13 research outputs found
On prevarieties of logic
It is proved that every prevariety of algebras is categorically equivalent to
a "prevariety of logic", i.e., to the equivalent algebraic semantics of some
sentential deductive system. This allows us to show that no nontrivial equation
in the language "meet, join, and relational product" holds in the congruence
lattices of all members of every variety of logic, and that being a
(pre)variety of logic is not a categorical property
Constructive Logic with Strong Negation is a Substructural Logic. II
The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL ew of the substructural logic FL ew . The main result of Part I of this series [41] shows that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL ew (namely, a certain variety of FL ew -algebras) are term equivalent. In this paper, the term equivalence result of Part I [41] is lifted to the setting of deductive systems to establish the definitional equivalence of the logics N and NFL ew . It follows from the definitional equivalence of these systems that constructive logic with strong negation is a substructural logi
Discriminator logics (Research announcement)
A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work
Classical logic with n truth values as a symmetric many-valued logic
We introduce Boolean-like algebras of dimension n (nBA s) having n constants e1, ⌠, en, and an (n+ 1) -ary operation q (a âgeneralised if-then-elseâ) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of nBA s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The nBA s provide the algebraic framework for generalising the classical propositional calculus to the case of nâperfectly symmetricâtruth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, nCL , and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in nCL , and, via the embeddings, in all the finite tabular logics
Discriminator logics (Research announcement)
A discriminator logic is the 1-assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator logic S can be presented (up to definitional equivalence) as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work
Intuitionistic logic as a connexive logic
We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic (CHL), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for CHL ; moreover, we suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsnerâs idea of superconnexivity
On some properties of quasi-MV algebras and square root quasi-MV algebras, IV
In the present paper, which is a sequel to
[20, 4, 12], we investigate further the structure theory of quasiMV
algebras and â0quasi-MV algebras. In particular: we provide
a new representation of arbitrary â0qMV algebras in terms
of â0qMV algebras arising out of their MV* term subreducts of
regular elements; we investigate in greater detail the structure
of the lattice of â0qMV varieties, proving that it is uncountable,
providing equational bases for some of its members, as well as
analysing a number of slices of special interest; we show that the
variety of â0qMV algebras has the amalgamation property; we
provide an axiomatisation of the 1-assertional logic of â0qMV
algebras; lastly, we reconsider the correspondence between Cartesian
â0qMV algebras and a category of Abelian lattice-ordered
groups with operators first addressed in [10]
Order algebraizable logics
AbstractThis paper develops an order-theoretic generalization of Blok and PigozziĘźs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)â(iv)