Embedding theorems and finiteness properties for residuated structures and substructural logics

Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2008.Paper 1. This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which either treated fewer subsignatures or focussed on the conservation of theorems only. Paper 2. It is proved that the variety of relevant disjunction lattices has the finite embeddability property (FEP). It follows that Avron’s relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron’s result that RMImin is decidable. Paper 3. An idempotent residuated po-monoid is semiconic if it is a subdirect product of algebras in which the monoid identity t is comparable with all other elements. It is proved that the quasivariety SCIP of all semiconic idempotent commutative residuated po-monoids is locally finite. The lattice-ordered members of this class form a variety SCIL, which is not locally finite, but it is proved that SCIL has the FEP. More generally, for every relative subvariety K of SCIP, the lattice-ordered members of K have the FEP. This gives a unified explanation of the strong finite model property for a range of logical systems. It is also proved that SCIL has continuously many semisimple subvarieties, and that the involutive algebras in SCIL are subdirect products of chains. Paper 4. Anderson and Belnap’s implicational system RMO can be extended conservatively by the usual axioms for fusion and for the Ackermann truth constant t. The resulting system RMO is algebraized by the quasivariety IP of all idempotent commutative residuated po-monoids. Thus, the axiomatic extensions of RMO are in one-to-one correspondence with the relative subvarieties of IP. It is proved here that a relative subvariety of IP consists of semiconic algebras if and only if it satisfies x (x t) x. Since the semiconic algebras in IP are locally finite, it follows that when an axiomatic extension of RMO has ((p t) p) p among its theorems, then it is locally tabular. In particular, such an extension is strongly decidable, provided that it is finitely axiomatized

Non-Boolean classical relevant logics I

Under embargo until: 2020-12-13Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing.acceptedVersio

Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape

Converse Ackermann property and constructive negation defined with a negation connective

The Converse Ackermann Property is the unprovability of formulas of the form (A -> B) -> C when C does contain neither -> nor ¬. Intuitively, the CAP amounts to rule out the derivability of pure non-necessitive propositions from non-necessitive ones. A constructive negation of the sort historically defined by, e.g., Johansson is added to positive logics with the CAP in the spectrum delimited by Ticket Entailment and Dummett’s logic LC

Relevant Logics Obeying Component Homogeneity

This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi for S*fde, dS*fde, crossS*fde. Among the other accomplishments of the paper, we generalize the semantics from Bochvar, Hallden, Deutsch and Daniels, we provide a general recipe to define containment logics, we explore the single-premise/single-conclusion fragment of S*fde, dS*fde, crossS*fdeand the connections between crossS*fde and the logic Eq of equality by Epstein. Also, we present S*fde as a relevant logic of meaninglessness that follows the main philosophical tenets of Goddard and Routley, and we briefly examine three further systems that are closely related to our main logics. Finally, we discuss Routley's criticism to containment logic in light of our results, and overview some open issues

"A Smack of Irrelevance" in Inconsistent Mathematics?

Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature