The Substitutional Analysis of Logical Consequence

Transcription

1 The Substitutional Analysis of Logical Consequence Volker Halbach Consequence and Paradox: Between Truth and Proof Tübingen 3rd March 2017 Consequentia formalis vocatur quae in omnibus terminis valet retenta forma consimili. Vel si vis expresse loqui de vi sermonis, consequentia formalis est cui omnis propositio similis in forma quae formaretur esset bona consequentia [...] John Buridan, Tractatus de Consequentiis, ca (Hubien, 1976, i.3, p.22f)

2 logical validity formal validity All men are mortal. Socrates is a man. Therefore Socrates is mortal. analytic validity John is a bachelor. Therefore John is unmarried. metaphysical validity There is H 2 O in the beaker. Therefore there is water in the beaker. Logical validity has become formal validity (but see, e.g, Read 1994).

3 logical validity formal validity All men are mortal. Socrates is a man. Therefore Socrates is mortal. analytic validity John is a bachelor. Therefore John is unmarried. metaphysical validity There is H 2 O in the beaker. Therefore there is water in the beaker. Logical validity has become formal validity (but see, e.g, Read 1994).

4 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956)

5 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions In Tübingen: truth-theoretic conceptions? 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956)

6 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions In Tübingen: truth-theoretic conceptions? 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956) An argument is logically valid iff the conclusion is true under all interpretations under which its premisses are true.

7 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions In Tübingen: truth-theoretic conceptions? 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956) An argument is logically valid iff the conclusion is true under all interpretations under which its premisses are true.

8 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions In Tübingen: truth-theoretic conceptions? 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956) An argument is logically valid iff the conclusion is true under all interpretations under which its premisses are true.

9 logical validity 1. Proof-theoretic conceptions 2. Semantic conceptions In Tübingen: truth-theoretic conceptions? 2.1 Substitutional analyses (Buridan, Bolzano?, Quine 1986) 2.2 Analyses in essentially richer metalanguage (Tarski, 1936b; Williamson, 2013) 2.3 Model-theoretic definition since (Tarski and Vaught, 1956) An argument is logically valid iff the conclusion is true under all interpretations under which its premisses are true.

10 logical validity I assume that we use a language formulated in classical first-order predicate logic that is an extension of Zermelo Fraenkel set theory possibly with urelements. The target is the analysis of the notion of logical consequence for this language. 40

11 logical validity How do we argue informally for the non-validity of the following argument? Substitutional counterexamples: Some horses are animals. Therefore all horses are animals. Substitutional counterexamples: Some horses are white. Therefore all horses are white.

12 logical validity The following argument isn t valid, because the premiss is true and the conclusion false. Everything is a set or it isn t. Therefore everything is a set or everything is not a set. x (Set(x) Set(x)) / x Set(x) x Set(x) Problem: This cannot trivially be turned into a set-theoretic counterexample. We cannot say that the premiss is true and the conclusion false in the intended model. Model-theoretic logical consequence is not obviously truth-preserving (in contrast to Tarski s (1936b) approach).

13 logical validity Can we have for all models M: if M γ for all γ Γ, then M ϕ while ϕ isn t a logical consequence of Γ because there is a class-sized counterexample, for instance, the intended interpretation? Assumption: We work in a one-sorted language without class quantifiers or the like (but we may have relativizing quantifiers for sets and urelements). There is nothing wrong with class theories but I would have to amend some definitions.

14 logical validity Of course we don t have to worry because of the Gödel Completeness theorem. Kreisel (1965, 1967a,b) argued for the extensional correctness of the model-theoretic definition for first-order logic with his squeezing argument: ϕ follows informally from Γ intuitive soundness every set-theoretic countermodel is an intuitive counterexample Γ PC ϕ Gödel completeness Γ ϕ Γ ϕ means that ϕ follows model-theoretically from Γ; Γ PC ϕ means that ϕ is provable from Γ (in, say, Natural Deduction).

15 logical validity For the Completeness theorem and the Squeezing argument we don t have to quantify over all set-theoretic models. We could equally use the following definitions: Γ ϕ iff (for all countable models M: if M γ for all γ Γ, then M ϕ). Γ ϕ iff (for all 0 2-definable models M: if M γ for all γ Γ, then M ϕ) (Γ r.e.). These analyses are not adequate analyses, although they are extensionally correct.

16 logical validity The Completeness theorem shows the extensional adequacy of the model-theoretic definitions, but we don t have an analysis of what Kreisel calls informal validity. Problems with the model-theoretic definitions: Model-theoretic logical truth doesn t obviously imply truth. Model-theoretic consequence doesn t obviously preserve truth. Model-theoretic validity doesn t obviously imply informal validity; it isn t obviously sound.

17 the substitutional analysis of logical validity Back to Buridan and the naive formality conception! Rough idea: A sentence is logically valid iff all its substitution instances are true. An argument with premisses in Γ and conclusion ϕ is valid iff for all substitutional interpretations: if all substitution instances of sentences in Γ are true, then the substitution instance of ϕ is true. 30

18 the substitutional analysis of logical validity Back to Buridan and the naive formality conception! Rough idea: A sentence is logically valid iff all its substitution instances are true. An argument with premisses in Γ and conclusion ϕ is valid iff for all substitutional interpretations: if all substitution instances of sentences in Γ are true, then the substitution instance of ϕ is true. 30

19 the substitutional analysis of logical validity I speculate that the reasons for the demise of the substitutional account are the following: Tarski s distinction between object and metalanguage in (Tarski, 1936a,b) and the dependence of the substitutional definition on the language set-theoretic reductionism (especially after Tarski and Vaught 1956) and resistance against primitive semantic notions usefulness of the set-theoretic analysis for model theory availability of squeezing arguments (even before Kreisel)

20 the substitutional analysis of logical validity General worry: The substitutional account is worse than the model-theoretic analysis, because in the definition of logical consequence we quantify only over substitution instances. This was exactly the reason why Tarski (1936b) rejected it and developed a precursor of the model-theoretic account. My aim is not to obtain only an extensionally correct substitutional definition of logical consequence, but an adequate definition.

21 the substitutional analysis of logical validity Rough idea: A sentence is logically valid iff all its substitution instances are true. Required notions: substitution instance : substitution functions true : axioms for a primitive satisfaction predicate Sat

22 the substitutional analysis of logical validity Substitution functions are functions that uniformly replace the nonlogical vocabulary in all formulae of the language with suitable expressions. I focus on the binary predicate symbols and Sat as nonlogical symbols; other predicate and function symbols, including constants, can be dealt with by the usual methods. I ll allow free variables in the substitution instances. This will overcome the problems of other substitutional accounts.

23 the substitutional analysis of logical validity Substitution functions are functions that uniformly replace the nonlogical vocabulary in all formulae of the language with suitable expressions. I focus on the binary predicate symbols and Sat as nonlogical symbols; other predicate and function symbols, including constants, can be dealt with by the usual methods. I ll allow free variables in the substitution instances. This will overcome the problems of other substitutional accounts.

24 substitution functions A function I from the set of all formulae Form Sat into Form Sat is a substitution function iff there are formulae σ (x, y), σ Sat (x, y) and possibly δ(x) such that the following two conditions hold: 1. If the variable other than the displayed ones in σ (x, y), σ Sat (x, y) and δ(x) with the highest index is v n, then I uniformly substitutes every variable v k with v k+n, and 2. If ϕ has been obtained by the substitution above, I has the following property: I(ϕ) = σ (x, y) if ϕ is x y, x = y if ϕ is x = y, σ Sat (x, y) if ϕ is Sat(x, y), I(ψ) if ϕ is ψ, I(ψ) I(χ) if ϕ is ψ χ, and x ([δ(x) ]I(ψ)) if ϕ is x ψ In the last line x (δ(x) I(ψ)) is meant if there is a δ(x); otherwise it s x ψ(x).

25 substitution functions Assume we have a binary symbols R, S and unary symbols P and Q in the language. x ( Px y Ryx ) (original formula) x ( Qx v y z (Q y v Sx z) ) (substitution instance) x ( Rxx y Ryx ) (original formula) x (Rzx ( Sx x y (Rzy Sy x )) (subst. inst.) The underlined formula is the relativizing formula.

26 substitution functions Assume we have a binary symbols R, S and unary symbols P and Q in the language. x ( Px y Ryx ) (original formula) x ( Qx v y z (Q y v Sx z) ) (substitution instance) x ( Rxx y Ryx ) (original formula) x (Rzx ( Sx x y (Rzy Sy x )) (subst. inst.) The underlined formula is the relativizing formula.

27 substitution functions original argument All men are mortal. Socrates is a man. Therefore Socrates is mortal. substitution instance All starfish live in the sea. That (animal) is a starfish. Therefore that (animal) lives in the sea. substitution instance All objects in the box are smaller than that (object). The pen is in the box. Therefore it is smaller than that (object). 20

28 satisfaction The axioms for Sat are added to our overall theory, an extension of ZF. Syntax is coded as usual. Schemata of the base theory are extended to the language with Sat (and possibly D). The quantifiers for a and b range over variables assignments, the quantifiers for ϕ and ψ over formulae of the entire language (including Sat and D). The following axioms are obligatory: a v w (Sat( Rvw, a) Ra(v)a(w)) and similarly for all predicate symbols other than Sat a ϕ (Sat( ϕ, a) Sat( ϕ, a)) a ϕ ψ (Sat( ϕ ψ, a) (Sat( ϕ, a) Sat( ψ, a))) a v ϕ (Sat( v ϕ, a) b ( b is v-variant of a (Sat( ϕ, b))

29 satisfaction As an interlude I list some optional axioms: Determinateness axioms: a v w (D( Rvw, a)) and similarly for predicate symbols other than Sat and D a ϕ (D( ϕ, a) D( ϕ, a)) a ϕ ψ (D( ϕ ψ, a) (D( ϕ, a) D( ψ, a))) a v ϕ (D( v ϕ, a) b ( b is v-variant of a (D( ϕ, b)) Sat-iteration axiom: a ϕ v (D( ϕ, a(v)) (Sat( Sat( ϕ, v), a) Sat( ϕ, a(v))))

30 the substitutional definition of logical validity A sentence is logically valid iff all its substitution instances are true. substitutional definition of logical truth ϕ (Val( ϕ ) I a Sat(I ϕ, a))) Here I ranges over substitution functions.

31 the substitutional definition of logical validity Similarly, an argument is logically valid iff there is no substitution function and no variable assignment that make the premisses true and the conclusion false. substitutional definition of logical validity Γ S ϕ iff I a ( γ Γ Sat(I(γ), a) Sat(I(ϕ), a))

32 properties of the substitutional definition The identity function with I(ϕ) = ϕ on sentences is a substitution function. Thus we have: Logical validity (logical truth) trivially implies truth, i.e., x (Val(x) a Sat(x, a)). Similarly, logical consequence preserves truth. On the substitutional account, the intended interpretation is completely trivial. There is no need for mysterious intended models (whose existence is refutable), indefinite extensibility, or the impossibility of unrestricted quantification.

33 properties of the substitutional definition I turn to Tarski s worry that on the substitutional account many set-theoretic models do not correspond to any substitution instance. 10

34 properties of the substitutional definition Let D, E, S be a model with domain D, extension E for and S for Sat. Lemma ST ϕ a D ω ( D, E, S ϕ[a] Sat(I 1 (ϕ), a I 1 M )) I 1 (ϕ) = x, y v 2 if ϕ is x y, x = y if ϕ is x = y, x, y v 3 if ϕ is Sat(x, y), I 1 (ψ) if ϕ is ψ, I 1 (ψ) I 1 (χ) x (x v 1 I 1 (ψ)) a I 1 M (v 1) = D a I 1 M (v 2) = E a I 1 M (v 3) = S a I 1 M (v n+3) = a(v n ) if ϕ is ψ χ, and if ϕ is x ψ

35 properties of the substitutional definition A substitutional interpretation is a pair I, a of a substitution function and a variable assignment. Every set-theoretic model corresponds to the substitutional interpretation I 1, a I 1 M. But not every substitutional interpretation corresponds to a set-theoretic model (for instance, if there is no relativizing formula δ(x) or δ(x) doesn t define a set).

36 properties of the substitutional definition ϕ follows informally from Γ intuitive soundness every set-theoretic countermodel is an intuitive counterexample Γ PC ϕ Gödel completeness Γ ϕ

37 properties of the substitutional definition Γ S ϕ formal soundness proof Lemma above Γ PC ϕ Gödel completeness Γ ϕ

38 properties of the substitutional definition formal soundness proof Γ S ϕ Γ PC ϕ Gödel completeness Γ ϕ The squeezing argument is now a reduction of a semantic notion defined from an irreducible semantic satisfaction predicate to a purely mathematical notion.

39 properties of the substitutional definition The substitutional notion of validity isn t very sensitive to the choice of the signature, if free variables are admitted in the substitution instances and the base theory is sufficiently strong. The substitutional notion of validity isn t very sensitive to the choice of Sat-axioms.

40 properties of the substitutional definition The substitutional notion of validity isn t very sensitive to the choice of the signature, if free variables are admitted in the substitution instances and the base theory is sufficiently strong. The substitutional notion of validity isn t very sensitive to the choice of Sat-axioms.

41 worries 1. inexpressible interpretations: 1.1 On the substitutional account we don t quantify over all class-sized models (or over all pluralities). 1.2 If the language is expanded, we quantify over more substitution functions. 2. A primitive notion of validity could be used instead of the primitive notion Sat (cf. Field 2015). 3. It s difficult to apply the substitutional account to arbitrary formal languages. Tarski (1936b) was able to apply his account to weak languages. 4. On the substitutional account logical validity is tied to a primitive and problematic notion of satisfaction, which isn t needed on the model-theoretic account.

42 worries 1. inexpressible interpretations: 1.1 On the substitutional account we don t quantify over all class-sized models (or over all pluralities). 1.2 If the language is expanded, we quantify over more substitution functions. 2. A primitive notion of validity could be used instead of the primitive notion Sat (cf. Field 2015). 3. It s difficult to apply the substitutional account to arbitrary formal languages. Tarski (1936b) was able to apply his account to weak languages. 4. On the substitutional account logical validity is tied to a primitive and problematic notion of satisfaction, which isn t needed on the model-theoretic account.

43 worries 1. inexpressible interpretations: 1.1 On the substitutional account we don t quantify over all class-sized models (or over all pluralities). 1.2 If the language is expanded, we quantify over more substitution functions. 2. A primitive notion of validity could be used instead of the primitive notion Sat (cf. Field 2015). 3. It s difficult to apply the substitutional account to arbitrary formal languages. Tarski (1936b) was able to apply his account to weak languages. 4. On the substitutional account logical validity is tied to a primitive and problematic notion of satisfaction, which isn t needed on the model-theoretic account.

44 worries 1. inexpressible interpretations: 1.1 On the substitutional account we don t quantify over all class-sized models (or over all pluralities). 1.2 If the language is expanded, we quantify over more substitution functions. 2. A primitive notion of validity could be used instead of the primitive notion Sat (cf. Field 2015). 3. It s difficult to apply the substitutional account to arbitrary formal languages. Tarski (1936b) was able to apply his account to weak languages. 4. On the substitutional account logical validity is tied to a primitive and problematic notion of satisfaction, which isn t needed on the model-theoretic account.

45 summary Main Thesis: (Informal) logical validity is substitutional validity. The intended interpretation and other class sized models are among the interpretations we quantify over in the definition of validity. Every set-theoretic model corresponds to the substitutional interpretation I 1, a I 1 M. But not every substitutional interpretation corresponds to a set-theoretic model (for instance, if there is no relativizing formula δ(x) or δ(x) doesn t define a set). The squeezing argument for substitutional validity naturally slots into the place of Kreisel s (1965; 1967a) informal logical validity. There is no need to postulate an elusive informal notion of logical validity.

46 summary Main Thesis: (Informal) logical validity is substitutional validity. The intended interpretation and other class sized models are among the interpretations we quantify over in the definition of validity. Every set-theoretic model corresponds to the substitutional interpretation I 1, a I 1 M. But not every substitutional interpretation corresponds to a set-theoretic model (for instance, if there is no relativizing formula δ(x) or δ(x) doesn t define a set). The squeezing argument for substitutional validity naturally slots into the place of Kreisel s (1965; 1967a) informal logical validity. There is no need to postulate an elusive informal notion of logical validity.

47 summary Main Thesis: (Informal) logical validity is substitutional validity. The intended interpretation and other class sized models are among the interpretations we quantify over in the definition of validity. Every set-theoretic model corresponds to the substitutional interpretation I 1, a I 1 M. But not every substitutional interpretation corresponds to a set-theoretic model (for instance, if there is no relativizing formula δ(x) or δ(x) doesn t define a set). The squeezing argument for substitutional validity naturally slots into the place of Kreisel s (1965; 1967a) informal logical validity. There is no need to postulate an elusive informal notion of logical validity.

48 summary Main Thesis: (Informal) logical validity is substitutional validity. The intended interpretation and other class sized models are among the interpretations we quantify over in the definition of validity. Every set-theoretic model corresponds to the substitutional interpretation I 1, a I 1 M. But not every substitutional interpretation corresponds to a set-theoretic model (for instance, if there is no relativizing formula δ(x) or δ(x) doesn t define a set). The squeezing argument for substitutional validity naturally slots into the place of Kreisel s (1965; 1967a) informal logical validity. There is no need to postulate an elusive informal notion of logical validity.

49 extensions free logic (negative, positive or neutral) nonclassical logics: e.g., K3; use PKF constant domain semantics; see Williamson (2000) identity as logical constant second-order logic

50 extensions free logic (negative, positive or neutral) nonclassical logics: e.g., K3; use PKF constant domain semantics; see Williamson (2000) identity as logical constant second-order logic

51 extensions free logic (negative, positive or neutral) nonclassical logics: e.g., K3; use PKF constant domain semantics; see Williamson (2000) identity as logical constant second-order logic

52 extensions free logic (negative, positive or neutral) nonclassical logics: e.g., K3; use PKF constant domain semantics; see Williamson (2000) identity as logical constant second-order logic

53 extensions free logic (negative, positive or neutral) nonclassical logics: e.g., K3; use PKF constant domain semantics; see Williamson (2000) identity as logical constant second-order logic

54 bibliography Hartry Field. What is logical validity? In Colin R. Caret and Ole T. Hjortland, editors, Foundations of Logical Consequence, pages Oxford University Press, Oxford, Hubert Hubien. Iohannis Buridani Tractatus de Consequentiis, volume XVI of Philosophes Médievaux. Publications Universitaires, Louvain, Georg Kreisel. Mathematical logic. In T. L. Saaty, editor, Lectures on Modern Mathematics III, pages Wiley, New York, Georg Kreisel. Informal rigour and completeness proofs. In Imre Lakatos, editor, The Philosophy of Mathematics, pages North Holland, Amsterdam, 1967a. Georg Kreisel. Mathematical logic: what has it done for the philosophy of mathematics? In Ralph Schoenman, editor, Bertrand Russell: Philosopher of the Century, pages George Allen & Unwin, London, 1967b. Willard V. Quine. Philosophy of Logic. Harvard University Press, second edition, Stephen Read. Formal and material consequence. Journal of Philosophical Logic, 23: , Alfred Tarski. Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica Commentarii Societatis Philosophicae Polonorum, 1, 1936a. Alfred Tarski. Über den Begriff der logischen Folgerung. In Actes du Congrès International de Philosophie Scientifique, volume 7, Paris, 1936b. Alfred Tarski and Robert Vaught. Arithmetical extensions of relational systems. Compositio Mathematica, 13:81 102, Timothy Williamson. Existence and contingency. Proceedings of the Aristotelian Society, 100: , Timothy Williamson. Modal Logic as Metaphysics. Oxford University Press, 2013.