formula by applying the schemes (α), (β), (...). >Derivation, >Derivability. I 114 N.B.: then by the dissolution of a derivative into evidence threads, one can put back the insertions into the initial formulas. Inserting/insertion rules/variables/evidence threads/Hilbert: we can do without rules of insertion by putting back the insertions (by means of evidence threads). From the derivation of formulas which do not contain a formula variable, we can eliminate the formula variables altogether, so that the formally deductive treatment of axiomatic theories can take place without any formula variables. >Proofs, >Provability. Hilbert: the rule that identical formulas of the propositional calculus are allowed as initial formulas is modified in such a way that each formula which results from an identical formula of the propositional calculus is permitted as an initial formula. Evidence threads(s): the rule of insertion also becomes superfluous by the fact that one can study the practical application in the course of time. That is, each case is documented, so you do not need a rule for non-current cases. Hilbert: In the place of the basic formula (x)A(x) > (A(a) is now: (x)A(x) > A(t) and in the place of (Ex)A(x) is now: A(t) > (Ex)A(x) t: term. Formulas are replaced by formula schemes. Axioms are replaced by axiom schemata. In the axiom schemata, the previously free individual variables are given by designations of arbitrary terms, and in the formula schemes, the preceding formula variables are replaced by arbitrary formulas(1). >Axioms, >Axiom systems. 1. D. Hilbert & P. Bernays: Grundlagen der Mathematik, I, II, Berlin 1934-1939 (2. Aufl. 1968-1970). |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Kripke Semantics | Bigelow | Bigelow I 109 Kripke Semantics/Bigelow/Pargetter: has only recently shown that K, D, T B are the right preliminary stages for S4 and S5. Before, they thought it was S1, S2 and S3. >Systems S4/S5, >S.A. Kripke, >Logic, >Accessibility relation, >Axioms, >Axiom systems, >Semantics. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Kripke Semantics | Hintikka | II XIII Kripke Semantics/HintikkaVsKripke: Kripke semantics is not a viable model for the theory of logical modalities (logical necessity and logical possibility). Problem: the right logic cannot be axiomatized. Solution: to interpret Kripke semantics as a non-standard semantics,... II XIV ...in the sense of Henkin's non-standard interpretation of the logic of higher levels, while the correct semantics for logical modalities would be analogous to a standard interpretation. >Logical possibility, >Logical necessity, >Modal logic, >Modalities. --- II 1 Kripke Semantics/Hintikka: Kripke semantics is a modern model-theoretic approach that is misleadingly called Kripke semantics. E.g.: F: is a framework consisting of SF: a set of models or possible worlds and R: a two-digit relation, a kind of alternative relation. Possible Worlds: w1 is supposed to be an alternative, which could legitimately be realized instead of w0 (the actual world). R: the only limitation we impose on it is reflexivity. Truth Conditions/modal logic/Kripke semantics/Hintikka: the truth conditions for modal sentences are then: II 2 (TN) Given a frame F, Np is true in w0 ε SF iff. P is true in every alternative wi ∈ SF to w0. (T.M) Given a frame F, Mp is true in w0 ε SF iff. P is true in at least one alternative wi ∈ SF to w0. Model Theory/modal logic/Hintikka: Kanger, Guillaume and later Kripke have seen that when we add reflexivity, transitivity, and symmetry, we get a model theory for axiom systems of the Lewis type for modal propositional logic. Kripke Semantics/modal logic/logical possibility/logical necessity/HintikkaVsKripke/HintikkaVsKripke semantics: problem: if we interpret the operators N, P as expressing logical modalities, they are inadequate: we need more than one arbitrary selection for logical possibility and necessity of possible worlds. We need truth in every logically possible world. But in the Kripke semantics it is not necessary that all such logically possible worlds are contained in the set of alternatives ((s) that is, there may be logically possible worlds that are not considered). (See below the logical possibility forms the largest class of possibilities). Problem: Kripke semantics is therefore inadequate for logical modalities. II 12 Kripke/Hintikka: Kripke has avoided epistemic logic and the logic of propositional attitudes, concentrating on pure modalities. >Epistemic logic. Therefore, it is strange that he uses non-standard logic. But somehow it seems clear to him that this is not possible for logical modalities. Metaphysical Possibility/Kripke/HintikkaVsKripke: Kripke has never explained what these mystical possibilities actually are. II 13 Worse: Kripke has not even shown that they are so restrictive that he can use his extremely liberal non-standard semantics. |
Hintikka I Jaakko Hintikka Merrill B. Hintikka Investigating Wittgenstein German Edition: Untersuchungen zu Wittgenstein Frankfurt 1996 Hintikka II Jaakko Hintikka Merrill B. Hintikka The Logic of Epistemology and the Epistemology of Logic Dordrecht 1989 |
| Loewenheim | Thiel | I 321 For example, the paradox of Loewenheim-Skolem: The fact, which can be proven by all axiom systems formulated in classical quantifier logic (with identity), that they can be fulfilled, if at all, then already in a countable individual realm, is quite rightly inferred from this, I 322 that therefore also such an axiom system for the real numbers must already be countably fulfillable, contrary to the underlying intention to characterize just the not countable totality of the real numbers. >Real numbers, >Satisfaction, >Models, >Model theory. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Logical Truth | Bigelow | I 132 Logical Truth/Bigelow/Pargetter: Problem: Logical and non-logical truths are not easy to distinguish. For example, you could simply add a to the axioms, then Na would be a theorem! (Because of the rule of necessitating, necessitation, see above.). Problem: the truth of "a" ultimately depends on our interpretation of the predicates. >Interpretation, >Valuation, >Predicates. Theorems: on the other hand, remain true with every interpretation. For them, it only depends on the interpretation of the other symbols (not the names and predicates). >Variables, >Symbols, >Logical constants. Logical truth/Bigelow/Pargetter: can be characterized in two ways a) axiomatically (true from the list of axioms). b) semantically (true by interpreting the logical symbols). >Axioms, >Axiom Systems, >Semantics. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Modal Logic | Bigelow | I 101 Modal logic/Language/Bigelow/Pargetter: at the end we get an orthodox language of modal logic: it is an extension of the classical language of Tarski in two respects: >A. Tarski, >Reference, >Individual constants, >Possibilia. I 102 1. extension of the referents of individual constants so that they can also include Possibilia 2. addition of rules for modal operators. That does not mean that this is the only right way. Possibilia we do not claim their existence for semantic reasons either. But there are good non-semantic reasons for believing in them. I 119 Modal Logic/Modality/Intuition/Bigelow/Pargetter: our intuitions are deceptive here. Some of our intuitions even contradict each other: E.g. Principle of the distribution of disjunction: ((a v b) would be > would be g) > ((a would be > would be g) u (b would be > would be g)). That seems to be true. For example, "If you ate or drink, you would be my prisoner." "So if you ate, you would be my prisoner and if you drank, you would be my prisoner." >Disjunction, >Counterfactual conditional. Problem: this principle cannot be added to our axiom system. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Model Theory | Model Theory: The model theory investigates whether an axiom system is fulfilled and thus provides one (or more) models. Model theory belongs to the semantics because it uses the concept of truth, while the proof theory belongs to the domain of syntax by asking for the existence of finite character string (of proofs). One problem is the exclusion of unintended models. |
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| Natural Deduction | Natural deduction, logic: is a calculus by Gerhard Gentzen (Gentzen, “Untersuchungen über das logische Schließen“. In Mathematische Zeitschrift Band 39, 1935, pp. 176–210, 405–431), which largely manages without axioms and instead works with introductory and eliminating rules for the operators used. Assumptions that are needed in the course of time can be partly eliminated later. See also axiomatization, axiom systems, axioms, inference. |
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| Necessity | Peacocke | II 313 Necessity/necessary/modification of predicates/Wiggins/Peacocke: Problem: 'big' cannot modify like 'nec' predicates of any fine degree. ((s) "nec" : Operator for necessary"). >Operators. That means, we get a finite axiomatized theory for 'big' but not for 'nec'. - There can only be an infinite number of modifications here. >Axioms, >Axiom systems, >Axiomatizability. Problem: 'nec' can be iterated in the object language, but Grandy's representational content cannot treat the iterations because the performance is not defined. Solution: 1. syntactical variabel 't>' is about series of terms of the form (t1 ... tn) 2. separate recursion for abstracts of the object language in the theory, that specifies inductively the conditions under which a sequence has the property correlated with the abstract('Corr'). II 316 Then the truth conditions turn the predications into sequences - so the theory is not entirely homophonic. >Homophony, >Truth conditions, >Predication. II 324 Necessity/satisfaction/language/Peacocke: the satisfaction and evaluation axioms not only express contingent truths about the language - necessarily in German each sequence fulfils x1 'is greater than Hesperus' in L, if their first element is greater than Hesperus. >Satisfaction. |
Peacocke I Chr. R. Peacocke Sense and Content Oxford 1983 Peacocke II Christopher Peacocke "Truth Definitions and Actual Languges" In Truth and Meaning, G. Evans/J. McDowell Oxford 1976 |
| Number Theory | Quine | IX 81 Elementary Number Theory/Quine: this is the theory that can only be expressed with the terms "zero, successor, sum, power, product, identity" and with the help of connections from propositional logic and quantification using natural numbers. One can omit the first four of these points or the first two and the fifth. But the more detailed list is convenient, because the classical axiom system fits directly to it. Quine: our quantifiable variables allow other objects than numbers. However, we will now tacitly introduce a limitation to "x ε N". Elementary Number Theory/Quine: less than/equal to: superfluous here. "Ez(x + z = y)" - x ε N > Λ + x = x. - x,y ε N >{x} + y = {x+y}. IX 239 Relative Strength/Proof Theory/Theory/Provability/Quine: Goedel, incompleteness theorem (1931)(1). Since number theory can be developed in set theory, this means that the class of all theorems IX 239 (in reality, all the Goedel numbers of theorems) of an existing set theory can be defined in that same set theory, and different things can be proved about it in it. >Set Theory/Quine. Incompleteness Theorem: as a consequence, however, Goedel showed that set theory (if it is free of contradiction) cannot prove one thing through the class of its own theorems, namely that it is consistent, i.e., for example, that "0 = 1" does not lie within it. If the consistency of one set theory can be proved in another, then the latter is the stronger (unless both are contradictory). Zermelo's system is stronger than type theory. >Type theory, >Strength of theories, >Set theory, >Provability. 1.Kurt Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In: Monatshefte für Mathematik und Physik. 38, 1931, S. 173–198, doi:10.1007/BF01700692 II 178 Elementary number theory is the modest part of mathematics that deals with the addition and multiplication of integers. It does not matter if some true statements will remain unprovable. This is the core of Goedel's theorem. He has shown how one can form a sentence with any given proof procedure purely in the poor notation of elementary number theory, which can be proved then and only then if it is wrong. But wait! The sentence cannot be proved and still be wrong. So it is true, but not provable. Quine: we used to believe that mathematical truth consists in provability. Now we see that this view is untenable to mathematics as a whole. II 179 Goedel's incompleteness theorem (the techniques applied there) has proved useful in other fields: Recursive number theory, or recursion theory for short. Or hierarchy theory. >Goedel/Quine. III 311 Elementary Number Theory/Quine: does not even have a complete proof procedure. Proof: reductio ad absurdum: suppose we had it with which to prove every true sentence in the spelling of the elementary number theory, III 312 then there would also be a complete refutation procedure: to refute a sentence one would prove its negation. But then we could combine the proof and refutation procedure of page III 247 to a decision procedure. V 165 Substitutional Quantification/Referential Quantification/Numbers/Quine: Dilemma: the substitutional quantification does not help elementary number theory to any ontological thrift, for either the numbers run out or there are infinitely many number signs. If the explanatory speech of an infinite number sign itself is to be understood again in the sense of insertion, we face a problem at least as serious as that of numbers - if it is to be understood in the sense of referential quantification, then one could also be satisfied from the outset uncritically with object quantification via numbers. >Quantification/Quine. V 166 Truth conditions: if one now assumes substitutional quantification, one can actually explain the truth conditions for them by numbers by speaking only of number signs and their insertion. Problem: if numerals are to serve their purpose, they must be as abstract as numbers. Expressions, of which there should be an infinite number, could be identified by their Goedel numbers. No other approach leads to a noticeable reduction in abstraction. Substitutional quantification: forces to renounce the law that every number has a successor. A number would be the last, but the substitutional quantification theorist would not know which one. It would depend on actual inscriptions in the present and future. (Quine/Goodman 1947). This would be similar to Esenin Volpin's theory of producible numbers: one would have an unknown finite bound. V 191 QuineVsSubstitutional Quantification: the expressions to be used are abstract entities as are the numbers themselves. V 192 NominalismVsVs: one could reduce the ontology of real numbers or set theory to that of elementary number theory by establishing truth conditions for substitutional quantification on the basis of Goedel numbers. >Goedel Numbers/Quine. QuineVs: this is not nominalistic, but Pythagorean. It is not about the high estimation of the concrete and disgust for the abstract, but about the acceptance of natural numbers and the rejection of most transcendent numbers. As Kronecker says: "The natural numbers were created by God, the others are human work". QuineVs: but even that is not possible, we saw above that the subsitutional quantification over classes is basically not compatible with the object quantification over objects. V 193 VsVs: one could also understand the quantification of objects in this way. QuineVs: that wasn't possible because there aren't enough names. You could teach space-time coordination, but that doesn't explain language learning. X 79 Validity/Sentence/Quantity/Schema/Quine: if quantities and sentences fall apart in this way, there should be a difference between these two definitions of validity about schema (with sentences) and models (with sentences). But it follows from the Löwenheim theorem that the two definitions of validity (using sentences or sets) do not fall apart as long as the object language is not too weak in expression. Condition: the object language must be able to express (contain) the elementary number theory. Object Language: In such a language, a scheme that remains true in all insertions of propositions is also fulfilled by all models and vice versa. >Object Language/Quine The requirement of elementary number theory is rather weak. Def Elementary Number Theory/Quine: speaks about positive integers by means of addition, multiplication, identity, truth functions and quantification. Standard Grammar/Quine: the standard grammar would express the functors of addition, multiplication, like identity, by suitable predicates. X 83 Elementary Number Theory/Quine: is similar to the theory of finite n-tuples and effectively equivalent to a certain part of set theory, but only to the theory of finite sets. XI 94 Translation Indeterminacy/Quine/Harman/Lauener: ("Words and Objections"): e.g. translation of number theory into the language of set theory by Zermelo or von Neumann: both versions translate true or false sentences of number theory into true or false sentences of set theory. Only the truth values of sentences like e.g. "The number two has exactly one element", which had no sense before translation, differ from each other in both systems. (XI 179: it is true in von Neumann's and false in Zermelo's system, in number theory it is meaningless). XI 94 Since they both serve all purposes of number theory in the same way, it is not possible to mark one of them as a correct translation. |
Quine I W.V.O. Quine Word and Object, Cambridge/MA 1960 German Edition: Wort und Gegenstand Stuttgart 1980 Quine II W.V.O. Quine Theories and Things, Cambridge/MA 1986 German Edition: Theorien und Dinge Frankfurt 1985 Quine III W.V.O. Quine Methods of Logic, 4th edition Cambridge/MA 1982 German Edition: Grundzüge der Logik Frankfurt 1978 Quine V W.V.O. Quine The Roots of Reference, La Salle/Illinois 1974 German Edition: Die Wurzeln der Referenz Frankfurt 1989 Quine VI W.V.O. Quine Pursuit of Truth, Cambridge/MA 1992 German Edition: Unterwegs zur Wahrheit Paderborn 1995 Quine VII W.V.O. Quine From a logical point of view Cambridge, Mass. 1953 Quine VII (a) W. V. A. Quine On what there is In From a Logical Point of View, Cambridge, MA 1953 Quine VII (b) W. V. A. Quine Two dogmas of empiricism In From a Logical Point of View, Cambridge, MA 1953 Quine VII (c) W. V. A. Quine The problem of meaning in linguistics In From a Logical Point of View, Cambridge, MA 1953 Quine VII (d) W. V. A. Quine Identity, ostension and hypostasis In From a Logical Point of View, Cambridge, MA 1953 Quine VII (e) W. V. A. Quine New foundations for mathematical logic In From a Logical Point of View, Cambridge, MA 1953 Quine VII (f) W. V. A. Quine Logic and the reification of universals In From a Logical Point of View, Cambridge, MA 1953 Quine VII (g) W. V. A. Quine Notes on the theory of reference In From a Logical Point of View, Cambridge, MA 1953 Quine VII (h) W. V. A. Quine Reference and modality In From a Logical Point of View, Cambridge, MA 1953 Quine VII (i) W. V. A. Quine Meaning and existential inference In From a Logical Point of View, Cambridge, MA 1953 Quine VIII W.V.O. Quine Designation and Existence, in: The Journal of Philosophy 36 (1939) German Edition: Bezeichnung und Referenz In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Quine IX W.V.O. Quine Set Theory and its Logic, Cambridge/MA 1963 German Edition: Mengenlehre und ihre Logik Wiesbaden 1967 Quine X W.V.O. Quine The Philosophy of Logic, Cambridge/MA 1970, 1986 German Edition: Philosophie der Logik Bamberg 2005 Quine XII W.V.O. Quine Ontological Relativity and Other Essays, New York 1969 German Edition: Ontologische Relativität Frankfurt 2003 Quine XIII Willard Van Orman Quine Quiddities Cambridge/London 1987 |
| Ordinal Numbers | Neumann | Thiel I 205 Ordinal numbers/Neumann/Thiel: Today, ordinal numbers are not only introduced differently than in Cantor and Dedekind, but are also defined differently. >Numbers. John v. Neumann: Axiomatic construction of the set theory. In the foundation of logic certain formulas are recognized as "excellent formulas". >Axioms, >Axiom systems, >Set theory, >Sets. I 206 The rules allow us to form unreservedly new sentential connective-logical propositional schemas, in which we can recognize excellent ones and not a. But this does not provide us with a real overview of the sentences of the sentential connectives logic, nor a systematic insight into their connections. We must distinguish between the logical framework and the sentences themselves in an axiomatic structure. >Logic, >Statements. I 207 Axiomatization allows a potentially infinite set of sentences by representing them as a conclusion set from finitely many sentences. >Axiomatization, cf. >Are there infinitely many possible sentences?/Researchgate. |
NeumJ I J. v. Neumann The Computer and the Brain New Haven 2012 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Proof Theory | Hilbert | Berka I 384 Proof Theory/Hilbert: first, the concepts and propositions of the theory to be examined are represented by a formal system, and treated without reference to their meaning only formally. I 385 Proof Theory: this (subsequent) investigation is dependent on the logical meaning of its concepts and conclusions. Thus formal theory is compared with a meaningful meta theory (proof theory)(1). Berka I 395 Proof Theory/Hilbert: basic thought, thesis: everything that makes up existing mathematics is strictly formalized, so that the actual mathematics becomes a set of formulas. New: the logical signs "follow" (>) and "not". Final scheme: S S › T T Where each time the premises, i.e. (S and S > T) are either an axiom, or are created by inserting an axiom or coincide with the final formula. Definition provable/Hilbert: a formula is provable if it is either an axiom or an axiom by insertion from it, or if it is the final formula of a proof. >Proofs, >Provability. Meta-Mathematics/proof theory/Hilbert: meta mathematics is now added to the actual mathematics: in contrast to the purely formal conclusions of the actual mathematics, the substantive conclusion is applied here. However, only to prove the consistency of axioms. >Axioms, >Axiom systems, >Axioms/Hilbert. In this meta-mathematics, the proofs of the actual mathematics are operated upon, and these themselves form the subject of the substantive investigation. >Meta-Mathematics. Thus the development of the mathematical totality of knowledge takes place in two ways: A) by obtaining new provable formulas from the axioms by formal concluding and B) by adding new axioms together with proof of the consistency by substantive concluding. >Consistency, >Material implication. Berka I 395 Truth/absolute truth/Hilbert: axioms and provable propositions are images of the thoughts which make up the method of the previous mathematics, but they are not themselves the absolute truths. >Truth/Hilbert. Def absolute truth/Hilbert: absolute truths are the insights provided by my proof theory with regard to the provability and consistency of the formula systems. Through this program, the truth of the axioms is already shown for our theory of proof(2). 1. K. Schütte: Beweistheorie, Berlin/Göttingen/Heidelberg 1960, p. 2f. 2. D. Hilbert: Die logischen Grundalgen der Mathematik, in: Mathematische Annalen 88 (1923), p. 151-165. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Propositional Logic | Wessel | I 35 Propositional Logic can be built in three different ways: 1) semantic (truth-functional), >Truth functions, >Semantics. 2) as a system of natural deduction, >Natural deduction, >G. Gentzen. 3) as axiomatic structure. >Axioms, >Axiom systems. Cf. >Predicate logic, >Logic. |
Wessel I H. Wessel Logik Berlin 1999 |
| Rational Choice | Political Economy | Mause I 62 Rational decisions/rational choice/VsEconomic Theory/VsPolitical Science/Political Economy: Economically oriented political science was confronted with problems because it initially assumed that the actors had complete information. Problem: the empirical significance of this approach is limited, since due to the axiomatics (individuals act rationally) every action must necessarily provide the greatest benefit to an actor.(1)(2) >Utility, >Axiomatic Utility theory, >Axioms, >Axiom Systems. 1. D. P. Green, I. Shapiro, Pathologies of rational-choice theory. A critique of applications in political science. New Haven 1994 2. J. S. Coleman,Th.J. Fararo (Eds) Rational-choice theory. Advocacy and critique. Newbury Park 1992. |
Mause I Karsten Mause Christian Müller Klaus Schubert, Politik und Wirtschaft: Ein integratives Kompendium Wiesbaden 2018 |
| Rationality | Political Economy | Mause I 62 Rationality/Economy/Economic Theory/Political Economy: Economically oriented political science was confronted with problems because it initially assumed that the actors had complete information. Problem: the empirical significance of this approach is limited, since due to the axiomatics (individuals act rationally) every action must necessarily provide the greatest benefit to an actor.(1)(2) >Utility, >Axiomatic Utility theory, >Axioms, >Axiom Systems. Solution: In the 1970s and 1980s - both in political science and in economics - the rigid axiomatic of rationality was therefore softened in favour of the idea of a "restricted, bounded rationality." (3) This takes into account the limited rationality and processing power of humans. >Bounded Optimality. 1. D. P. Green, I. Shapiro, Pathologies of rational-choice theory. A critique of applications in political science. New Haven 1994 2. J. S. Coleman,Th.J. Fararo (Eds) Rational-choice theory. Advocacy and critique. Newbury Park 1992. 3. Cf. Herbert A. Simon, Homo Rationalis. Die Vernunft im menschlichen Leben. Frankfurt a. M. 1993 |
Mause I Karsten Mause Christian Müller Klaus Schubert, Politik und Wirtschaft: Ein integratives Kompendium Wiesbaden 2018 |
| Reason | Minsky | Münch III 125 Everyday reason/everyday problems/MinskyVsAristotle: rather logical approaches do not work. Syllogisms cannot deal with everyday complexity when solving problems. >Problem solving, cf. >Syllogisms, >Artificial intelligence, Axioms: "One does not go undressed out of the house", etc. >Human Level AI. Since logicians are not concerned with systems that can be extended later, they must design axioms that allow only permitted conclusions. >Axioms, >Axiom systems, >Logic. This is different with intelligence. Marvin Minsky, “A framework for representing knowledge” in: John Haugeland (Ed) Mind, design, Montgomery 1981, pp. 95-128 |
Minsky I Marvin Minsky The Society of Mind New York 1985 Minsky II Marvin Minsky Semantic Information Processing Cambridge, MA 2003 Mü III D. Münch (Hrsg.) Kognitionswissenschaft Frankfurt 1992 |
| Riemann | d’Abro | A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967 41 Riemann/d'Abro: Riemann's great contribution to the development of non-Euclidean geometry was precisely the replacement of the restricted axiomatic method of the decomposition of postulates (as Bolyai and Lobachevski practiced it) by the more effective methods of analysis. Thus, Riemann has generalized the different types of non-Euclidean geometry. >Geometry, >Generalization. By rejecting it, it was also recognized that the axioms of Euclid could at least not be a priori. The mathematicians were forced to reject the Kantian assumption for all axioms, including the implied assumptions. >Axioms, >Axiom systems, >Euclid, >a priori. |
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| S 4 / S 5 | S 4 / S 5, logic, philosophy: S 4 and S 5 are modal logical systems that differ in terms of what is expressible in them. The increase in expressiveness is achieved by adding axioms. S 5 results from S 4 by the added axiom Mp > NMp. "What is possible is necessarily possible". See also axioms, axiom systems, modal logic, modalities, stronger/weaker. |
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| S 4 / S 5 | Bigelow | I 107 System S4/Bigelow/Pargetter: contains T and in addition: A10 Axiom S4 (Na > Nna) Everyday language translation: if something has to be true, it must be true that it has to be true. System B/Brouwer/Bigelow/Pargetter: contains T plus A11. Axiom B: (a > NMa) Everyday language translation: if something is true, it must be true that it is possible. System S4/Bigelow/Pargetter: some of his theorems are not theorems of B, and some of B are not theorems of S4. With an additional Axiom S5, we can prove both S4 and B as theorems: A12. Axiom S5: (Ma > NMa) System S5/Bigelow/Pargetter: contains all theorems of S4 and of B and nothing else. I 108 Systems/Proveability/Bigelow/Pargetter: T plus S5 can prove S4 and B, but also T plus S4 and B together can prove S5. Nevertheless: T plus S4 without B cannot prove S5 T plus B without S4 cannot prove S5. Logical necessity/S5/Bigelow/Pargetter: the system S5 is a plausible characterization of the logical necessity. System S4/Bigelow/Pargetter: when we interpret: Rhomb/diamond/possibility/M: "cannot be proved by logic alone" Box/Necessity/N: "can be proved by logic alone" Then S4 becomes: Everyday language translation: "If something can be proved by logic alone, then one can prove by logic alone that one can prove it by logic alone". Bigelow/Pargetter: that is plausible. System B/Bigelow/Pargetter: Everyday language translation: "If something is true, one can show with logic alone that it cannot be refuted by logic alone. System S5/Bigelow/Pargetter: Everyday language translation: If something cannot be refuted by logic alone, it can be proved by logic alone that it cannot be refuted by logic alone. >Axioms, >Axiom systems. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| S 4 / S 5 | Cresswell | Hughes I 39 S4/Hughes/Cresswell: T + Np> NNP - (weaker) - S5: T + M> NMP (stronger) - both are stronger than T. >Stronger/weaker, >Strength of theories, >Axiom systems. Hughes I 110 S4/Hughes/Cresswell: = T + (Lp> LLp). (s) What is necessary is necessary for logical reasons. Stronger than T, weaker than S5. S5: T + (Mp> LMp). ((s) what is possible, is necessarily possible.) Hughes I 69 Semantics/semantic models/Hughes/Cresswell: E.g. T, S4, S5 alone are insufficient to characterize the different meanings of necessity and possibility. Semantic graphs/Hughes/Cresswell: consider various worlds. >Accessibility. |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 Hughes I G.E. Hughes Maxwell J. Cresswell Einführung in die Modallogik Berlin New York 1978 |
| Semantic Ascent | Field | I 96 Semantic ascent/Field: Not a set of axioms (not logically true), but weaker: with a modal operator (diamond) "It is possible that" in front. - Then it is logically true that it is logically possible. >Logical truth, >Logical possibility. I 245 Semantic ascent: "not every axiom of this theory is true" is a solution for infinitely axiomatized theories. >Axioms, >Axiom systems. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Theoretical Entities | Craig | Field III 43 Theoretical Entities/Craig/Field: Re-interpretation of the sciences without theoretical entities. >Theoretical entities. FieldVsCraig: in contrast to numbers, these can very well be causally relevant, e.g. electrons. >Causality, >Causal relation, >Causation, >Measurements. Fraassen I 206 Craig/Craig's Theorem: we eliminate theoretical entities and replace a theory T by a description of an infinitely complex regularity which contains all the observation consequences of the original theory. >Regularities, >Description. The original theory is then finally axiomatized. Craig's transformation is infinitely axiomatized. >Axioms, >Axiom systems. SmartVsCraig: if the theory were only blackening, it would be a cosmic coincidence if the transformation were to work with names of theoretical entities instead of the theoretical entities themselves. >Blackening of the paper, >Cosmic coincidence. So the alleged recourse would only be the postulation of a coincidence. This means we do not need to go to infinity. >Regress, >Infinity, >Randomness. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 Fr I B. van Fraassen The Scientific Image Oxford 1980 |
| Theories | Thiel | Thiel I 73 Theory/Mathematics/Thiel: The term "meta mathematics" had already appeared in a different meaning in the 19th century, reformulated by Hilbert. Hilbert had proved that in Euclid not all of the properties used in the geometric propositions are really developed from the basic properties recorded in the axioms. So it was incomplete. >Axioms, >Axioms/Hilbert. After Cantor's work at the end of the 19th century, it looked as if one could actually find a complete axiom system. Admittedly, no meta mathematics would have been necessary for this. >Completeness, >Incompleteness. I 75 Meta mathematics makes a difference between the proof that a statement A cannot be refuted (the proof that its opposite is not justifiable) and a "positive" justification of A. The first is a rebuttal of ~A thus a proof of ~~A, the second a proof of A. I 76 New: in meta mathematics the existential statements are interpreted more strictly. Anyone who now claims the existence of evidence must also indicate a verifiable way of constructing such evidence. Def "effective" or "constructive" assertion of existence. >Proofs, >Provability, >Syntax. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Truth | Hilbert | Berka I 395 Truth/absolute truth/Hilbert: axioms and provable propositions are images of the thoughts which make up the method of the previous mathematics, but they are not themselves the absolute truths. >Axioms, >Axiom systems, >Axioms/Hilbert. Def absolute truth/Hilbert: absolute truths are the insights provided by my proof theory with regard to the provability and consistency of the formula systems. >Proof theory/Hilbert. Through this program, the truth of the axioms is already shown for our theory of proof(1). Berka I 486 Relative Truth/correctness in the domain/Tarski: the relative truth plays a much greater role than the (Hilbertian) concept of the absolute truth, which has so far been mentioned: Definition correct statement in the domain a/Tarski: every statement in domain a is correct, which then (in the usual sense (s)> Putnam would choose spelling with asterisks)) would be true if we limit the scope of the individuals to the given class a. That is, if we interpret the terms "individual" as "element of class a", "class of individuals" as "subclasses of class a", and so on. Class Calculation: here you would have to interpret expressions, e.g. of the type "Πxp" as "for each subclass x of class a:p" and, e.g. "Ixy" as "the subclass x of the class a is contained in the subclass y of the class a". Then we modify definition 22 and 23. As derived terms, we will introduce the concept of the statement, which in an individual domain with k elements is correct, and the assertion which is correct in each individual area(2). >Truth/Tarski, >Truth Definition/Tarski. 1. D. Hilbert: Die logischen Grundlagen der Mathematik, in: Mathematische Annalen 88 (1923), pp. 151-165. 2. A. Tarski: Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol 1, Lemberg 1935. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Belnap, Nuel | Cresswell Vs Belnap, Nuel | HCI 299 Paradoxes of implication/Hughes/Cresswell: are at worst harmless. In most cases, we wish to speak of entailment. VsEntailment/VsBelnap/VsAnderson: Their system E (see above) pays too high a price with the absence of the disjunctive syllogism (see below principle C). I 300 Problem: the mere construction of such an axiom system does not provide us with a clear notion of entailment. Paradoxes of implication/Hughes/Cresswell: are even desirable: we want to be able to say: "If you accept that, you can prove anything." I.e. in a contradictory system everything can be proven. aca |
Cr I M. J. Cresswell Semantical Essays (Possible worlds and their rivals) Dordrecht Boston 1988 Cr II M. J. Cresswell Structured Meanings Cambridge Mass. 1984 |
| Goedel, K. | Dennett Vs Goedel, K. | I 603 Gödel number/Dennett: Goedel numbers make it possible to arrange all possible axiom systems in alphabetical order. Goedel/Turing: showed that this set belongs to a different set in the Library of Babel: the set of all possible computers. Each Turing machine in which happens that a consistent algorithm runs for proving mathematical truths is associated with a Godel s theorem - with an arithmetic truth that it can not prove. Dennett: So what? Mind/Goedel: it shows that the mind can not simply be like machines. People can do things which may not be performed by machines. DennettVs! DennettVsGödel: problem: how can you find out, whether a mathematician has proved a theorem, or has only made a noise like a parrot? (> Proofs). |
Dennett I D. Dennett Darwin’s Dangerous Idea, New York 1995 German Edition: Darwins gefährliches Erbe Hamburg 1997 Dennett II D. Dennett Kinds of Minds, New York 1996 German Edition: Spielarten des Geistes Gütersloh 1999 Dennett III Daniel Dennett "COG: Steps towards consciousness in robots" In Bewusstein, Thomas Metzinger Paderborn/München/Wien/Zürich 1996 Dennett IV Daniel Dennett "Animal Consciousness. What Matters and Why?", in: D. C. Dennett, Brainchildren. Essays on Designing Minds, Cambridge/MA 1998, pp. 337-350 In Der Geist der Tiere, D Perler/M. Wild Frankfurt/M. 2005 |
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