Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 26 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| 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 |
| Beliefs | Prior | Cresswell II 146 Belief/Prior/Cresswell: Thesis: Belief should not be considered a predicate of that-sentence - but instead believes-that should be seen as a syntactic unit that is applied directly to a sentence. Cf. >That-sentences, >Predicates, >Beliefs, >Objects of thought, >Objects of belief. Prior I 6f Belief/Prior: no adequate approach without distinction between mind state of belief and that which is believed (state/content). >Belief state/Perry, >Mind state. Prior: in case of false beliefs: instead of non-existing object: attribution: E.g. Othello attributes infidelity to Desdemona. >`Attribution, >Predication, >Non-existence. PriorVsRussell: Problem: above it is abstract loyalty. >Abstract objects, >Abstractness. In case of falsity, the belief relation would then need to have an additional position (to the true fact). >Relation theory. I 11 False Belief/Russell: false facts fail in truth-making. >Truthmakers, >Facts. Montague: points in the wrong direction. >R. Montague. PriorVs: not for a neutral observer. >Intentionality, >Thinking. I 27 Belief/Prior: belief is no relation - E.g. ...that nothing is perfect: there is no object. >Generality, >Generalization. I 53 Belief Function/Prior: E.g. X believes that ... is not identical in identical propositions: e.g. ...is a bachelor/...is an unmarried man. although one may feel that the propositions are self-identical. I 81 Belief/Prior: you do not have to believe rightly that you believe something. >about/Prior) You can also simultaneously believe p and not-p. You can believe something contradictory. E.g. the fear that God will punish you for your disbelief. >Thinking, >Logic. You can find out that you did not believe what was thought you believed. - If someone believes what he says when he says that he mistakenly believes that it is raining, then this belief is not necessarily mistaken. >Error, >Deception, >Falsehood, >Levels/order, >Description levels. No epistemic logic ist necessary, propositional calculus is sufficient. >Epistemic logic, >Propositional calculus. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 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 |
| Calculus | Mates | I 63 artificial language/formal language/counterpart/Mates: the statements of the natural language correspond the artificial formulas, as a counterpart, not as abbreviations. >Symbols, >Equivalence, >Propositional forms, >Propositional functions, >Formal language, >Natural language. If symbols are associated with no sense, then it is an uninterpreted calculus. >Interpretation/Mates. I 115 Propositional calculus: the propositional calculus has no quantifiers. >Propositional calculus, >Quantifiers, >Quantification. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Calculus | Quine | VII (d) 71 Propositional Logic/propositional calculus/Quine: p, q relate to propositional terms, whatever they may be, but propositional terms like truth values are indistinguishable in terms of the calculus. ((s) only if they are interpreted (something is inserted). - p, q do not refer but you can also consider them as referring. >Propositions/Quine. IX 188 Predicate calculus 2nd order: are individuals and classes of individuals. |
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 |
| Constants | Tarski | Berka I 496 Names/variables/constants/Tarski: variables represent names constants are names. >Representation, > Proxy, >Variables. For each constant and each variable of the object language (except for the logical constants of the propositional calculus) we can constitute a fundamental feature that contains this sign. The statement variables enter into the fundamental functions neither as functors nor as arguments. Statement variable: any ((s) individual) of them is regarded as an independent fundamental function.(1) >Object language, >Metalanguage. 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 |
| Decidability | Cresswell | I HC 120 Decidability/propositional calculus/Hughes/Cresswell: although it is possible to give a clear view of the validity of the propositional calculus, the propositional calculus is still not a decidable system. - But there are a number of decidable fragments of the propositional calculus. The same goes for the modal expansion of the propositional calculus. >Predicate calculus, >Expansion, >Validity, >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 |
| 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 |
| Functional Calculus | Berka | Berka I 119 Extended function calculus/Hilbert: Extended function calculus is used to express the existence of the opposite of a statement. E.g. For every statement X there is a statement Y, so that at least one and only one is true. This saves the constraint of content representation. >Formalism, >Statements, >Validity, >Satisfiability. I 120 Then we can ask for a criterion for the correctness of formulas with arbitrary combinations of all- and existential quantifiers. >Universal quantification, >Existential quantification, >Quantification. Then there is the principal possibility of decidability about the provability of a mathematical theorem. >Decidability, >Provability, >Proofs. Narrow function calculus: The narrow function calculus is sufficient for the formalization of logical reasoning. >Formalization. Berka I 337 Function calculus/Hilbert/Ackermann: here (in contrast to the propositional calculus) the decision problem is still unsolved and difficult. - But for certain simple cases a procedure could be given. Simplest case: only function variable with one argument. >Decision problem, >Propositional calculus. I 337 Functional calculus: here the following circumstance has to be considered in particular: the generality or satisfiability of a logical expression may depend on how large the number of objects in the individual domain is. >Individual domain, >Domain. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Implication Paradox | Wessel | I 129 C.I.Lewis VsParadoxes of the implication: "strict implication": modal: instead of "from contradiction any statement": "from impossible ..." >Implication, strict, >Modalities, >Modal logic. WesselVsLewis, C.I.: circular: modal terms only from logical entailment relationship - 2.Vs: strict Implication cannot occur in provable formulas of propositional calculus as an operator. >Consequence, >Operators. I 140ff Paradoxes of implication: strategy: avoid contradiction as antecedent and tautology as consequent. >Tautologies, >Antecedent, >Consequent. I 215 Paradoxes of implication/quantifier logic: Additional paradoxes: for individual variables x and y may no longer be used as any singular terms - otherwise from "all Earth's moons move around the earth" follows "Russell moves around the earth". Solution: Limiting the range: all individuals of the same area, for each subject must be clear: P (x) v ~ P (x) - that is, each predicate can be meant as a propositional function - Wessel: but that is all illogical. >Logic, >Domain. |
Wessel I H. Wessel Logik Berlin 1999 |
| Implication, strict | Lewis | Berka I 154 Definition Strict Implication/CI. I. Lewis/Berka: (1918)(1): C'pq = NMKpNq - "It is not the case that p is true and q is false". - >Paradox of material implication: the for it responsible statement "p is true and q is false" is not free of self-contradiction - Implication: if it should have the meaning "q derivable from p", the above statement is obviously a contradiction. --- I 155 Paradox of strict implication: 1. An impossible statement implies any statement - 2. A necessary is implied by each statement. - It also follows that all impossibilities and all necessities are strictly equivalent. - Solution: enhanced propositional calculus. 1.C.I. Lewis: A Survey of Symbolic Logic. Berkeley 1918, Reprint, New York 1960. |
Lewis I David K. Lewis Die Identität von Körper und Geist Frankfurt 1989 Lewis I (a) David K. Lewis An Argument for the Identity Theory, in: Journal of Philosophy 63 (1966) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (b) David K. Lewis Psychophysical and Theoretical Identifications, in: Australasian Journal of Philosophy 50 (1972) In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis I (c) David K. Lewis Mad Pain and Martian Pain, Readings in Philosophy of Psychology, Vol. 1, Ned Block (ed.) Harvard University Press, 1980 In Die Identität von Körper und Geist, Frankfurt/M. 1989 Lewis II David K. Lewis "Languages and Language", in: K. Gunderson (Ed.), Minnesota Studies in the Philosophy of Science, Vol. VII, Language, Mind, and Knowledge, Minneapolis 1975, pp. 3-35 In Handlung, Kommunikation, Bedeutung, Georg Meggle Frankfurt/M. 1979 Lewis IV David K. Lewis Philosophical Papers Bd I New York Oxford 1983 Lewis V David K. Lewis Philosophical Papers Bd II New York Oxford 1986 Lewis VI David K. Lewis Convention. A Philosophical Study, Cambridge/MA 1969 German Edition: Konventionen Berlin 1975 LewisCl Clarence Irving Lewis Collected Papers of Clarence Irving Lewis Stanford 1970 LewisCl I Clarence Irving Lewis Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| 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 |
| 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 |
| Number Theory | Tarski | Berka I 532 Elementary number theory/Tarski: the science in which all variables represent names of natural numbers and as constants: (in addition to the characters of the propositional calculus and the functional calculus) the characters of zero, unity, equality, the sum and of the product may occur.(1) >Numbers, >Unity, >Equality, >Equal sign, >Variables, >Name of a number, >Natural numbers, >Real numbers, >One, >Zero. 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 |
| Proof Theory | McDowell | ad Hughes I 119 Validity/Propositional Calculus: truth tables are not sufficient for an evaluation of formulas in the propositional calculus - because we cannot assign specific individual variable and predicate variables. >Intuitionism, >Proof, cf. >Model theory. |
McDowell I John McDowell Mind and World, Cambridge/MA 1996 German Edition: Geist und Welt Frankfurt 2001 McDowell II John McDowell "Truth Conditions, Bivalence and Verificationism" In Truth and Meaning, G. Evans/J. McDowell Hughes I G.E. Hughes Maxwell J. Cresswell Einführung in die Modallogik Berlin New York 1978 |
| Proper Names | Tarski | Berka I 451 Def quotation name/Tarski: any name of a statement (or even meaningless expression) consisting of quotes and the expression, and which is precisely the signified through the considered name. E.g. the name ""it snows"". ((s) Quotation marks twice) N.B.: identical configured expressions must not be identified. - Therefore quotation names are general, not individual names (classes of character strings). >Description levels, >Quotation marks, cf. >Names of sentences. I 453 Syntactically simple expressions - such as letters - have no independent meaning. I 451 Def structural-descriptive name/Tarski: (different category than the quotation names): they describe, of what words the expression, designated by the name, consists and of which characters each individual word consists and in what order they follow one another. - This goes without quotation marks. Method: introduce single names for all letters and other characters (No quotation names). E.g. for letters f, j, P, etc.: Ef, Jay, Pee, ex - E.g. to the quotation name ""snow"" (quotation marks twice) corresponds the structural-descriptive name: word that consists of the six consecutive letters Es, En, O, double-u - (letter names without quotation marks). I 451 Semantically ambiguous/Russell/Tarski: E.g. name, designating: a) with respect to items b) to classes, relations, etc. I 464 Name/translation/metalanguage/object language/Tarski: difference: an expression of the object language in the metalanguage may a) be given a name, or b) a translation. >Object language, >Metalanguage. I 496 Names/variables/constants/Tarski: variables represent names constants are names. >Representation, >Proxy. For each constant and each variable of the object language (except for the logical constants of propositional calculus) can form a fundamental function that contains this character (the statement variables neither occur into the fundamental functions as functors nor as arguments). Statement variable: any ((s) individual) of them is regarded as an independent fundamental function.(1) >Constants/Tarski, >Functions/Tarski. 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 |
| Propositional Logic | Berka | Berka I 237 Propositional logic: has no subject variables - because it contains no quantifiers. >Variables, >Individual variables, >Quantifiers >Quantification, >Statements, >Propositional calculus, >Logic. |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Propositional Logic | Quine | VII (d) 71 Propositional Logic/Propositional Calculus/Quine: "p", "q" refer to propositional terms, whatever they may be, but propositional terms such as truth values are not distinguishable in terms of the calculus - ((s) only if they are interpreted (something is inserted)). - P, q do not refer -but it is also okay if you look at them as refering. >Truth Value/Quine |
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 |
| Recursion | Tarski | Skirbekk I 156 Recursion/recursive method/Tarski: starting from simple propositional calculus specifying the operations with which we construct composite functions. >Functions/Tarski, >Recursive rules. Skirbekk I 157 Recursion/Tarski: problem: composite statements are constructed from simpler propositional functions, but not always from simpler statements. >Propositional functions. Hence no general recursion is possible. Recursive definition of satisfaction is only possible in a much richer metalanguage (i.e. in metalanguage we have variables of a higher logical type than the in the object language.(1) >Expressivity, >Richness. 1. A.Tarski, „Die semantische Konzeption der Wahrheit und die Grundlagen der Semantik“ (1944) in: G. Skirbekk (ed.) Wahrheitstheorien, Frankfurt 1996 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Skirbekk I G. Skirbekk (Hg) Wahrheitstheorien In Wahrheitstheorien, Gunnar Skirbekk Frankfurt 1977 |
| Tautologies | Mates | I 116 Tautologies/Mates: Tautologies depend on the meaning of "and", "not", " if .... then ", etc. , but not the meaning of " all ", " some ", " people ", " mortal" , etc. - ((s) I.e. only from the logical constants, not from the quantifiers). >Logical constants, >Quantifiers, >Quantification. On the other hand: analyticity of a syllogism depends on the meaning of "all" , "some". >Analyticity/Syntheticity, >Synthetic, >Syllogisms. I 117 Tautology/Mates : can not be an atomic statement - because this also may not be valid. I 119 There are valid statements that are not to be tautological: E.g. "(x ) Fx > Fa " There are inferences that are not tautological. - In inferences only tautological inferences are needed. >Derivation, >Derivability. Def tautology: valid statement whose validity does not depend on the quantifiers. >Validity. I 119 Tautology/propositional calculus/Mates : since all statements of p.c. (propositional calculus) are quantifier-free, they are tautological, if they are valid. >Propositional calculus. That carries over to their inserting results. - There is still no decision procedure, if there is tautology. I 127 A tautology is the same as a consequence of any set of propositions. >Consequences. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Terminology | Simons | I 14 Product/mereology/Simons: the average equals the greatest lower bound. Total: "the individual that overlaps something if it at least overlaps one of x or y, is the total. It is not always equivalent to the least upper bound (lub). Lattice theory: the lattice theory is about the "smallest individual which contains both". Def difference: the largest individual that is contained in x, which has no part in common with y, exists only if x is not part of y. Def fusion/general sum: a fusion or general sum is the sum of all objects which satisfies a specific predicate Fx, denoted by the variable-binding operator s: sx [Fx]. There may be several fusions. The sum is the largest fusion. I 226 Fusion: fusion includes replacement of the former. E.g. a former F is replaced by two Fs. Def nucleus/general product: the nucleus is the product of the objects that meet a predicate px[Fx]. Universe: the universe is the sum of all objects. This corresponds to the unit element of the Boolean algebra. Atom: an individual that does not have any parts is an atom. An individual in general may have parts. A universe with 3 atoms (atom) may have 7 individuals. If there are c atoms, there are 2c-1 combinations. It follows that there cannot be even numbers. Combinations of individuals are individuals themselves again. I 32 Def upper bound/mereology/Simons: the individuals which fulfill a predicate fx are bound up if there is an individual from which they are all a part. Sum: "the individual that overlaps something if it at least overlaps one of x or y". ((S) Hasse diagram: the upper point is part of the bottom.) Universe: here, the upper bound is for everything. The existence of an upper bound does not imply the existence of sums or least upper bound, e.g. the set of subsets of natural numbers which are either non-empty or finite or infinite and have a finite complement. Each collection is upwardly limited by the entire set of natural numbers without a least upper bound. E.g.: collection of all finite sets of even numbers. E.g. open intervals on the real number strand: here each two open intervals have at least an upper bound, namely the interval of its endpoints. I 33 Their outer extreme points are, however, separate intervals with a gap between them and they do not have a sum. If a sum exists, then also a least upper bound but not vice versa. Being part of a wider whole means: having an upper bound. I 60 Def prosthetics/Lesniewski/Simons: ("first principles"): prosthetics is Lesniewski's counterpart to the propositional calculus, which it contains as a fragment. In addition, it includes variables for each type of statements and quantifiers - equivalent with systems of proposition types (statements types) by Church or Henkin. I 112 Definition upper bound/mereology/Simons: the individuals who are fulfilling a predicate fx are bound up if there is an individual from which they are all a part. Sum: "the individual that overlaps something if it overlaps at least one of x or y". I 211 Coincidence/Simons: equality of the elements is not sufficient for equality of the parts ((s) e.g. member-like bodies may have different chairpersons). Coincidence: the coincidence is temporarily indistinguishable. The class {Tib + Tail]} has only three parts. Tibbles can have a lot more. I 225 Permanent coincidence of F1 and F2: F1 and F2 are indistinguishable in the real world. At most by modal property. I 228 Coincidence principle/Simons: coincidence (all parts have in common) is necessary for superposition (two things at the same time in the same place). Composition/mereology/Simons: e.g. the ship, but not the wood is composed of planks. A human has parts that are not shared by the collection of atoms. I 334 Topology/mereology/Simons topological concepts that go beyond the mereology: adjacency and connectivity are used for the definition of "whole". |
Simons I P. Simons Parts. A Study in Ontology Oxford New York 1987 |
| Truth Definition | Tarski | Berka I 403 Truth-Definition/Tarski: in artificial languages: not solvable if they contain variables of an arbitrarily high order. >Levels, >Variables, cf. >Type theory. Solution: truth-concept as undefined basic concept - it can be used in a "deductive discipline".(1) Berka I 477 Truth/Truth-Definition/language/Tarski: would the language be finite, it took just a list to fill in the scheme.(2) 1. A.Tarski, „Der Wahrheitsbegriff in den Sprachen der deduktiven Disziplinen“, in: Anzeiger der Akademie der Wissenschaften in Wien, mathematisch-naturwissenschaftliche Klasse 69 (1932) pp 23-25 2. A.Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Commentarii Societatis philosophicae Polonorum. Vol. 1, Lemberg 1935 --- Horwich I 119 Truth-Definition/Tarski: has other interesting consequences: we can use it to prove the semantic sentence of contradiction and the semantic sentence of contradiction - but not the corresponding logical sentences, because these contain the term "true". (They belong to the propositional calculus). >Semantics, >Logic, >Excluded Middle, >Truth predicate, >Semantic closure, >Metalanguage, >Provability, >Propositional logic, >Propositional calculus. Also, it is shown that truth never coincides with provability - because there are true statements that are not provable.(3) 3. A. Tarski, The semantic Conceptions of Truth, Philosophy and Phenomenological Research 4, pp. 341-75 --- Skirbekk I 156 Truth/Tarski: we get the truth-definitions simply because of the definition of fulfillment: Definition fulfillment/Tarski: fulfillment is a relationship between any object and propositional function - an object satisfies a function when the function is a true statement, when replacing the free variable with the name of object - Snow satisfies the propositional function "x is white" - Vs: that is circular, because "true" occurs in the defintion of fulfillment - Solution: fulfillment itself must be defined recursively - if we have the fulfillment, it relates by itself on the statements themselves - a statement is either satisfied by all objects, or by none. Skirbekk I 162 Truth Definition/Tarski: not circular, because the conditions under which statements of the form "if ... then" are true, are extralogical. Skirbekk I 163 Truth-Schema/Tarski: correct: (T)X is true if and only if p. - wrong: (T") X is true if and only if p is true ((s) Vs: here 'true' occurs twice) - Tarsk: Confusion of name and object) statements and their names) - ((s) p is the statement itself, not assertion of its truth.) >Redundancy theory. Skirbekk I 169 Truth-Definition/Tarski: "actually" does not occur, because it does not concern the content - also no assertibility condition because the definition is not epistemologically - epistemologically would be "snow is white" not true.(4) 4. A.Tarski, „Die semantische Konzeption der Wahrheit und die Grundlagen der Semantik“ (1944) in: G. Skirbekk (ed.) Wahrheitstheorien, Frankfurt 1996 |
Tarski I A. Tarski Logic, Semantics, Metamathematics: Papers from 1923-38 Indianapolis 1983 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 Horwich I P. Horwich (Ed.) Theories of Truth Aldershot 1994 Skirbekk I G. Skirbekk (Hg) Wahrheitstheorien In Wahrheitstheorien, Gunnar Skirbekk Frankfurt 1977 |
| Truth Functions | Cresswell | Hughes I 24 Truth function/Hughes/Cresswell: only four different truth functions (tr.f.) of p: 1) Negation 2) p itself 3) true, no matter whether if p itself is true or false 4) wrong, no matter whether p itself is true or false. Hughes I 44 Truth-function/Hughes/Cresswell: each well formed formula (wff) of the propositional calculus is a truth function of its variables. - ((s) That is, it can be found in the truth table). Modal function: any well-formed formula (wff) containing a modal operator, is a modal function of its variables. >Truth tables, >Propositional calculus. |
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 |
| Truth Values | Quine | VII (d) 71 Propositional Calculus/indistinguishability/theoretical terms/Quine: "p", "q" etc. refer to propositional concepts, whatever they may be. But we know that propositional concepts like truth values are not distinguishable in terms of the calculus, the expressiveness of the calculus is limited. VII (f) 112 Truth Values/Quine: can be allowed as abstract entities. VII 115 Truth Value/Quine: is not an abstract entity to which we appeal with assertions. VII (h) 154 Range/Russell: a change in the range of a description is neutral to the truth value of any sentence. Quine: but only if the description designates something. Lauener XI 38 Quantification/Lauener/(s): truth values can only be attributed to quantified sentences. Quine I 226 Vagueness/Quine: leaves the truth values untouched. Therefore it can be useful. >Vagueness. I 263ff Truth Value/intension/extension/Quine: in extensional contexts a singular term may be replaced by a singular term with the same name without changing the truth value of the sentence. This is not possible in opaque (intensional) contexts. >Intensions, >Extensions, >Opacity. I 266 Opaque Contexts/Truth Value/Frege: in a construction with a propositional attitude, a sentence or term may not denote truth values, a class or an individual, but functions as the "name of a thought" or the name of a property or an "individual concept". ((s) In non-intensional contexts, a sentence in Frege's work designates a truth value, "The True," or "The False". > "Great Fact", >"Slingshot Argument"). II 192 From today's point of view, quantifier logic is nothing more than a further development of the logic of truth functions. The truth value of a truth function can be calculated on the basis of the truth values of the arguments. Why then does quantifier logic not become decidable by truth tables? This validity criterion would be too strict because the quantified sub-expressions are not always independent of each other. Some sub-expressions may turn out to be untrue, but are unworthy of a closer look at an assignment to truth values. See also >Truth tables. III 281 Truth value/Existence/Nonexistence/Ontology/Logic/Quine: which truth values have sentences like "Zerberus barking"? (See also >Unicorn example). The answer "wrong" would be premature. III 282 Problem: for all sentences that would be wrong, there would be a negation that would be true! Our derivation methods do not prove anything in case the object does not exist. What would have to be proved is based on an unfulfilled condition. Truth value gap/Quine: comes from everyday language, in logic we have to fill it. And be it arbitrary. Every sentence should have a truth value (true or false). >Everyday language. That was the reason for the convenient extension of the term conditional in § 3,m which generally allowed a truth value for the whole conditional. We now need a similar extension for singular terms, which do not describe anything. But this cannot be achieved by an all-encompassing decision. But this can be done for simple sentences, from which we derive rules for compound sentences. Def simple predicate: is a predicate if it does not explicitly have the form of a quantification, negation, conjunction, alternation etc. of shorter components. If a simple predicate is applied to a singular term that does not denote anything, the sentence in question is to be considered false. Then e.g. "Zerberus barks" is wrong, because it represents an application of the predicate "[1] barks" to "Zerberus". V 112 Truth values/Language learning/Quine: truth values correspond to a more advanced level of learning. Using different theories for different subject areas V 113 we finally learn (if at all) which judgement to make in the indeterminate cases of conjunction or alternation in the middle of the table. Logic/Learn languages/Quine: bivalent logic is a theoretical product which, like all theory, is only learned indirectly. How, we can only speculate about that. VI 128 Singular terms/truth value/sense/divalued logic/unicorn/Quine: in the case of unrelated singular terms or failed descriptions, we may not know the truth value. It is not profitable to describe such sentences as meaningless, since the existence of the object could turn out (e.g. Pluto). It is alright to leave the truth value open, but not the meaning of a sentence! VI 129 Singular terms/truth value/sense/divalued logic/unicorn/Quine: in the case of unrelated singular terms or failed descriptions, we may not know the truth value. It is not profitable to describe such sentences as meaningless, since the existence of the object could turn out (e.g. Pluto). It is alright to leave the truth value open, but not the meaning of a sentence! VI 131 Antirealism/Sentence of the excluded Middle/Dummett/Quine: Dummett turns against the sentence of the excluded middle with epistemological arguments. (Also Brouwer): No sentence is true or false, as long as no procedure for the determination of the truth value is known. |
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 Q XI H. Lauener Willard Van Orman Quine München 1982 |
| Validity | Cresswell | Hughes I 65ff Validity/Hughes/Cresswell: No structural property of formulas, no relation between formulas. - In contrast: derivability relation between formulas. >Derivability, >Formulas. Yet the set of the derived and the valid formulas in a system are identical. Hughes I 119 Validity/propositional calculus: truth tables are not sufficient for an evaluation of formulas in the propositional calculus. >Predicate calculus. Because we can not assign truth values to individual variables and predicate variables. >Truth values, >Valuation. |
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 |
| Valuation | Cresswell | ad Hughes I 119 Validity/propositional calculus/Cresswell: truth tables are not sufficient for an evaluation of formulas in the propositional calculus. - Because we cannot assign truth values to individual variables and to predicate variables. >Individual variable, >Individual constants, >Truth values, >Interpretation of variables, >Predicate calculus. |
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 |
| Variables | Cresswell | Hughes I 118 Variables/free/bound/Hughes/Cresswell: it is all about occurrences of variables. - Therefore one and the same variable can be in one and the same formula both bound also occur as free. (> mention / >use /> word /object, >Word/object, >Free variables, >Bound variables. A token of x can be free and once again bound in the same formula. Hughes I 120 Free variables/inserting/propositional calculus/Hughes/Cresswell: when evaluating a formula, we must assume that the other possibly occurring free variables are constant. >Valuation. |
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 |
| |||
| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Boole, G. | Frege Vs Boole, G. | Berka I 57 Classical Logic/Berka: bivalent extensional logic: mostly created by Frege. Frege: clear distinction between variables and constants, laws and rules, created the concepts of logic functions (propositional functions) and quantifiers, the semantics of sense and meaning, also the first axiomatic system of classical logic. Russell: was the first to recognize Frege’s importance. Notation/Russell: was rather a follower of Peano. FregeVsBoole: pro clear distinction between statements and class logic. Sets the AL as a foundation. Truth Functions/Frege: their theory is of central importance for the propositional calculus. Truth Value Tables: already known by Boole, Schröder, and Peirce. Systematically elaborated first by Post (1921)(1), Lukasiewicz (1921)(2), and Wittgenstein (1921)(3). First transferred to the PL by Foster (1931)(4), later by Wright (1957)(5). 1. E. L. Post, Introduction to a general theory of elemantary propositions, American Journal of Mathematics 43 (1921) , 163-185 2. J. Lukasiewicz, Logica dwuwartosciowa, PF 23 (1921), 189-205 3. L. Wittgenstein, Logisch-Philosophische Abhandlung, Ann. Naturphil. 14 (1921), 185-262 4. A. L. Forster, Formal Logic in finite terms Ann. Math. 32 (1931) 407-430 5. G. H. von Wright, Logical Studies, London 1957 Frege IV 92 SchröderVsBoole: Vs"Universe of Discourse" (> Quine: "You can still hatch enough..."). Sectors Calculus/Sector Calculus/Manifold/Schröder/Frege: division of an area into sectors, so that no point is at the same time in two sectors. (Boxes). ((s) no overlap). Most important relation between sectors: the containment of one in the others: "classification" (both can simply coincide). This corresponds to the relationship part/whole. IV 93 Instead of "sectors" we can also say "classes". Instead of manifold: "main sector". Manifold/Schröder/Frege/(s): classes of classes. ("main sector" box which comprises one or more boxes). Classes/Individuals/Schröder/Frege: Schröder also refers to an individual as class, but which only contains this single element, the individual. But also a class with several elements (individuals) can be considered a "thought thing" and thus can be presented as an individual. FregeVsSchröder: the difference between individual and class becomes fluent here. IV 9 FregeVsBoole: he tries to "embed abstract logic into the guise of algebraic signs". Frege against: tries to establish a uniform formula language of mathematics and logic. IV 13 FregeVsBoole: discards his concept of "universal class" ("universe", "universe of discourse" coined by de Morgan). SchröderVsBoole: the zero class is contained in every class. I.e. also in the class of the classes that are identical with the universal class. I.e. zero classes (in Boole designated with "0") and universal classes (in Boole designated with "1") are equal. So we have both: 1 = 1 and 0 = 1, and that is not possible. Schröder: the universal class cannot contain classes as elements among themselves, which in turn contain elements of the same manifold. ((s) anticipation of the theory of types). |
F I G. Frege Die Grundlagen der Arithmetik Stuttgart 1987 F II G. Frege Funktion, Begriff, Bedeutung Göttingen 1994 F IV G. Frege Logische Untersuchungen Göttingen 1993 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| De re | Wright, von Vs De re | Hughes I 162 Def de re/Hughes/Cresswell: a well formed formula (wff) a containing a modal operator expresses a modality de re if the range of a modal operator from a contains a free occurrence of an individual variable, otherwise a expresses a modality de dicto. WrightVsDe re/Hughes/Cresswell: (and other authors): wanted to eliminate de re in favor of de dicto. one should be able to construct a well formed formula (wff) a' to each well formed formula (wff) a, which does not contain a modality de re and whose equivalence with a can be proved. Hughes/CresswellVsWright: that does not even seem possible with propositional calculus + S5. But apparently nobody has proved that it is impossible. Wright's strategy can be called the "principle of predication" (the term does not come explicitly from him). |
Hughes I G.E. Hughes Maxwell J. Cresswell Einführung in die Modallogik Berlin New York 1978 |
| Extensionality | Prior Vs Extensionality | I 48 Extensionalism/Fallacy of/Extensionality/Extension/Extensional/Prior: Ontology/PriorVsQuine: existence as "being a value of a bound variable" is only a unproven dogma. Quantifiers: There is another unproven dogma: that mixed constructions like "__ is green and __" or "believes that __" cannot fall into the same category as the simple ones. In particular, it is said that "X believes that __" should not fall into the same category as "It is not the case, that __". I.e. supposedly they not both single-digit links. Resistance comes from the formal logicians who want to simplify their systems by saying that if the sentences S1 and S2 have the same truth value, then every composite sentence, which only differs in that it has S1 as a sub-sentence where the other one has S2 has as a sub-sentence, has the same truth value. This is the "law of extensionality". PriorVsExtensionality: if the law was true, the following two sentences would have to mean the same thing: a) "X thinks the grass is pink" b) "X thinks the grass is purple" But everyone knows that you can think one thing without thinking the other. Point: "X thinks the grass is pink" is not a true composite sentence with "grass is pink" as a component. Technically speaking: It is no real function with "grass is pink" as an argument. Extensionality/Prior: but, apart from a certain narrow-mindedness, I cannot derive from this that the law of extensionality is wrong. One must admit that there is a long and interesting history of logic in which it is true, just like classical mechanics in physics. I 49 On the other hand, if its defenders speak of intuitive and immediate knowledge of its truth, then I can only say that I have contrary intuitions. Extensionality/Extension/Lesniewski/Lukasiewicz/Prior: both schools tell us that if you drop extensionality, you must admit that some propositions are then neither true nor false. This is justified in classic logic by the fact that there are only four cases a) "true p" is always true, no matter if "p" is true or false, b) "false p": reversed c) not p: reverses the truth value d) "asserts p": true if p is true, otherwise false. Furthermore: if "p" and "q" have the same truth value, then function of "p" has the same truth value as the function of "q". Now, if a function does not obey the law of extensionality, it cannot be one of these four, and if there are other besides these, there must be more than two truth values. (PriorVs). Vs: the first step of this argument already presupposes what it is to prove: namely, that the only property of "p", on which its truth value depends, is its truth value. E.g. "If X thinks that p" was a function of "p". But there are no functions that are false with true arguments. I 50 But why should the truth value of a function "p" not depend on of other properties of "p" than its truth value? To say that this was impossible is to say that for each function fx of a number x, the question whether x > 0 depends on whether x is > 0, which is simply false. E.g. fx = x 1: because in some cases, where x > 0, e.g. x = 2, is x 1 > 0, while in other cases, e.g.: x = 1, x is 1 not > 0. So whether this function of x itself is > 0 does not depend on whether x itself is > 0, but whether x > 1. Likewise, whether X believes that p does not depend on whether it is the case or not that p. Prior: why ever not? ((s) Both are true, but the analogy does not need to be true.) I 101 Protothetics/Protothetic/Lesniewski/Prior: our system is a fragment of Lesniewski's "Protothetics". (20s). 1) normal propositional calculus, ((s) p,q..u,v,>,...) 2) quantifier logic 3) normal identity laws. Full protothetics also includes the law of extensionality. (Tarski seems to support it, because it has proved his independence.) PriorVsExtensionality. |
Pri I A. Prior Objects of thought Oxford 1971 Pri II Arthur N. Prior Papers on Time and Tense 2nd Edition Oxford 2003 |
| Frege, G. | Quine Vs Frege, G. | Quine I 425 VsFrege: tendency to object orientation. Tendency to align sentences to names and then take the objects to name them. I 209 Identity/Aristotle/Quine. Aristotle, on the contrary, had things right: "Whatever is predicated by one should always be predicated by the other" QuineVsFrege: Frege also wrong in "Über Sinn und Bedeutung". QuineVsKorzybski: repeated doubling: Korzybski "1 = 1" must be wrong, because the left and right side of the equation spatially different! (Confusion of character and object) "a = b": To say a = b is not the same, because the first letter of the alphabet cannot be the second: confusion between the sign and the object. Equation/Quine: most mathematicians would like to consider equations as if they correlated numbers that are somehow the same, but different. Whitehead once defended this view: 2 + 3 and 3 + 2 are not identical, the different sequence leads to different thought processes (QuineVs). I 264 according to Russell "Propositional Attitudes": believes, says, strives to, that, argues, is surprised, feares, wishes, etc. ... I 265 Propositional attitudes create opaque contexts into which quantification is not allowed. (>) It is not permissible to replace a singular term by an equally descriptive term, without stretching the truth value here. Nor a general term by an equally comprehensive one. Also cross-references out of opaque contexts are prohibited. I 266 Frege: in a structure with a propositional attitude a sentence or term may not denote truth values, a class nor an individual, but it works as "name of a thought" or name of a property or as an "individual term". QuineVsFrege: I will not take any of these steps. I do not forbid the disruption of substitutability, but only see it as an indication of a non-designating function. II 201 Frege emphasized the "unsaturated" nature of the predicates and functions: they must be supplemented with arguments. (Objections to premature objectification of classes or properties). QuineVsFrege: Frege did not realize that general terms can schematized without reifying classes or properties. At that time, the distinction between schematic letters and quantifiable variables was still unclear. II 202 "So that" is ontologically harmless. Despite the sad story of the confusion of the general terms and class names, I propose to take the notation of the harmless relative clause from set theory and to write: "{x:Fx} and "ε" for the harmless copula "is a" (containment). (i.e.the inversion of "so that"). Then we simply deny that we are using it to refer to classes! We slim down properties, they become classes due to the well-known advantages of extensionality. The quantification over classes began with a confusion of the general with the singular. II 203 It was later realized that not every general term could be allocated its own class, because of the paradoxes. The relative clauses (written as term abstracts "{x: Fx}") or so-that sentences could continue to act in the property of general terms without restrictions, but some of them could not be allowed to exercise a dual function as a class name, while others could. What is crucial is which set theory is to be used. When specifying a quantified expression a variable may not be replaced by an abstraction such as: "x} Fx". Such a move would require a premise of the form (1), and that would be a higher form of logic, namely set theory: (1) (Ey)(y = {x:Fx}) This premise tells us that there is such a class. And at this point, mathematics goes beyond logic! III 98 Term/Terminology/Quine: "Terms", here as a general absolute terms, in part III single-digit predicates. III 99 Terms are never sentences. Term: is new in part II, because only here we are beginning to disassemble sentences. Applying: Terms apply. Centaur/Unicorn/Quine: "Centaur" applies to any centaur and to nothing else, i.e. it applies to nothing, since there are no centaurs. III 100 Applying/Quine: Problem: "evil" does not apply to the quality of malice, nor to the class of evil people, but only to each individual evil person. Term/Extension/Quine: Terms have extensions, but a term is not the denotation of its extension. QuineVsFrege: one sentence is not the denotation of its truth value. ((s) Frege: "means" - not "denotes"). Quine: advantage. then we do not need to assume any abstract classes. VII (f) 108 Variables/Quine: "F", etc.: not bindable! They are only pseudo-predicates, vacancies in the sentence diagram. "p", "q", etc.: represent whole statements, they are sometimes regarded as if they needed entities whose names these statements are. Proposition: these entities are sometimes called propositions. These are rather hypothetical abstract entities. VII (f) 109 Frege: alternatively: his statements always denote one or the other of exactly two entities: "the true one" or "the false one". The truth values. (Frege: statements: name of truth values) Quine pro Frege: better suited to distinguish the indistinguishable. (see above: maxim, truth values indistinguishable in the propositional calculus (see above VII (d) 71). Propositions/Quine: if they are necessary, they should rather be viewed as names for statements. Everyday Language/Quine: it is best if we return to everyday language: Names are one kind of expression and statements are another! QuineVsFrege: sentences (statements) must not be regarded as names and "p", "q" is not as variables that assume entities as values that are entities denoted by statements. Reason: "p", "q", etc. are not bound variables! Ex "[(p>q). ~p]> ~p" is not a sentence, but a scheme. "p", "q", etc.: no variables in the sense that they could be replaced by values! (VII (f) 111) VII (f) 115 Name/QuineVsFrege: there is no reason to treat statements as names of truth values, or even as names. IX 216 Induction/Fregean Numbers: these are, other than those of Zermelo and of von Neumann, immune against the trouble with the induction (at least in the TT), and we have to work with them anyway in NF. New Foundations/NF: But NF is essentially abolishing the TT! Problem: the abolition of TT invites some unstratified formulas. Thus, the trouble with induction can occur again. NFVsFrege: is, on the other hand, freed from the trouble with the finite nature which the Fregean arithmetic touched in the TT. There, a UA was needed to ensure the uniqueness of the subtraction. Subtraction/NF: here there is no problem of ambiguity, because NF has infinite classes - especially θ - without ad-hoc demands. Ad 173 Note 18: Sentences/QuineVsFrege/Lauener: do not denote! Therefore, they can form no names (by quotation marks). XI 55 QuineVsFrege/Existence Generalisation/Modal/Necessary/Lauener: Solution/FregeVsQuine: this is a fallacy, because in odd contexts a displacement between meaning and sense takes place. Here names do not refer to their object, but to their normal sense. The substitution principle remains valid, if we use a synonymous phrase for ")". QuineVsFrege: 1) We do not know when names are synonymous. (Synonymy). 2) in formulas like e.g. "(9>7) and N(9>7)" "9" is both within and outside the modal operaotor. So that by existential generalization (Ex)((9>7) and N(9>7)) comes out and that's incomprehensible. Because the variable x cannot stand for the same thing in the matrix both times. |
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 |
| Intuitionism | Wessel Vs Intuitionism | I 239 WesselVsIntuitionism: the limitation of negation to a specific field destroys logic as an independent science. But this can be solved in a universal system of rules. (see below). I 269 WesselVsIntuitionism: Main defect: that the universal character of logic is denied. Different logics for finite and infinite domains. Also the representatives of microphysics (quantum mechanics) propagate different domain logics. I 270 Wessel: this has to do with a wrong understanding of the object of logic: Logic/Wessel: a special science that investigates the properties of the rules of language. Science: Understands by the object of logic (erroneously) any extra-linguistic object (e.g. quantum, elementary particle, etc.). WesselVs: Dilemma: that this considered object is not directly given to the view, it must be constructed linguistically. But for this you need logic, circular. Negation/Intuitionism/Wessel: the intuitionists reject the negation of the classical calculus, but they should apply (our) non-traditional predication theory, which already takes into account the problem of undecidability. For example, the question whether a certain sequence of numbers occurs at some point in the development of the number π: here there are three possibilities: 1. it can occur (A) 2. it cannot occur (B) 3. it is impossible to determine (C) Suppose someone claims A, then two different negations are possible: 1. the assertion of B 2. the explanation that it is not right. Negation/WesselVsIntuitionism: confuses two different types of negation: the propositional (outer) and the negation in the operator of awarding predicates ( I 271 Intuitionists/Logic/Wessel: accepts, like most classic logicians, the bisubjunction ~(s< P) ↔ (s We now compare some formulas, using the character combinations that are actually meaningless: -i p, ?p etc. -i p: shall be ~(s <--) u ~( I 272 In the class logic the de Morgan laws apply, the IntuitionismVsDe Morgan: Vs 3. and 4. law . 3. ~(p u q) > ~p v ~q 4 ~(~p v ~q) > p u q. Intuitionism/Wessel: is a hidden epistemic logic: "It is provable that p is provable or that ~p is provable". WesselVs: but first you have to have logical basic systems that are not dependent on empiricism! Epistemic predicates ("provable") must not be confused with logical operators! The classic paradoxes occur for the most part also in intuitionistic logic. I 273 There is evidence to show that there must be a number, but not the number itself! Example + One need not be a follower of intuitionism to prefer evidence that constructively provide the number. I 274 MT5. There is a group of formulae provable in the IPC (intuitionist propositional calculus) for which the following applies: some of their P-R are provable in PT and others are not...e.g. p > ~p > ~p p > ~q > (q > _p) ++ I 275 MT6. There is a group of formulas that can be proven in the IPC (intuitionist propositional calculus) to which applies; all their P-R are not provable in PT. E.g. ~(p v q) > ~p u ~q, ~~(p u ~p) WesselVsIntuitionism: MT5 and MT6 show that the intuitionists are inconsistent: if they identify s |
Wessel I H. Wessel Logik Berlin 1999 |
| |||