What is a genuine intuitionistic notion of falsity?

I highlight the importance of the notion of falsity for a semantical consideration of intuitionistic logic. One can find two principal (and non-equivalent) versions of such a notion in the literature, namely, falsity as non-truth and falsity as truth of a negative proposition. I argue in favor of the first version as the genuine intuitionistic notion of falsity

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

Weak Arithmetical Interpretations for the Logic of Proofs

Artemov established an arithmetical interpretation for the Logics of Proofs LPCS, which yields a classical provability semantics for the modal logic S4. These Logics of Proofs are parameterized by so-called constant specifications CS that state which axioms can be used in the reasoning process, and the arithmetical interpretation relies on the constant specifications being finite. In this paper, we remove this restriction by introducing weak arithmetical interpretations that are sound and complete for a wide class of constant specifications, including infinite ones. In particular, they interpret the full Logic of Proofs LP. Keywords: Logic of Proofs, arithmetical interpretation, BHK semantic

Is game semantics necessary?

We discuss the extent to which game semantics is implicit in the formalism of linear logic and in the intuitions underlying linear logic

A Pragmatic Interpretation of Quantum Logic

Scholars have wondered for a long time whether the language of quantum mechanics introduces a quantum notion of truth which is formalized by quantum logic (QL) and is incompatible with the classical (Tarskian) notion. We show that QL can be interpreted as a pragmatic language of assertive formulas which formalize statements about physical systems that are empirically justified or unjustified in the framework of quantum mechanics. According to this interpretation, QL formalizes properties of the metalinguistic notion of empirical justification within quantum mechanics rather than properties of a quantum notion of truth. This conclusion agrees with a general integrationist perspective that interprets nonstandard logics as theories of metalinguistic notions different from truth, thus avoiding incompatibility with classical notions and preserving the globality of logic. By the way, some elucidations of the standard notion of quantum truth are also obtained. Key words: pragmatics, quantum logic, quantum mechanics, justifiability, global pluralism.Comment: Third version: 20 pages. Sects. 1, 2, and 4 rewritten and improved. Explanations adde

On formal aspects of the epistemic approach to paraconsistency

This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed