4 research outputs found
A fundamental non-classical logic
We give a proof-theoretic as well as a semantic characterization of a logic
in the signature with conjunction, disjunction, negation, and the universal and
existential quantifiers that we suggest has a certain fundamental status. We
present a Fitch-style natural deduction system for the logic that contains only
the introduction and elimination rules for the logical constants. From this
starting point, if one adds the rule that Fitch called Reiteration, one obtains
a proof system for intuitionistic logic in the given signature; if instead of
adding Reiteration, one adds the rule of Reductio ad Absurdum, one obtains a
proof system for orthologic; by adding both Reiteration and Reductio, one
obtains a proof system for classical logic. Arguably neither Reiteration nor
Reductio is as intimately related to the meaning of the connectives as the
introduction and elimination rules are, so the base logic we identify serves as
a more fundamental starting point and common ground between proponents of
intuitionistic logic, orthologic, and classical logic. The algebraic semantics
for the logic we motivate proof-theoretically is based on bounded lattices
equipped with what has been called a weak pseudocomplementation. We show that
such lattice expansions are representable using a set together with a reflexive
binary relation satisfying a simple first-order condition, which yields an
elegant relational semantics for the logic. This builds on our previous study
of representations of lattices with negations, which we extend and specialize
for several types of negation in addition to weak pseudocomplementation; in an
appendix, we further extend this representation to lattices with implications.
Finally, we discuss adding to our logic a conditional obeying only introduction
and elimination rules, interpreted as a modality using a family of
accessibility relations.Comment: added topological representation of bounded lattices with
implications in Appendi
Rewriting for Fitch style natural deductions
Contains fulltext :
104040.pdf (author's version ) (Open Access)15th Internat. Conference on Rewriting Techniques and Applications, RTA 2004, Aache
Rewriting for Fitch style natural deductions
Logical systems in natural deduction style are usually presented in the Gentzen style. A different definition of natural deduction, that corresponds more closely to proofs in ordinary mathematical practice, is given in [Fitch 1952]. We define precisely a Curry-Howard interpretation that maps Fitch style deductions to simply typed terms, and we analyze why it is not an isomorphism. We then describe three reduction relations on Fitch style natural deductions: one that removes garbage (subproofs that are not needed for the conclusion), one that removes repeats and one that unshares shared subproofs. We also define an equivalence relation that allows to interchange independent steps. We prove that two Fitch deductions are mapped to the same ¿-term if and only if they are equal via the congruence closure of the aforementioned relations (the reduction relations plus the equivalence relation). This gives a Curry-Howard isomorphism between equivalence classes of Fitch deductions and simply typed ¿-terms. Then we define the notion of cut-elimination on Fitch deductions, which is only possible for deductions that are completely unshared (normal forms of the unsharing reduction). For conciseness, we restrict in this paper to the implicational fragment of propositional logic, but we believe that our results extend to full first order predicate logic
Rewriting for Fitch style natural deductions
Abstract. Logical systems in natural deduction style are usually presented in the Gentzen style. A different definition of natural deduction, that corresponds more closely to proofs in ordinary mathematical practice, is given in [Fitch 1952]. We define precisely a Curry-Howard interpretation that maps Fitch style deductions to simply typed terms, and we analyze why it is not an isomorphism. We then describe three reduction relations on Fitch style natural deductions: one that removes garbage (subproofs that are not needed for the conclusion), one that removes repeats and one that unshares shared subproofs. We also define an equivalence relation that allows to interchange independent steps. We prove that two Fitch deductions are mapped to the same λ-term if and only if they are equal via the congruence closure of the aforementioned relations (the reduction relations plus the equivalence relation). This gives a Curry-Howard isomorphism between equivalence classes of Fitch deductions and simply typed λ-terms. Then we define the notion of cut-elimination on Fitch deductions, which is only possible for deductions that are completely unshared (normal forms of the unsharing reduction). For conciseness, we restrict in this paper to the implicational fragment of propositional logic, but we believe that our results extend to full first order predicate logic.