A semantics for the logic of proofs

J. of Automated Reasoning, 2009

Boolean Models and Methods in Mathematics, Computer Science and Engineering ed. by Crama and Hammer, Chapter 3., 2010

Mathematical Structures in Computer Science, 1997

The purpose of this note is to show that the correctness of a multiplicative proof net with mix is equivalent to its semantic correctness: a proof structure is a proof net if and only if its semantic interpretation is a clique, where one given nite coherence space interprets all propositional variables. This is just an example of what can be done with these kinds of semantic techniques; for more information and further results, the reader is referred to .

2001

Abstract. The Logic of Proofs LP solved long standing Gödel's problem concerning his provability calculus (cf.[4]). It also opened new lines of research in proof theory, modal logic, typed programming languages, knowledge representation, etc. The propositional logic of proofs is decidable and admits a complete axiomatization. In this paper we show that the first order logic of proofs is not recursively axiomatizable.

1997

We provide a $\mathrm{n}\mathrm{a}\mathrm{t}_{\mathrm{U}1}' \mathrm{a}1$ deduction systenls $\lambda_{exr}$ . of classical propositiollal logic and prove proof theoretical and colnputational properties of the systelll. The introits correctness; and a computational use of the logical inconsistency in $\lambda_{exc}^{v}$ , extended with a certain signature. 1010 1997 7-34 this, we introduce the notion of $LJK$ proofs with invariantsl. On the other hand, structural rules in logic are so important and fundamental that they draffiically change logical systems without logical symbols.and the decidability of logical systems depend on them. This notion is obtained by carefully c.onsidering the use of right contraction rules. Careful consideration naturally leads to separation of the succedent into two parts, i.e., a contraction allowed part and a forbidden part. In one of them, we can expect some disjunction property. We discuss that the right contraction rules can be applied to certain subformulae among the given formulae. The subformulae to which the right contraction rules are applied are specified in terms of the notion "invariant" of $LJK$ prooffi. Moreover, the invariant notion plays an $\mathrm{i}_{\mathrm{I}\mathrm{n}_{\mathrm{P}^{\mathrm{o}\mathrm{r}}}}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$ role in embedding classical proofs into intuitionistic prooffi. $\frac{\overline{arrow_{\wedge}4,A\supset B}arrow\supset-4arrow_{\wedge}4}{\frac{}{arrow((A\supset B)\supset A)\supset\wedge 4}\frac{(_{A}4\supset B)\supset\wedge 4arrow A,\wedge 4}{(A\supset B)\supset Aarrow_{\wedge}4}arrow}\supsetarrowarrow\supset c$ $proof\mathit{2}.\cdot$ $\frac{Aarrow A}{(_{\wedge}4\supset B)\supset_{A}4,\wedge 4arrow\wedge 4}u'arrow$ $\underline{\overline{Aarrow((\lrcorner 4\supset B)\supset A)\supset\lrcorner 4}}arrow\supsetarrow\iota\iota$ ' $A4arrow((A\supset B)\supset A)\supset_{\mathrm{s}}4,$ $B$ $\frac{\overlinearrow((A\supset B)\supset\wedge 4)\supset\wedge 4,A\supset B\wedge 4,arrowarrow\supset 4A}{\frac{(A\supset B)\supset\wedge 4arrow((\wedge 4\supset B)\supset\wedge 4)\supset A\wedge 4}{},\frac{arrow((\wedge 4\supset B)\supset\wedge 4)\supset_{4}4,((\wedge 4\supset B)\supset\wedge 4)\supset A4}{arrow((\wedge 4\supset B)\supset\wedge 4)\supset A4}}\supsetarrowarrow carrow\supset$

2011

The propositional logic of proofs LP revealed an explicit provability reading of modal logic S4 which provided an indented provability semantics for the propositional intuitionistic logic IPC and led to a new area, Justification Logic. In this paper, we find the first-order logic of proofs FOLP capable of realizing first-order modal logic S4 and, therefore, the first-order intuitionistic logic HPC. FOLP enjoys a natural provability interpretation; this provides a semantics of explicit proofs for first-order S4 and HPC compliant with Brouwer-Heyting-Kolmogorov requirements. FOLP opens the door to a general theory of first-order justification.

2001

In 1933 Godel introduced a calculus of provability (also known as modal logic S4) and left open the question of its exact intended semantics. In this paper we give a solution to this problem. We find the logic LP of propositions and proofs and show that Godel's provability calculus is nothing but the forgetful projection of LP.

1998

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The Philosophical Quarterly, 2008

Proof-theoretic semantics is an approach to logical semantics based on two ideas, of which the first is that the meaning of a logical connective can be explained by stipulating that some mode of inference, e.g., a natural deduction introduction or elimination rule, is permissible. The second idea is that the soundness of rules which are not stipulated outright may be deduced by some proof-theoretic argument from properties of the rules which are stipulated outright. I examine the first idea. My main conclusion is that the idea is more problematic, and requires more discussion, than has been generally realized. I mention five problems which will have to be overcome before the idea can be accepted as definitely viable.