Kripke Semantics A Semantic for Intuitionistic Logic By Jonathan Prieto-Cubides Updated: 2016/11/18 Supervisor: Andr´esSicard-Ram´ırez Logic and Computation Research Group EAFIT University Medell´ın,Colombia History Saul Kripke He was born on November # 13, 1940 (age 75) Philosopher and Logician # Emeritus Professor at # Princeton University In logic, his major # contributions are in the field of Modal Logic Saul Kripke In Modal Logic, we # attributed to him the notion of Possible Worlds Its notable ideas # ◦ Kripke structures ◦ Rigid designators ◦ Kripke semantics Kripke Semantics The study of semantic is the study of the truth Kripke semantics is one of many (see for instance # (Moschovakis, 2015)) semantics for intuitionistic logic It tries to capture different possible evolutions of the world # over time The abstraction of a world we call a Kripke structure # Proof rules of intuitionistic logic are sound with respect to # krikpe structures Intuitionistic Logic Derivation (proof) rules of the ^ connective Γ ` ' ^ ^-elim Γ ` ' 1 Γ ` ' Γ ` ^-intro Γ ` ' ^ Γ ` ' ^ ^-elim Γ ` 2 Intuitionistic Logic Derivation (proof) rules of the _ connective Γ ` ' _-intro1 Γ ` ' _ Γ ` ' _ Γ;' ` σ Γ; ` σ _-elim Γ ` σ Γ ` _-intro Γ ` ' _ 2 Intuitionistic Logic Derivation (proof) rules of the ! connective Γ;' ` Γ ` ' Γ ` ' ! !-intro !-elim Γ ` ' ! Γ ` Intuitionistic Logic Derivation (proof) rules of the : connective where :' ≡ ' !? Γ;' ` ? Γ ` ? :-intro explosion Γ ` :' Γ ` ' Intuitionistic Logic Other derivation (proof) rules Γ ` ' unit assume weaken Γ ` > Γ;' ` ' Γ; ` ' Classical Logic The list of derivation rules are the same above plus the following rule Γ; :' ` ? RAA Γ ` ' Kripke Model Def. A Kripke model is a quadruple K = hK; Σ; C; Di where K is a (non-empty) partially ordered set # C is a function defined on the constants of L # D is a set-valued function on K # Σ is a function on K such that # ◦ C(c) 2 D(l) for all k 2 K ◦ D(k) 6= ; for all k 2 K ◦ Σ(k) ⊂ Atk for all k 2 K where Atk is the set of all atomic sentences of L with constants for the elements of D(k). (See the full description (van Dalen, 2004) or review a short description on (Moschovakis, 2015)) Kripke Model (Cont.) D and Σ satisfy the following conditions: (i) k ≤ l ) D(k) ⊆ D(l) (ii) ? 62 Σ(k), for all k (iii) k ≤ l ) Σ(k) ⊆ Σ(l) k3 ' ' k 1 k2 ' ' ' o Figure: A Kripke model example Lemma 1 Lemma 1 Σ(k) ⊆ Sentk , Σ(k) satisfies : (i) ' _ 2 Σ(k) , ' 2 Σ(k) or 2 Σ(k) (ii) ' ^ 2 Σ(k) , ' 2 Σ(k) and 2 Σ(k) (iii) ' ! 2 Σ(k) , for all l ≥ k (' 2 Σ(l) ) 2 Σ(l)) (iv) 9x'(x) 2 Σ(k) , there is an a 2 D(k) such that '(a) 2 Σ(k) (v) 8x'(x) 2 Σ(k) , for all l ≥ k and a 2 D(l) '(a) 2 Σ(l) Proof. Immediate. (Also see (van Dalen, 2004, p. 165)) 1Set of all sentences with parameters in D(k). Notation k ' Notation We write k ' for ' 2 Σ(k) to say \k forces '" Using this notation, we can reformulate on terms of . For instance look at the last two items (iv) k 9x'(x) , there is an a 2 D(k) such that k '(a) (v) k 8x'(x) , for all l ≥ k and a 2 D(l) l '(a) Corollary Corollary (i) k :' , for all l ≥ k; l 1 ' (ii) k ::' , for all l ≥ k, there exists a p ≥ l such that (p ') Proof. (i) k :' , k ' !?, for all l ≥ k (l ' ) l ?)), for all l ≥ k (l 1 ') (ii) White-board Monotonicity Lemma Lemma k ≤ l; k ' ) l ' Proof. If ' is an atom, we are done by definition above # If ' is '1 _ '2 # Rest. White-board # If ' is 8x'1(x), then let k 8x'1(x) and l ≥ k Suppose # p ≥ l and a 2 D(p), then, since p ≥ k, p '1(a) Therefore, l 8x'1(x) Example 1 In the bottom node k0 no atoms are known, in the node k1 only ' is known k1 ' k0 1 ' _:' k0 Figure: diagram example 1 Example 2 k1 ' k2 k0 k0 1 :(' ^ ) !:' _: Figure: diagram example 2 Example 3 k3 '; k1 ' k2 k0 1 ( ! ') ! (: _ ') k0 Figure: diagram example 3 Example 4 In the bottom node the following implications are forced: '2 ! '1, '3 ! '2, '3 ! '1 k1 '1;'2 k0 1('1 $ φ2) _ ('2 $ φ3 k0 '1 _ ('1 $ φ3) Figure: diagram example 4 Example 5 Note that in this example, D(k0) = f0g and D(k1) = f0; 1g k1 0 1 '; (0); σ(1) k0 1 (' ! 9xσ(x)) ! 9x(' ! σ(x)) k0 0 Figure: diagram example 5 Example 6 k 3 0 1 2 3 '(0);'(1);'(2) k '(0);'(1) 2 0 1 2 0 1 k1 '(0) k0 1 8x::'(x) ! ::8x'(x) k0 0 Figure: diagram example 6 Theorem (Soundness Theorem) Γ ` ' ) Γ ' Proof (See (van Dalen, 2004)) Use induction on the derivation D of ' from Γ. We will abbreviate \k (~a) for all 2 Γ” by \k Γ(~a)". The model K is fixed in the proof (1) D consists of just ', then obviously k Γ(~a) ) k '(~a) for all k and (~a) 2 D(k) (2) D ends with application of a derivation rule (^I) Induction hypothesis: 8k8~a 2 D(k)(k Γ(~a) ) k 'i (~a)), for i = 1; 2. Now choose a k 2 K and ~a 2 D(k) such that k Γ(~a), then k '1(~a) and k '2(~a), so k ('1 ^ '2)(~a) (^E) Immediate (_I) Immediate. (_E) Induction hypothesis: 8k(k Γ ) k ' _ ), 8k(k Γ ' ) k σ), 8k(k Γ ) k σ). Now let k Γ, then by the ind.hyp. k φ _ , so k ' or k ). In the first case k Γ;', so k σ. In the second case k Γ; , so k σ. In both cases k σ, so we are done (!I) Induction hypothesis: (8k)(8~a 2 D(k))(k Γ(~a);'(~a) ) k (~a)). Now let k Γ(~a) for some ~a 2 D(k). We want to show k (' ! )(~a), so let l ≥ k and l '(~a). By monotonicity l Γ(~a), and ~a 2 D(l), so ind. hyp. tell us that l (~a). Hence 8l ≥ k(l '(~a) ) l (~a)), so k (' ! )(~a) (!E) Immediate (?) Induction hypothesis: 8k(k Γ ) k ?). Since, evidently, no k can force Γ, 8k(k Γ ) k ') is correct (8I) The free variables in Γ are ~x and z does not occur in the sequence ~x. Induction hypothesis: (8k)(8~a; b 2 D(k))(k Γ(~a ) k '(~a; b)). Now let k Γ(~a) for some ~a 2 D(k), we must show k 8z'(~a; z). So let l ≥ k and b 2 D(l). By monotonicity l Γ(~a) and ~a 2 D(l), so by the ind. hyp. l '(~a; b). This shows (8l ≥ k)(8b 2 D(l))(l '(~a; b)), and hence k 8z'(~a; z) (8E) Immediate (9I) Immediate (9E) Induction hypothesis: (8k)(8~a 2 D(k))(k Γ(~a) ) k 9z'(~a; z)) and (8k)(8~a; b 2 D(k))(k '(~a; b); k Γ(~a) ) k σ(~a)). Here the variables in Γ and σ are ~x, and z does not occur in the sequence ~x. Now let k Γ(~a), for some ~a 2 D(k), then k 9z'(~a; z). So let k '(~a; b) for some b 2 D(k). By the induction hypothesis k σ(~a) Theorem (Completeness Theorem) Γ ` ' , Γ ' (Γ and ' closed) Proof. (See (van Dalen, 2004) for the lemma mention below) We have already shown ). For the converse we assume Γ 0 ' and apply Lemma 6.3.9, which yields a contradiction. References [1] van Dalen, Dirk. Logic and Structure. 4th ed. Springer, 2004. [2] Moschovakis, Joan. Intuitionistic Logic, The Stanford Encyclopedia of Philosophy, Edward N. Zalta, 2015..