A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Co-constructive logics for proofs and refutations

This paper considers logics which are formally dual to intuition- istic logic in order to investigate a co-constructive logic for proofs and refu- tations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely- held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for state- ments for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems

Co-constructive logic for proofs and refutations

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems

The intended interpretation of the intuitionistic first-order logical operators.

The present thesis is an investigation on an open problem in mathematical logic: the problem of devising an explanation of the meaning of the intuitionistic first-order logical operators, which is both mathematically rigorous and faithful to the interpretation intended by the intuitionistic mathematicians who invented and have been using them. This problem has been outstanding since the early thirties, when it was formulated and addressed for the first time. The thesis includes a historical, expository part, which focuses on the contributions of Kolmogorov, Heyting, Gentzen and Kreisel, and a long and detailed discussion of the various interpretations which have been proposed by these and other authors. Special attention is paid to the decidability of the proof relation and the introduction of Kreisel's extra-clauses, to the various notions of 'canonical proof' and to the attempt to reformulate the semantic definition in terms of proofs from premises. In this thesis I include a conclusive argument to the effect that if one wants to withdraw the extra-clauses then one cannot maintain the concept of 'proof as the basic concept of the definition; instead, I describe an alternative interpretation based on the concept of a construction 'performing' the operations indicated by a given sentence, and I show that it is not equivalent to the verificationist interpretation. I point out a redundancy in the internal-pseudo-inductive-structure of Kreisel's interpretation and I propose a way to resolve it. Finally, I develop the interpretation in terms of proofs from premises and show that a precise formulation of it must also make use of non-inductive clauses, not for the definition of the conditional but -surprisingly enough- for the definitions of disjunction and of the existential quantifier

A Theory of Factfinding: The Logic for Processing Evidence

Academics have never agreed on a theory of proof. The darkest corner of anyone’s theory has concerned how legal decisionmakers logically should find facts. This Article pries open that cognitive black box. It does so by employing multivalent logic, which enables it to overcome the traditional probability problems that impeded all prior attempts. The result is the first-ever exposure of the proper logic for finding a fact or a case’s facts. The focus will be on the evidential processing phase, rather than the application of the standard of proof as tracked in my prior work. Processing evidence involves (1) reasoning inferentially from a piece of evidence to a degree of belief and of disbelief in the element to be proved, (2) aggregating pieces of evidence that all bear to some degree on one element in order to form a composite degree of belief and of disbelief in the element, and (3) considering the series of elemental beliefs and disbeliefs to reach a decision. Zeroing in, the factfinder in step #1 should connect each item of evidence to an element to be proved by constructing a chain of inferences, employing multivalent logic’s usual rules for conjunction and disjunction to form a belief function that reflects the belief and the disbelief in the element and also the uncommitted belief reflecting uncertainty. The factfinder in step #2 should aggregate, by weighted arithmetic averaging, the belief functions resulting from all the items of evidence that bear on any one element, creating a composite belief function for the element. The factfinder in step #3 does not need to combine elements, but instead should directly move to testing whether the degree of belief from each element’s composite belief function sufficiently exceeds the corresponding degree of disbelief. In sum, the factfinder should construct a chain of inferences to produce a belief function for each item of evidence bearing on an element, and then average them to produce for each element a composite belief function ready for the element-by-element standard of proof. This Article performs the task of mapping normatively how to reason from legal evidence to a decision on facts. More significantly, it constitutes a further demonstration of how embedded the multivalent-belief model is in our law