Nissim Francez

I show that in the context of proof-theoretic semantics, Dummett’s distinction between the assertoric meaning of a sentence and its ingredient sense can be seen as a distinction between two proof-theoretic meanings of a sentence: 1.Meaning as a conclusion of an introduction rule in a meaning-conferring natural-deduction proof system. 2.Meaning as a premise of an introduction rule in a meaning-conferring natural-deduction proof system. The effect of this distinction on compositionality of proof-theoretic meaning is discussed

We define an automata-theoretic counterpart of grammars based on the Lambek-calculus L, a prominent formalism in computational linguistics. While the usual push-down automaton has the same weak generative power as the L-based grammars , there is no direct relationship between the computations of a PDA for some language L and the derivations of an L-based grammar for L. In the Lambek-automaton, on the other hand, there is a tight relation between automaton computations and grammar derivations. The automaton exhibits a novel mode of operation, using hypothetical steps, directly inspired by the hypothetical reasoning embodied by L

The paper introduces a proof-theoretic semantics for adjectival modification as an alternative to the traditional model-theoretic semantics basing meaning on truth-conditions. The paper considers the proof-theoretic meaning of modification by means of the three traditional adjective classes: intersective, subsective and privative. It does so by introducing a meaning-conferring natural-deduction proof system for such modification. The PTS theory of meaning is not polluted by ontological commitments, for example, a scale for beauty and a yardstick for being beautiful. It only uses syntactic artefacts of the proof language. The paper also defines, by suitable rules, iterated modification, shedding light on the relationship between iteration and adjectival classes. Modification via coordinated adjectives is covered too. An appendix delineates briefly the main ingredients of PTS.

The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: negative introduction and elimination rules, and positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality, using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible.

We study structural rules in the context of multi-valued logics with finitely-many truth-values. We first extend Gentzen’s traditional structural rules to a multi-valued logic context; in addition, we propos some novel structural rules, fitting only multi-valued logics. Then, we propose a novel definition, namely, structural rules completeness of a collection of structural rules, requiring derivability of the restriction of consequence to atomic formulas by structural rules only. The restriction to atomic formulas relieves the need to concern logical rules in the derivation.

In this paper I introduce the notion of bilogics, namely logics interpreted over a pair of structures, in contrast to classical logic and many of its variations, the formulae of which are interpreted over one structure. In particular, I introduce and study Contrastive Logic, suitable for expressing contrast and conformity between the two structures involved.A major reason for this study is striving towards an extension of truth-conditional semantics to cover several natural-language particles, which have been hitherto considered not to be amenable to such an extensional treatment, and were delegated to the level of non-extensional pragmatics. Examples of such particles are but and already

The paper highlights proof-theoretic semantics as providing natural resources for capturing semantic variation in natural language. The semantic variations include:Distinction between extensional predication and attribution to intensional transitive verbs a non-specific object.Omission of a verbal argument in a transitive verb.Obtaining sameness of meaning of sentences with transitive verbs with omitted object and existentially quantified object.Blocking unwarranted entailments in adjective–noun combinations.Capturing quantifier scope ambiguity.Obtaining context dependent quantifier domain restriction. The proof-theoretic resources employed to capture the above semantic variations include:The use of different kinds of formal parameters and different ways of binding them to predicates.Omission of a premise from a proof-rule.Basing meanings on derivations in meaning-conferring proof-systems.Appealing to substructurality of I/E-rules.Controlling the order of rule applications in derivations.Controlling the open formulas participating in a context within a sequent.

The paper presents a contra-classical dialectic logic, inspired and motivated by Hegel s dialectics. Its axiom schemes are 0.1 Thus, in a sense, this dialectic logic is a kind of “mirror image“ of connexive logic. The informal interpretation of ‘ $$\rightarrow $$ ’ emerging from the above four axiom schemes is not of a conditional (or implication); rather, it is the relation of determination in the presence of truth-value gaps: $$\varphi \rightarrow \psi $$ is read as $$\varphi $$ determines $$\psi $$, namely, necessarily, if $$\varphi $$ is true, then $$\psi $$ is either true or false, not gappy. As far as I know, such a connective has not been considered before in the literature.

This paper investigates the meaning of restricted quantification when the embedded conditional is taken as the conditional of some first-order connexive logics. The study is carried out by checking the suitability of RQ for defining a connexive class theory, in analogy to the definition of Boolean class theory by using RQ in classical logic. Negative results are obtained for Wansing’s first-order connexive logic QC and one variant of Priest’s first-order connexive logic QP. A positive result is obtained for another variant of QP.

I argue in favour of object languages of logics to be diversely-generated, that is, not having identical immediate sub-formulas. In addition to diversely-generated object languages constituting a more appropriate abstraction of the use of sentential connectives in natural language, I show that such language lead to a simplifications w.r.t. some specific issues: the identity of proofs, the factual equivalence and the Mingle axiom in Relevance logics. I also point out that some of the properties of classical logic based on freely-generated object languagest.

I presented a sub-classical relating logic based on a relating via an NL-inspired relating relation Rcss. The relation Rcss is motivated by the NL-phenomenon of phrasal (subsentential) coordination, exhibiting an important aspect of contents relating among the arguments of binary connectives. The resulting logic Lcss can be viewed as a relevance logic exhibiting a contents related relevance, stronger than the variable-sharing property of other relevance logics like R. Note that relating here is not “tailored” to justify some predetermined logic; rather, the relating relation is independently justified, and induces a logic not previously investigated.

In some presentations of classical and intuitionistic logics, the objectlanguage is assumed to contain (two) truth-value constants: ⊤ (verum) and ⊥ (falsum), that are, respectively, true and false under every bivalent valuation. We are interested to define and study analogical constants ‡, 1 ≤ i ≤ n, that in an arbitrary multi-valued logic over truth-values V = {v1,..., vn} have the truth-value vi under every (multi-valued) valuation. As is well known, the absence or presence of such constants has a significant deductive impact on the logics studied. We define such constants proof-theoretically via their associated I/E-rules in a natural-deduction proof system. In particular, we propose a generalization of the notions of contradiction and explosiveness of a logic to the context of multi-valued logics.