On Axioms and Rexpansions

Caleiro, Carlos; Marcelino, Sérgio

doi:10.1007/978-3-030-71258-7_3

Cited by 11 publications

(17 citation statements)

References 29 publications

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“…) never uses the 1 2 value of matrix K. As K behaves classically for the 0, 1 values, we conclude that all bivaluations in BVal(K ω * L ω ) are classical. On the other hand, we know from Lemmas 11,20 and…”

Section: Representing Pairs Of Pairs As Four-tuples We Get the Set Of...mentioning

confidence: 99%

“…Often, one works with a single-conclusion logic which can then be streghtened by the addition of new axioms [6,26,20] (and possibly also new syntax). Concretely, let Σ 1 , ⊢ 1 be a single-conclusion logic, Σ 2 be a signature, Ax ⊆ L Σ2 (P ) be a set of axiom schemata, and define Ax inst = {A σ : A ∈ Ax and σ : P → L Σ1∪Σ2 (P )}.…”

Section: Adding Axiomsmentioning

confidence: 99%

“…△ Nicer, finite-valued, semantics can be obtained in particularly well-behaved scenarios, which (unsurprisingly) do not include the examples above. We refer to reader to [20] for further details.…”

Section: Example 43 (Intuitionistic Logic)mentioning

confidence: 99%

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Modular many-valued semantics for combined logics

Caleiro¹,

Marcelino²

2022

Preprint

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We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial nondeterministic logical matrices. Our constructions preserve finite-valuedness in the context of multiple-conclusion logics whereas, unsurprisingly, it may be lost in the context of single-conclusion logics.Besides illustrating our constructions over a wide range of examples, we also develop concrete applications of our semantic characterizations, namely regarding the semantics of strengthening a given many-valued logic with additional axioms, the study of conditions under which a given logic may be seen as a combination of simpler syntactically defined fragments whose calculi can be obtained independently and put together to form a calculus for the whole logic, and also general conditions for decidability to be preserved by the combination mechanism.

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Section: Representing Pairs Of Pairs As Four-tuples We Get the Set Of...mentioning

confidence: 99%

Section: Adding Axiomsmentioning

confidence: 99%

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Modular many-valued semantics for combined logics

Caleiro¹,

Marcelino²

2022

Preprint

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“…Given a Σ-matrix M = A, D , a set of axioms Ax ⊆ F m Σ and a set of rules R ⊆ ℘(F m Σ ) × F m Σ , we write Val Ax M for the set of valuations on M such that v(ϕ σ ) ⊆ D for every ϕ ∈ Ax and every substitution σ, and Val R M for the set of valuations on M such that v(Γ σ ) ⊆ D implies ϕ σ ∈ D for every Γ ϕ ∈ R and substitution σ. The following result (whose simple proof we omit) is a corollary of [4, Lemma 2.7] and will be very useful to show relative axiomatization results (this technique is used in [5] to obtain general modular semantics for axiomatic extensions of a given logic). In item (ii), M ω is a shorthand for i<ω M. Proposition 5.15.…”

Section: 3mentioning

confidence: 99%

Logics of Involutive Stone Algebras

Marcelino

Rivieccio

2021

Preprint

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An involutive Stone algebra (IS-algebra) is a structure that is simultaneously a De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributive lattice satisfying the well-known Stone identity, ∼ x ∨ ∼ ∼ x ≈ 1). IS-algebras have been studied algebraically and topologically since the 1980’s, but a corresponding logic (here denoted IS ≤ ) has been introduced only very recently. The logic IS ≤ is the departing point for the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that IS ≤ is a conservative expansion of the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic so as to obtain an extension of IS ≤ . We show that every logic thus defined can be axiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ≤ that cannot be obtained in the above- described way, but can nevertheless be axiomatized finitely by other methods. Most of our axiomatization results are obtained in two steps: through a multiple-conclusion calculus first, which we then reduce to a traditional one. The multiple-conclusion axiomatizations introduced in this process, being analytic, are of independent interest from a proof-theoretic standpoint. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of IS ≤ . Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ≤ are already uncountably many.

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“…The added expressiveness allows for finite characterizations of logics that do not admit finite semantics based on logical matrices [23,24] and provide valuable insight about proof theoretical properties of said logics [2]. It also allows for general recipes for various practical problems in logic, including procedures to constructively updating semantics when imposing new axioms [9,12], including language extensions; or effectively combining the semantics of two logics capturing the effect of joining their axiomatizations [8,21]. Recently, PNmatrices also provided new interpretations of quantum states as valuations [15].…”

Section: Introductionmentioning

confidence: 99%

Computational Properties of Partial Non-deterministic Matrices and Their Logics

Marcelino

Caleiro

Filipe

2021

Logical Foundations of Computer Science

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Incorporating non-determinism and partiality in the traditional notion of logical matrix semantics has proven to be decisive in a myriad of recent compositionality results in logic. However, several important properties which are known to be computable for finite matrices have not been studied in this wider context of partial nondeterministic matrices (PNmatrices).This paper is dedicated to understanding how this generalization of the considered semantical structures affects the computational properties of basic problems regarding their induced logics, in particular their sets of theorems.We will show that the landscape is quite rich, as some problems keep their computational status, for others the complexity increases, and for a few decidability is lost. Namely, we show that checking if the logics defined by two finite PNmatrices have the same theorems is undecidable. This latter result is obtained by reduction from the undecidable problem of checking universality of term-DAG-automata.

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On Axioms and Rexpansions

Cited by 11 publications

References 29 publications

Modular many-valued semantics for combined logics

Modular many-valued semantics for combined logics

Logics of Involutive Stone Algebras

Computational Properties of Partial Non-deterministic Matrices and Their Logics

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