From Many-Valued Consequence to Many-Valued Connectives

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but on what we call Gentzen-regular connectives (including negation, conjunction, and disjunction). For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations.Comment: Updated version [corrections of an incorrect claim in first version; two bib entries added

Knowledge, Justification, and Reason-Based Belief

Is knowledge definable as justified true belief ("JTB")? We argue that one can legitimately answer positively or negatively, depending on how the notion of justification is understood. To facilitate our argument, we introduce a simple propositional logic of reason-based belief. We show that this logic is sufficiently flexible to accommodate various useful features, including quantification over reasons. We use our framework to contrast two notions of JTB: one internalist, the other externalist. We argue that Gettier cases essentially challenge the internalist notion but not the externalist one. In particular, we may equate knowledge and JTB if the latter is grounded in what we call "adequate" reasons.Comment: v3 edits acknowledgment

Suszko's Problem: Mixed Consequence and Compositionality

Suszko's problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski's structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. In this paper we give a more systematic perspective on Suszko's problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that (intersective) mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos and Wansing, for logics with various structural properties (reflexivity, transitivity, none, or both). We strengthen these results into exact rank results for non-permeable logics (roughly, those which distinguish the role of premises and conclusions). We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties.Comment: Keywords: Suszko's thesis; truth value; logical consequence; mixed consequence; compositionality; truth-functionality; many-valued logic; algebraic logic; substructural logics; regular connective

Postcripts to "A pragmatic approach to the problem of logical omniscience"

Postcripts to "A pragmatic approach to the problem of logical omniscience

Question-Embedding and Factivity [paper]

First version. (as of 11/18/2007) NB. Some references are still missing).Attitude verbs fall in different categories depending on the kind of complements which they can embed. In English, a verb like know takes both declarative and interrogative complements. By contrast, believe takes only declarative complements and wonder takes only interrogative complements. The present paper examines the hypothesis, originally put forward by Hintikka 1975, that the only verbs that can take both that-complements and whether-complements are the factive verbs. I argue that at least one half of the hypothesis is empirically correct, namely that all veridical attitude verbs taking that-complements take whether-complements. I distinguish veridical verbs from factive verbs, and present one way of deriving the generalization. Counterexamples to both directions of the factivity hypothesis are discussed, in particular the case of emotive factive verbs like regret, and the case of nonveridical verbs that licence whether-complements, in particular tell, guess, decide and agree. Alternative accounts are discussed along the way, in particular Zuber (1982), Ginzburg (1995) and Saebo (2006)

Comments on Ede Zimmermann's "Monotonicity in Opaque Verbs"

This is the handout of my comments on E. Zimmermann's paper "Monotonicity in Opaque Verbs", which I prepared for the workshop on Intensional Verbs and Non-Referential Terms held at IHPST on January 14, 2006

Question-Embedding and Factivity (abstract)

This is the abstract of the talk I gave at the conference "Journées Internationales de Sémantique et Modélisation" held in Paris, march 2005. The paper discusses the link between the factivity of predicates taking declarative complements and their question-embedding behavior, arguing that the question-embedding behavior of these verbs can be predicted from their factive or non-factive feature

A pragmatic approach to the problem of logical omniscience

A pragmatic approach to the problem of logical omniscienc

Reliability, Margin for Error and Self-Knowledge

Forthcoming in D.H. Pritchard & V. Hendricks (eds.), New Waves in Epistemology, (Adelshot: Ashgate Publishing).Margin for error principles play a central role in the epistemology of Timothy Williamson, both in his account of vague knowledge (Williamson 1992, 1994), and in his attack against the luminosity of knowledge (Williamson 2000). The present paper pursues two objectives: the first is an attempt to refine and systematize the modal analysis of the reliability of knowledge given by Williamson, and to delimit the scope of margin for error principles. The second is a criticism of Williamson's thesis that knowledge is not luminous, elaborating on previous work by Dokic & Egre (2004), based on the intuition that knowledge is modular and that a representation of this modularity is needed at the logical level in order to avoid the paradoxical conclusions which result rom Williamson's assumptions