Nissim Francez

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.

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.

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.

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.

The paper presents a plan for negation, proposing a paradigm shift from the Australian plan for negation, leading to a family of contra-classical logics. The two main ideas are the following: Instead of shifting points of evaluation (in a frame), shift the evaluated formula. Introduce an incompatibility set for every atomic formula, extended to any compound formula, and impose the condition on valuations that a formula evaluates to true iff all the formulas in its incompatibility set evaluate to false. Thus, atomic sentences are not independent in their truth-values. The resulting negation, in addition to excluding the negated formula, provides a positive alternative to the negated formula. I also present a sound and complete natural deduction proof systems for those logics. In addition, the kind of negation considered in this paper is shown to provide an innovative notion of grounding negation.

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.

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.

The paper exposes the relevance of permuting conversions (in natural-deduction systems) to the role of such systems in the theory of meaning known as proof-theoretic semantics, by relating permuting conversion to harmony, hitherto related to normalisation only. This is achieved by showing the connection of permuting conversion to the general notion of canonicity, once applied to arbitrary derivations from open assumption. In the course of exposing the relationship of permuting conversions to harmony, a general definition of the former is proposed, generalising the specific cases of disjunction and existential quantifiers considered in the literature.

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.

In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules.

This paper develops an inference system for natural language within the ‘Natural Logic’ paradigm as advocated by van Benthem, Sánchez and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek -based system we propose extends the system by Fyodorov et~al., which is based on the Ajdukiewicz/Bar-Hillel calculus Bar Hillel,. This enables the system to deal with new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated by using normalization axioms.