# Dictionary of Arguments

Philosophical and Scientific Issues in Dispute

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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
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.

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
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
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. >Arithmetic, >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

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
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:
(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
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.
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
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

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 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

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

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

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.

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

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
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.

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.

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

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
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

The author or concept searched is found in the following controversies.
Disputed term/author/ism Author Vs Author
Entry
Reference
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

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