Paraconsistency in classical logic

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized.     

Quine and Slater on Paraconsistency and Deviance

In a famous and controversial paper, B. H. Slater has argued against the possibility of paraconsistent logics. Our reply is centred on the distinction between two aspects of the meaning of a logical constant ^*c^* in a given logic: its operational meaning, given by the operational rules for ^*c^* in a cut-free sequent calculus for the logic at issue, and its global meaning, specified by the sequents containing ^*c^* which can be proved in the same calculus. Subsequently, we use the same strategy to counter Quine's meaning variance argument against deviant logics. In a nutshell, we claim that genuine rivalry between (similar) logics ^*L^* and ^*L^* is possible whenever each constant in ^*L^* has the same operational meaning as its counterpart in ^*L^* although differences in global meaning arise in at least one case.     

Paraconsistency in Categories: Case of Relevance Logic

Categorical-theoretic semantics for the relevance logic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.     

On the Treatment of Incomparability in Ordering Semantics and Premise Semantics

In his original semantics for counterfactuals, David Lewis presupposed that the ordering of worlds relevant to the evaluation of a counterfactual admitted no incomparability between worlds. He later came to abandon this assumption. But the approach to incomparability he endorsed makes counterintuitive predictions about a class of examples circumscribed in this paper. The same underlying problem is present in the theories of modals and conditionals developed by Bas van Fraassen, Frank Veltman, and Angelika Kratzer. I show how to reformulate all these theories in terms of lower bounds on partial preorders, conceived of as maximal antichains, and I show that treating lower bounds as cutsets does strictly better at capturing our intuitions about the semantics of modals, counterfactuals, and deontic conditionals.     

Varieties of Three-Valued Heyting Algebras with a Quantifier

This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q ₃ of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q ₃ and we construct the lattice of subvarieties (Q ₃ ) of the variety Q ₃ .     

Semantics and the sacred

ABSTRACT

This article looks at four different scholarly perspectives on ‘sacred’ – the ineffable sacred, the experienced sacred, the polarized sacred and the contextualized sacred – in order to draw out their implicit presuppositions about meaning. The first two stances presuppose that meaning depends on what bits of language are about (referentialism), and the other two stances presuppose that meaning depends on relations between bits of language (holism). The article concludes three things: these prominent views of ‘sacred’ rest on usually implicit or unrecognized assumptions about the nature of meaning; some of those assumptions explain why certain theories are contentious and problematic and others ground more promising and productive approaches.     

Semantics and supervenience

I have presented much of this research in talks at the University of Costa Rica and the University of Texas at Austin. I am grateful to my audiences for their comments and advice. I would like especially to thank Luis Camacho, Nicholas Asher, and Robert Koons. Many of the ideas in the paper stem from an informal seminar on type-free theories held at the University of Texas's Center for Cognitive Science from 1984 to 1987. I am grateful to the participants in that seminar — Ignacio Angelelli, Nicholas Asher, Herbert Hochberg, Hans Kamp, Frederick Kronz, Per Lindström and Mark Sainsbury — for their many insights into type-free semantics, and to the Center for Cognitive Science for providing such a hospitable environment for this work. I have also profited from the criticisms of two anonymous referees. Finally, I am indebted to the University of Texas's University Research Institute and to the National Science Foundation's Information Science and History and Philosophy of Science programs for grant support.     

Semantics and the computational paradigm in cognitive psychology

There is a prevalent notion among cognitive scientists and philosophers of mind that computers are merely formal symbol manipulators, performing the actions they do solely on the basis of the syntactic properties of the symbols they manipulate. This view of computers has allowed some philosophers to divorce semantics from computational explanations. Semantic content, then, becomes something one adds to computational explanations to get psychological explanations. Other philosophers, such as Stephen Stich, have taken a stronger view, advocating doing away with semantics entirely. This paper argues that a correct account of computation requires us to attribute content to computational processes in order to explain which functions are being computed. This entails that computational psychology must countenance mental representations. Since anti-semantic positions are incompatible with computational psychology thus construed, they ought to be rejected. Lastly, I argue that in an important sense, computers are not formal symbol manipulators.     

Trivalent Semantics and the Vaguely Vague

Michael Tye responds to the problem of higher-order vagueness for his trivalent semantics by maintaining that truth-value predicates are “vaguely vague”: it’s indeterminate, on his view, whether they have borderline cases and therefore indeterminate whether every sentence is true, false, or indefinite. Rosanna Keefe objects (1) that Tye’s argument for this claim tacitly assumes that every sentence is true, false, or indefinite, and (2) that the conclusion is any case not viable. I argue – contra (1) – that Tye’s argument needn’t make that assumption. A version of her objection is in fact better directed against other arguments Tye advances, though Tye can absorb this criticism without abandoning his position’s core. On the other hand, Keefe’s second objection does hit the mark: embracing ‘vaguely vague’ truth-value predicates undermines Tye’s ability to support validity claims needed to defend his position. To see this, however, we must develop Keefe’s remarks further than she does.     

Ontologies and Construals in Lexical Semantics

The purpose of this paper is to propose a framework of lexical meaning, broadly along the lines of Cognitive Semantics (Langacker 1987a). Within the proposed model, all aspects of meaning are to be explained in terms of properties of ontologies in conceptual space, i.e. properties of content ontologies and schematic ontologies and construals which are imposed on the conceptual structures on the occasion of use. It is through the operations of construals on ontological structures that different readings of lexical expressions arise. Lexical meanings are dynamic and sensitive to contextual demands, rather than fixed and stable. In a dynamic, usage-based model like this, polysemy and multiple readings emerge as a natural consequence of the human ability to think flexibly. Another more specific purpose of this paper is to draw attention to the usefulness of ontologies in linguistic research in general and semantic modelling in particular.     

Abstract Events in Semantics

Philosophia - Here, we defend the thesis whereby the event (conceived as a mental and social abstract entity) plays a main role of sense in the meaning of certain sentences. This thesis is based on...     

Referentiality and Matrix Semantics

Referential semantics importantly subscribes to the programme of theory of logical calculi. Defined by Wójcicki in [8], it has been subsequently studied in a series of papers of the author, till the full exposition of the framework in [9] and its intuitive characterisation in [10].     

Semantics in Banach spaces

A new approach to semantics, based on ordered Banach spaces, is proposed. The Banach spaces semantics arises as a generalization of the four particular cases: the Giles' approach to belief structures, its generalization to the non-Boolean case, and “fuzzy extensions” of Boolean as well as of non-Boolean semantics.     

Bilateralism in Proof-Theoretic Semantics

The paper suggests a revision of the notion of harmony, a major necessary condition in proof-theoretic semantics for a natural-deduction proof-system to qualify as meaning conferring, when moving to a bilateral proof-system. The latter considers both forces of assertion and denial as primitive, and is applied here to positive logics, lacking negation altogether. It is suggested that in addition to the balance between (positive) introduction and elimination rules traditionally imposed by harmony, a balance should be imposed also on: (i) negative introduction and elimination rules, and (ii) positive and negative introduction rules. The paper suggests a proof-theoretical definition of duality (not referring to truthtables), using which double harmony is defined. The paper proves that in a doubly-harmonious system, the coordination rule, typical to bilateral systems, is admissible.