Implicational Partial Galois Logics: Relational Semantics

2021 ◽  
Author(s):  
Eunsuk Yang ◽  
J. Michael Dunn
Keyword(s):  
Relational Semantics

Relational semantics and a relational proof system for full Lambek calculus

1998 ◽  
Vol 63 (2) ◽  
pp. 623-637 ◽  
Author(s):  
Wendy MacCaull
Keyword(s):  
Proof Theory ◽  
Kripke Semantics ◽  
Substructural Logic ◽  
Lambek Calculus ◽  
Relational Semantics ◽  
Complete Proof ◽  
Proof Search ◽  
Proof Tree ◽  
Relational Logic ◽  
Completeness Theorems

AbstractIn this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). We start with the Kripke semantics for FL as discussed in [11] and develop a second Kripke-style semantics, RelKripke semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with two distinguished elements, two ternary relations and a list of conditions on the relations. It is accompanied by a Kripke-style valuation system analogous to that in [11]. Soundness and completeness theorems with respect to FL hold for RelKripke models. Then, in the spirit of the work of Orlowska [14], [15], and Buszkowski and Orlowska [3], we develop relational logic RFL. The adjective relational is used to emphasize the fact that RFL has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in FL is provable if and only if a translation, t(γ1 ● … ● γn ⊃ α)ευu, has a cut-complete fundamental proof tree. This result is constructive: that is, if a cut-complete proof tree for t(γ1 ● … ● γn ⊃ α)ευu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ1 ● … ● γn ⊃ α)ευu and from this, build a RelKripke countermodel for γ1 ● … ● γn ⊃ α. These results allow us to add FL, the basic substructural logic, to the list of those logics of importance in computer science with a relational proof theory.

scholarly journals Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

Studia Logica ◽  
2012 ◽  
Vol 101 (3) ◽  
pp. 505-545 ◽  
Author(s):  
Georges Hansoul ◽  
Bruno Teheux
Keyword(s):  
Algebraic Approach ◽  
Relational Semantics ◽  
Łukasiewicz Logics

scholarly journals A relational semantics for parallelism and non-determinism in a functional setting

2012 ◽  
Vol 163 (7) ◽  
pp. 918-934 ◽  
Author(s):  
Antonio Bucciarelli ◽  
Thomas Ehrhard ◽  
Giulio Manzonetto
Keyword(s):  
Relational Semantics ◽  
Functional Setting

scholarly journals Non-uniform (hyper/multi)coherence spaces

2010 ◽  
Vol 21 (1) ◽  
pp. 1-40 ◽  
Author(s):  
PIERRE BOUDES
Keyword(s):  
Relational Semantics ◽  
Game Semantics ◽  
Stable Function ◽  
Free Construction ◽  
Single Piece ◽  
Coherence Spaces

In (hyper)coherence semantics, proofs/terms are cliques in (hyper)graphs. Intuitively, the vertices represent the results of computations and the edge relation witnesses the ability to carry out the computation assembled into a single piece of data or a single (strongly) stable function, at arrow types.In (hyper)coherence semantics, the argument of a (strongly) stable functional is always a (strongly) stable function. As a consequence, compared to the relational semantics where there is no edge relation, some vertices are missing. Recovering these vertices is essential if we are to reconstruct proofs/terms from their interpretations. It will also be useful for comparing with other semantics, such as game semantics.Bucciarelli and Ehrhard (2001) introduced a non-uniform coherence space semantics, where no vertex is missing. By constructing the co-free exponential, we get a new version of this semantics, together with non-uniform versions of hypercoherences and multicoherences. This provides a new semantics in which an edge is a finite multiset. Thanks to the co-free construction, these non-uniform semantics are deterministic in the sense that the intersection of a clique and an anti-clique contains at most one vertex, which is a result of interaction, and they then extensionally collapse onto the corresponding uniform semantics.

scholarly journals INSTANTIAL NEIGHBOURHOOD LOGIC

2016 ◽  
Vol 10 (1) ◽  
pp. 116-144 ◽  
Author(s):  
JOHAN VAN BENTHEM ◽  
NICK BEZHANISHVILI ◽  
SEBASTIAN ENQVIST ◽  
JUNHUA YU
Keyword(s):  
Fine Structure ◽  
Normal Form ◽  
Axiom System ◽  
Topological Spaces ◽  
Relational Semantics ◽  
Basic Model ◽  
Semantic Tableaux ◽  
Dynamic Logics ◽  
A Current

AbstractThis paper explores a new language of neighbourhood structures where existential information can be given about what kind of worlds occur in a neighbourhood of a current world. The resulting system of ‘instantial neighbourhood logic’ INL has a nontrivial mix of features from relational semantics and from neighbourhood semantics. We explore some basic model-theoretic behavior, including a matching notion of bisimulation, and give a complete axiom system for which we prove completeness by a new normal form technique. In addition, we relate INL to other modal logics by means of translations, and determine its precise SAT complexity. Finally, we discuss proof-theoretic fine-structure of INL in terms of semantic tableaux and some expressive fine-structure in terms of fragments, while discussing concrete illustrations of the instantial neighborhood language in topological spaces, in games with powers for players construed in a new way, as well as in dynamic logics of acquiring or deleting evidence. We conclude with some coalgebraic perspectives on what is achieved in this paper. Many of these final themes suggest follow-up work of independent interest.

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