On the Expressive Power of IF-Logic with Classical Negation

Lecture Notes in Computer Science, 2017

We analyze from a global point of view the expressive resources of IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular IF sentences, this amounts to the study of the fragment of IF logic which is individuated by the game-theoretical property of Action Recall. We prove that the fragment of Action Recall can express all existential second-order (ESO) properties. This can be accomplished already by the prenex fragment of Action Recall, whose only second-order source of expressiveness are the so-called signalling patterns. The proof shows that a complete set of Henkin prefixes is explicitly definable in the fragment of Action Recall. In the more general case, in which also irregular IF sentences are allowed, we show that full ESO expressive power can be achieved using neither Henkin nor signalling patterns.

2006

A sentence of IF-logic is nondetermined in a model if neither player has a winning strategy in the two-player semantic game on that model. It is well-known that there are sentences 1 of IF-logic that are nondetermined in every model with at least two elements. In fact, only the first-order definable sentences of IF-logic are determined in all models. 2 Thus every non first-order IF-sentence is nondetermined in some models. In this paper we take a closer look at some familiar examples of sentences of IF-logic and models in which they are determined. It turns out that if we want to use the familiar examples of IF-sentences to characterize well-known mathematical structures, we observe that the relevant IF-sentences are nondetermined in exactly the standard * This paper is based on a talk in the seminar "Logic and games" of Rohit Parikh in the Graduate Center of CUNY, July 2001. /x(x = y).

Computer Science Logic, 2000

Studia Logica, 2015

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap-Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev.

Notre Dame Journal of Formal Logic, 2013

It is argued that the `inner' negation $\nega$ familiar from 3-valued logic can be interpreted as a form of `conditional' negation: $\nega A$ is read ``$A$ is false \emph{if it has a truth value}''. It is argued that this reading squares well with a particular 3-valued interpretation of a conditional that in the literature has been seen as a serious candidate for capturing the truth conditions of the natural language indicative conditional (e.g. ``If Jim went to the party he had a good time''). It is shown that the logic induced by the semantics shares many familiar properties with classical negation, but is orthogonal to both intuitionistic and classical negation: it differs from both in validating the inference from $A\rightarrow \nega B$ to $\nega(A\rightarrow B)$.

2017

Studia Logica, 2005

This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t)) for some term t, where n denotes negation. The first question asks for conditions on the hypotheses that, if satisfied, guarantee the existence of a double-negation-free proof when the conclusion is free of double negation. The second question asks about the existence of an axiom system for classical propositional calculus whose use, for theorems with a conclusion free of double negation, guarantees the existence of a double-negation-free proof. After giving conditions that answer the first question, we answer the second question by focusing on the Łukasiewicz three-axiom system. We then extend our studies to infinite-valued sentential calculus and to intuitionistic logic and generalize the notion of being double-negation free. The double-negation proofs of interest rely exclusively on the inference rule condensed detachment, a rule that combines modus ponens with an appropriately general rule of substitution. The automated reasoning program OTTER played an indispensable role.

Logique Et Analyse, 2017

We refine the interpolation property of the {&, v, ~, A, E}-fragment of classical first-order logic, showing that if [G is satisfiable] and [D is ot logically true] and G|- D then there is an interpolant c, constructed using only non-logical vocabulary common to both members of G and members of D, such that (i) G entails c in the first-order version of Kleene’s strong three-valued logic (K3), and (ii) c entails D in the first-order version of Priest’s Logic of Paradox (LP). The proof proceeds via a careful analysis of derivations in a cut-free sequent calculus for first-order classical logic. Lyndon’s strengthening falls out of an observation regarding such derivations and the steps involved in the construction of interpolants. The proof is then extended to cover the {&, v, ~, A, E}-fragment of classical first-order logic with identity. Keywords: Craig–Lyndon Interpolation Theorem (for classical first-order logic); Kleene’s strong 3-valued logic;Priest’s Logic of Paradox; Belnap’s f...

Reports on Mathematical Logic, 2013

Reports on Mathematical Logic, 2002