k-Provability in PA

We study the decidability of k-provability in PA —the relation ‘being provable in PA with at most k steps’—and the decidability of the proof-skeleton problem—the problem of deciding if a given formula has a proof that has a given skeleton (the list of axioms and rules that were used). The decidability of k-provability for the usual Hilbert-style formalisation of PA is still an open problem, but it is known that the proof-skeleton problem is undecidable for that theory. Using new methods, we present a characterisation of some numbers k for which k-provability is decidable, and we present a characterisation of some proof-skeletons for which one can decide whether a formula has a proof whose skeleton is the considered one. These characterisations are natural and parameterised by unification algorithms.publishersversionpublishe

Variations on a Montagovian theme

What are the objects of knowledge, belief, probability, apriority or analyticity? For at least some of these properties, it seems plausible that the objects are sentences, or sentence-like entities. However, results from mathematical logic indicate that sentential properties are subject to severe formal limitations. After surveying these results, I argue that they are more problematic than often assumed, that they can be avoided by taking the objects of the relevant property to be coarse-grained (“sets of worlds”) propositions, and that all this has little to do with the choice between operators and predicates

A New Arithmetically Incomplete First- Order Extension of Gl All Theorems of Which Have Cut Free Proofs

Reference [12] introduced a novel formula to formula translation tool (“formulators”) that enables syntactic metatheoretical investigations of first-order modal logics, bypassing a need to convert them first into Gentzen style logics in order to rely on cut elimination and the subformula property. In fact, the formulator tool, as was already demonstrated in loc. cit., is applicable even to the metatheoretical study of logics such as QGL, where cut elimination is (provably, [2]) unavailable. This paper applies the formulator approach to show the independence of the axiom schema _A ! _8xA of the logics M3 and ML3 of [17, 18, 11, 13]. This leads to the conclusion that the two logics obtained by removing this axiom are incomplete, both with respect to their natural Kripke structures and to arithmetical interpretations. In particular, the so modified ML3 is, similarly to QGL, an arithmetically incomplete first-order extension of GL, but, unlike QGL, all its theorems have cut free proofs. We also establish here, via formulators, a stronger version of the disjunction property for GL and QGL without going through Gentzen versions of these logics (compare with the more complex proofs in [2, 8]).This research was partially supported by NSERC grant No. 8250

Three Essays in Intuitionistic Epistemology

We present three papers studying knowledge and its logic from an intuitionistic viewpoint. An Arithmetic Interpretation of Intuitionistic Verification Intuitionistic epistemic logic introduces an epistemic operator to intuitionistic logic which reflects the intended BHK semantics of intuitionism. The fundamental assumption concerning intuitionistic knowledge and belief is that it is the product of verification. The BHK interpretation of intuitionistic logic has a precise formulation in the Logic of Proofs and its arithmetical semantics. We show here that this interpretation can be extended to the notion of verification upon which intuitionistic knowledge is based. This provides the systems of intuitionistic epistemic logic extended by an epistemic operator based on verification with an arithmetical semantics too. This confirms the conception of verification incorporated in these systems reflects the BHK interpretation. Intuitionistic Verification and Modal Logics of Verification The systems of intuitionistic epistemic logic, IEL, can be regarded as logics of intuitionistic verification. The intuitionistic language, however, has expressive limitations. The classical modal language is more expressive, enabling us to formulate various classical principles which make explicit the relationship between intuitionistic verification and intuitionistic truth, implicit in the intuitionistic epistemic language. Within the framework of the arithmetic semantics for IEL we argue that attempting to base a general verificationism on the properties of intuitionistic verification, as characterised by IEL, yields a view of verification stronger than is warranted by its BHK reading. Intuitionistic Knowledge and Fallibilism Fallibilism is the view that knowledge need not guarantee the truth of the proposition known. In the context of a classical conception of truth fallibilism is incompatible with the truth condition on knowledge, i.e. that false propositions cannot be known. We argue that an intuitionistic approach to knowledge yields a view of knowledge which is both fallibilistic and preserves the truth condition. We consider some problems for the classical approach to fallibilism and argue that an intuitionistic approach also resolves them in a manner consonant with the motivation for fallibilism