S. Artemov. Logic of Proofs. Annals of Pure and Applied Logic, v. 67, pp. 29--59, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that a proof derives only the end formula (sequent) of a proof tree. On the other hand, the same systems may be regarded as multi conclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly single conclusion proof systems was studied in [6], 15] and in [60] 61] where it received a complete axiomatization (system FLP) However, FLP is not compatible with any modal logic. For example, FLP derives : x : x : which has the forgetful projection : 2 2( The latter is false in any normal modal logic. Therefore, provability ....

....first order logic of proofs is hyperarithmetical (in fact, Pi 0 1 (TA) complete, where TA is the set of all true arithmetical sentences) In particular, this means that this logic does not admit a complete axiomatization. The logic of proofs based on the standard Godel numbering were studied in [6], 16] 106] The latter paper finds some axiomatizable version of the first order logic of proofs with the standard Godel numbering satisfying special monotonicity restrictions on its Godel numbering. x12. Discussion. 1. There are two provability models each having its own areas of ....

, Logic of proofs, Annals of Pure and Applied Logic, vol. 67 (1994), no. 1, pp. 29--59.

....A 2 T , and 2A 2B 2 T implies 2A 2 T or 2B 2 T ) Note that faithfully interpretable theories in a finite number of propositional variables are necessarily r.e. The theorem was given a more compact proof and at the same time generalized to all r.e. theories extending I Delta 0 EXP by Zambella [1994]. If one applies the theorem to the minimal L propositional theory, an earlier proved strengthening of Solovay s theorem (Artemov [1980] Avron [1984] Boolos [1982] Montagna [1979] Visser [1980] rolls out. 5.2. Corollary. Uniform arithmetic completeness theorem) There exists a sequence of ....

....of PA) The logic of the provability predicate of HA with regard to HA certainly contains additional principles beyond the obvious intuitionistic version of L: the intuitionistic propositional calculus IPC plus 2(2A A) 2A. The situation has been discussed by Visser in several papers (Visser [1985,1994], Visser et al. 1995] It is unknown what the real logic is, for all we know it may even be complete Pi 0 2 . In any case it contains the additional principles: ffl 2: 2A 22A ffl 2( 2A 2A) 22A ffl 2(A B) 2(A 2B) Leivant s Principle 2 ) But this is not an exhaustive list. It ....

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Logic of proofs, Annals of Pure and Applied Logic, 67, pp. 29--59.

....A 2 T , and 2A 2B 2 T implies 2A 2 T or 2B 2 T ) Note that faithfully interpretable theories in a finite number of propositional variables are necessarily r.e. The theorem was given a more compact proof and at the same time generalized to all r.e. theories extending I Delta 0 EXP by Zambella [1994]. If one applies the theorem to the minimal L propositional theory, an earlier proved (Artemov [1980] Avron [1984] Boolos [1982] Montagna [1979] Visser [1980] strengthening of Solovay s theorem rolls out. 5.2. Corollary. Uniform arithmetic completeness theorem) There exists a sequence of ....

....I Sigma k but with the induction for Sigma k formulas formulated as a rule. The corresponding provability logic can be axiomatized over CSM by the Pi 1 essential reflexivity schema 4A 4(2(A S) S) where S is as before. We also know two natural provability logics of type A (Beklemishev [1994]) The first system corresponds to pairs of theories (T; U) such that U is an extension of T by finitely many Pi 1 sentences and proves times iterated consistency of T , such as e.g. the pairs (PA; PA Con(ZF) I Sigma 1 ; I Sigma 1 Con(I Sigma 2 ) etc. This logic can be axiomatized ....

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Logic of proofs, Annals of Pure and Applied Logic, 67, pp. 29--59.

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S. Artemov. Logic of Proofs. Annals of Pure and Applied Logic, v. 67, pp. 29--59, 1994.

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Artemov, S.: Logic of Proofs. Annals of Pure and Applied Logic 67 (1994) 29-59

....the length of m and c m. i.e. very small . 2.8 Example. A list of c 1 ; c 2 ; c n may be described as the ground reference structure R over M = f1; 2; n 1g and C = fc 1 ; c 2 ; c n g as R = f(1; 1 ) n; n ) n 1; g for 1 = A 1 [ 2] 2 ; 2 = A 2 [[3]] 3 ; n = A n [ n 1] Here works as a marker of the end node. The list can be represented by the formula [ 1] 1 ) It does not mean, however that we intend to store the entire list in one cell 1. We will see now how a regular reference structure list looks like: b R = f(i; ....

....The classical propositional axioms together with constants fc 1 ; c 2 ; c 3 ; g adopted as new axioms, A2) p] A A, A3) p] A [ p] B (C D) if C = D (mod b p = A = B) Rule modus ponens. Axiom (A3) is similar to the unification axiom from [1] and the functionality axiom from [3]. 3.5 Example. The following is provable in LR: ffl : p 1 ] A 1 : p n ] A n ) if the system 8 : b p k = A k (k = 1; n) p = q ) b p = b q for all cell variables p; q occurring in [ p 1 ] A 1 : p n ] A n (3) is not unifiable. ffl [ p 1 ] A 1 : p n ....

S. Artemov. Logic of proofs. Annals of Pure and Applied Logic, v.67, No. 1, pp. 29-59, 1994.

....: So, the relation C = D (mod b p = A = B) is decidable. Definition 3.5 Axioms of d PU : A1) The classical propositional calculus. A2) 2 p A A. A3) 2 p A 2 p B (C D) if C = D (mod b p = A = B) Axiom (A3) is similar to the unification axiom from [1] and the functionality axiom from [3]. Lemma 3.6 If A 2 b L and A, then A is valid for all interpretations . Proof. A straightforward induction on the proof of A. Lemma 3.7 The following is provable in d PU : A3 0 ) 2 p 1 A 1 : 2 pn A n ) if a substitution oe with the property b p k oe = A k oe; k = 1; ....

S. Artemov. Logic of proofs. Annals of Pure and Applied Logic, v.67, 1994 (to appear).

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Logic of proofs, Annals of Pure and AppliedLogic, 67, pp. 29--59.

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, Logic of proofs, Annals of Pure and Applied Logic, vol. 67 (1994), no. 1, pp. 29--59.

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