Basic Constructive Connectives, Determinism and Matrix-Based Semantics

Ciabattoni, Agata; Lahav, Ori; Zamansky, Anna

doi:10.1007/978-3-642-22119-4_11

Cited by 3 publications

(1 citation statement)

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“…In this section we use the soundness and completeness theorems to establish a connection between determinism of the semantics of a certain connective, and the fact that this connective admits axiom-expansion. A similar connection was shown in [8] for canonical single-conclusions sequent systems. Roughly speaking, an n-ary connective is deterministic in a system G if for every formula (ψ 1 , .…”

Section: Determinism and Axiom-expansionsupporting

confidence: 76%

Semantic investigation of canonical Gödel hypersequent systems

Lahav¹

2013

J Logic Computation

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We define a general family of hypersequent systems with well-behaved logical rules, of which the known hypersequent calculus for (propositional) Gödel logic, is a particular instance. We present a method to obtain (possibly, non-deterministic) many-valued semantics for every system of this family. The detailed semantic analysis provides simple characterizations of cut-admissibility and axiom-expansion for the systems of this family.Well-behaved logical rules, as those of HG, have a great philosophical benefit. Indeed, according to a guiding principle in the philosophy of logic, attributed to Gentzen, the meanings of the connectives are determined by their derivation rules. To achieve this in (cut-free) sequent or hypersequent calculi, one should have "ideal" logical rules, each of which introduces a unique connective and does not involve other connectives. The notion of canonical sequent rules, introduced in [5], provides a precise formulation of the structure of these "ideal" rules in the framework of multiple-conclusion sequent calculi. Canonical sequent systems were in turn defined as sequent calculi that include all standard structural rules, the two identity rules, and an arbitrary set of canonical sequent rules. Clearly, LK, the well-known calculus for classical logic, is the most important example of a canonical sequent system. However, infinitely many new calculi with various new connectives can be defined in this framework. In [4] the single-conclusion counterparts of canonical rules and canonical systems were introduced, and provided a proof-theoretical approach to define constructive connectives.In the current paper, we define canonical Gödel systems. First, we adopt the notion of (single-conclusion) canonical sequent rule to the hypersequents framework, and define the family of canonical hypersequent rules. Canonical Gödel systems are defined as (single-conclusion) hypersequent calculi that include all standard structural rules, the identity rules, the communication rule, and an arbitrary set of canonical hypersequent rules. Here HG is the prototype example, but again, a variety of new connectives can be defined, and be added to (or replace) the usual connectives of Gödel logic. Then, we study canonical Gödel systems from a semantic point of view. First and foremost, this includes a general method to obtain a (strongly) sound and complete manyvalued semantics for every canonical Gödel system. As in [5] and [4], a major key here is the use of non-deterministic semantics, which intuitively occurs whenever the right-introduction rules and the left-introduction rules for a certain connective do not match, leaving some options undetermined. In addition, we also consider the semantic effect of the identity rules and provide finer semantics for canonical Gödel systems in which the identity rules are restricted to apply only on some given set of formulas. This semantics is then used to identify the "good" canonical Gödel systems, namely those that enjoy (strong) cut-admissibility. In fact, we show that the si...

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Section: Determinism and Axiom-expansionsupporting

confidence: 76%