Axiomatic extensions of the constructive logic with strong negation and the disjunction property

We study axiomatic extensions of the propositional constructive logic with strong negation having the disjunction property in terms of corresponding to them varieties of Nelson algebras. Any such varietyV is characterized by the property: (PQWC) ifA,B εV, thenA×B is a homomorphic image of some well-connected algebra ofV. We prove:

• each varietyV of Nelson algebras with PQWC lies in the fibre σ^?1(W) for some varietyW of Heyting algebras having PQWC,
• for any varietyW of Heyting algebras with PQWC the least and the greatest varieties in σ^?1(W) have PQWC,
• there exist varietiesW of Heyting algebras having PQWC such that σ^?1(W) contains infinitely many varieties (of Nelson algebras) with PQWC.

     

Subformula semantics for strong negation systems

We present a semantics for strong negation systems on the basis of the subformula property of the sequent calculus. The new models, called subformula models, are constructed as a special class of canonical Kripke models for providing the way from the cut-elimination theorem to model-theoretic results. This semantics is more intuitive than the standard Kripke semantics for strong negation systems.     

The Craig interpolation theorem for prepositional logics with strong negation

This paper deals with, prepositional calculi with strong negation (N-logics) in which the Craig interpolation theorem holds. N-logics are defined to be axiomatic strengthenings of the intuitionistic calculus enriched with a unary connective called strong negation. There exists continuum of N-logics, but the Craig interpolation theorem holds only in 14 of them.     

A weak intuitionistic propositional logic with purely constructive implication

We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.This work was partially supported by NSF Grant DCR85-13417     

Curry's paradox in contractionless constructive logic

We propose contractionless constructive logic which is obtained from Nelson's constructive logic by deleting contractions. We discuss the consistency of a naive set theory based on the proposed logic in relation to Curry's paradox. The philosophical significance of contractionless constructive logic is also argued in comparison with Fitch's and Prawitz's systems.     

Notes on conditional logic

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems. These notes were originally drafted for a course given at the University of Auckland in 1987 (Philosophy 29.308). The work reported in Section 5 is due to Michael Strevens, who took the course and presented his conception of a filtration in a term paper [5]. The author wishes to thank Strevens for many stimulating discussions and for permission to include his results here     

A constructive proof of Craig's interpolation lemma for m-valued logic

The algebraic proof of Craig's interpolation lemma for m-valued logic was given by Rasiowa in [1]. We present here a constructive proof of this lemma, based on a Gentzen type formalization.     

Impact of negation salience and cognitive resources on negation during attitude formation

Because of the increased cognitive resources required to process negations, past research has shown that explicit attitude measures are more sensitive to negations than implicit attitude measures. The current work demonstrated that the differential impact of negations on implicit and explicit attitude measures was moderated by (a) the extent to which the negation was made salient and (b) the amount of cognitive resources available during attitude formation. When negations were less visually salient, explicit but not implicit attitude measures reflected the intended valence of the negations. When negations were more visually salient, both explicit and implicit attitude measures reflected the intended valence of the negations, but only when perceivers had ample cognitive resources during encoding. Competing models of negation processing, schema-plus-tag and fusion, were examined to determine how negation salience impacts the processing of negations.     

A note on the decidability of a strong relevant logic

A modified filtrations argument is used to prove that the relevant logic S of [2] is decidable.     

Anaphora and negation

Philosophical Studies - One of the central questions of discourse dynamics is when an anaphoric pronoun is licensed. This paper addresses this question as it pertains to the complex data involving...     

Changes in activation levels with negation

Three experiments investigated the effects of negation during on-line language processing. It was hypothesized that negation of a noun (e.g., no bread) would affect the activation level of the mental representation of that noun. Experiment 1 manipulated the location of the negation in sentences that were followed by a probe recognition task. Subjects were slower to indicate that a probe had been in the sentence when the probe corresponded to a negated noun. Experiment 2 replicated these results with a probe naming task. Experiment 3 replicated the result that reading the phrase no bread inhibits responses to bread in the probe task but found no evidence of inhibition of the response to an associate probe, such as butter. The results of these three studies suggest that negation affects the discourse focus of a noun phrase, and hence the activation level of its representation.     

Boolean negation and all that

Conclusion We have seen that proofs of soundness of (Boolean) DS, EFQ and of ABS — and hence the legitimation of these inferences — can be achieved only be appealing to the very form of reasoning in question. But this by no means implies that we have to fall back on classical reasoning willy-nilly. Many logical theories can provide the relevant boot-strapping. Decision between them has, therefore, to be made on other grounds. The grounds include the many criteria familiar from the philosophy of science: theoretical integrity (e.g., paucity of ad hoc hypotheses), adequacy to the data (explaining the data of inference —all inferences, not just those chosen from consistent domains!) and so on. This paper has not attempted to address these issues in general. All it demonstrates is that the charge that a dialetheist solution to the semantic paradoxes can be maintained only by making some intelligible notion ineffable cannot be made to stick. The dialetheist has a coherent position, endorsing the T-scheme, but rejecting DS, EFQ (even Boolean DS and EFQ) and ABS. And any argument to the effect that the relevant notions are both ineffable and intelligible begs the question. The case against consistent solutions to the semantic paradoxes therefore remains intact.     

Moral expressivism and sentential negation

This paper advances three necessary conditions on a successful account of sentential negation. First, the ability to explain the constancy of sentential meaning across negated and unnegated contexts (the Fregean Condition). Second, the ability to explain why sentences and their negations are inconsistent, and inconsistent in virtue of the meaning of negation (the Semantic Condition). Third, the ability of the account to generalize regardless of the topic of the negated sentence (the Generality Condition). The paper discusses three accounts of negation available to moral expressivists. The first—the dominant commitment account—fails to meet the Fregean Condition. The two remaining accounts—commitment semantics and the expression account—satisfy all three conditions. A recent argument that the dominant commitment account is the only option available to expressivists is considered and rejected.     

Rumfitt on truth-grounds,negation, and vagueness

In The Boundary Stones of Thought (2015), Rumfitt defends classical logic against challenges from intuitionistic mathematics and vagueness, using a semantics of pre-topologies on possibilities, and a topological semantics on predicates, respectively. These semantics are suggestive but the characterizations of negation face difficulties that may undermine their usefulness in Rumfitt’s project.