A logic is said to be paraconsistent if it does not allow everything to follow from contradictory premises. There are several approaches to paraconsistency. This paper is concerned with several philosophical positions on paraconsistency. In particular, it concerns three ‘schools’ of paraconsistency: Australian, Belgian and Brazilian. The Belgian and Brazilian schools have raised some objections to the dialetheism of the Australian school. I argue that the Australian school of paraconsistency need not be closed down on the basis of the Belgian (...) and Brazilian schools’ objections. In the appendix of the paper, I also argue that the Brazilian school’s view of logic is not coherent. (shrink)

This paper first offers a standard modal extension of dialetheic logics that respect the normal semantics for negation and conjunction, in an attempt to adequately model absolutism, the thesis that there are true contradictions at metaphysically possible worlds. It is shown, however, that the modal extension has unsavoury consequences for both absolutism and dialetheism. While the logic commits the absolutist to dialetheism, it commits the dialetheist to the impossibility of the actual world. A new modal logic AV is then proposed (...) which avoids these unsavoury consequences by invalidating the interdefinability rules for the modal operators with the use of two valuation relations. However, while using AV carries no significant cost for the absolutist, the same isn't true for the dialetheist. Although using AV allows her to avoid the consequence that the actual world is an impossible world, it does so only on the condition that the dialetheist admits that she cannot give a dialetheic solution to all self-refe.. (shrink)

Since its first appearance in 1966, the notion of a supervaluation has been regarded by many as a powerful tool for dealing with semantic gaps. Only recently, however, applications to semantic gluts have also been considered. In previous work I proposed a general framework exploiting the intrinsic gap/glut duality. Here I also examine an alternative account where gaps and gluts are treated on a par: although they reflect opposite situations, the semantic upshot is the same in both cases--the value of (...) some expressions is not uniquely defined. Other strategies for generalizing supervaluations are considered and some comparative facts are discussed. (shrink)

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

One might ask of two or more texts—what can be inferred from them, taken together? If the texts happen to contradict each other in some respect, then the unadorned answer of standard logic is EVERYTHING. But it seems to be a given that we often successfully reason with inconsistent information from multiple sources. The purpose of this paper is to attempt to develop an adequate approach to accounting for this given.

There are two ways of understanding the notion of a contradiction: as a conjunction of a statement and its negation, or as a pair of statements one of which is the negation of the other. Correspondingly, there are two ways of understanding the Law of Non-Contradiction (LNC), i.e., the law that says that no contradictions can be true. In this paper I offer some arguments to the effect that on the first (collective) reading LNC is non-negotiable, but on the second (...) (distributive) reading it is perfectly plausible to suppose that LNC may, in some rather special and perhaps undesirable circumstances, fail to hold. (shrink)

According to Fogelin’s account of deep disagreements, disputes caused by a clash in framework propositions are necessarily rationally irresolvable. Fogelin’s thesis is a claim about real-life, and not purely hypothetical, arguments: there are such disagreements, and they are incapable of rational resolution. Surprisingly then, few attempts have been made to find such disputes in order to test Fogelin’s thesis. This paper aims to rectify that failure. Firstly, it clarifies Fogelin’s concept of deep disagreement and shows there are several different breeds (...) of such disagreements. Thus, to fully assess Fogelin’s thesis, it will be necessary to seek out cases of each breed to evaluate their rational irresolvability. Secondly, it begins this task by looking at a significant debate within the logical literature over the truth of contradictions. We demonstrate that, while the debate exemplifies a breed of deep disagreement, the parties involved can supply one another with rationally compelling reasons. (shrink)

Liberals who want to support multiculturalism need to be able to justify the parental authority to instill cultural value systems or worldviews into children. However, such authority may be at odds with liberal demands that citizens be autonomous. This paper argues that parents do not have the legitimate authority to instill in their children a specific value system, contrary to the complex and intriguing arguments of Robert Noggle (2002). Noggle’s argument, which draws heavily on key ideas in Rawls’ theory of (...) justice, is that children are not moral agents and that parents are in a special kind of fiduciary relationship vis-à-vis their children. Noggle’s position is contrasted with the more limited conception of parental authority advanced by David Archard (2002). I argue that we can accept that parents are agents of their children, but contra Noggle, this does not entitle them to impose their parochial value systems onto their children. I argue that while children have an interest in acquiring values, they do not have an interest in acquiring a value system. (shrink)

David Lewis has argued that impossible worlds are nonsense: if there were such worlds, one would have to distinguish between the truths about their contradictory goings-on and contradictory falsehoods about them; and this--Lewis argues--is preposterous. In this paper I examine a way of resisting this argument by giving up the assumption that ‘in so-and-so world’ is a restricting modifier which passes through the truth-functional connectives The outcome is a sort of subvaluational semantics which makes a contradiction ‘A & ~A’ false (...) even when both ‘A’ and ‘~A’ are true, just as supervaluational semantics makes a tautology ‘A v ~A’ true even when neither ‘A’ nor ‘~A’ are. Connections with discussive logics and complications of the account are discussed, and some general morals are drawn. (shrink)

According to Suszko’s Thesis, there are but two logical values, true and false. In this paper, R. Suszko’s, G. Malinowski’s, and M. Tsuji’s analyses of logical twovaluedness are critically discussed. Another analysis is presented, which favors a notion of a logical system as encompassing possibly more than one consequence relation. [A] fundamental problem concerning many-valuedness is to know what it really is. [13, p. 281].

According to Fogelin’s account of deep disagreements, disputes caused by a clash in framework propositions are necessarily rationally irresolvable. Fogelin’s thesis is a claim about real-life, and not purely hypothetical, arguments: there are such disagreements, and they are incapable of rational resolution. Surprisingly then, few attempts have been made to find such disputes in order to test Fogelin’s thesis. This paper aims to rectify that failure. Firstly, it clarifies Fogelin’s concept of deep disagreement and shows there are several different breeds (...) of such disagreements. Thus, to fully assess Fogelin’s thesis, it will be necessary to seek out cases of each breed to evaluate their rational irresolvability. Secondly, it begins this task by looking at a significant debate within the logical literature over the truth of contradictions. We demonstrate that, while the debate exemplifies a breed of deep disagreement, the parties involved can supply one another with rationally compelling reasons. (shrink)

This paper extends David Lewis' result that all first degree modal logics are complete to weakly aggregative modal logic by providing a filtration-theoretic version of the canonical model construction of Apostoli and Brown. The completeness and decidability of all first-degree weakly aggregative modal logics is obtained, with Lewis's result for Kripkean logics recovered in the case k = 1.

This paper articulates the structure of a two species of weakly aggregative necessity in a common idiom, neighbourhood semantics, using the notion of a k-filter of propositions. A k-filter on a non-empty set I is a collection of subsets of I which contains I, is closed under supersets on I, and contains ∪{Xi ≤ Xj : 0 ≤ i < j ≤ k} whenever it contains the subsets X0,…, Xk. The mathematical content of the proof that weakly aggregative modal logic (...) is complete relative to k-ary frame theory, the standard semantic idiom for weakly aggregative modal logic is presented in language-independent terms as a representation theorem for k-filters: every non-trivial k-filter is included in the union of ≤ k non-trivial filters. The elementary theory of k-filters is developed and then applied in the form of an ultrafilter extension result for k-ary frame theory. (shrink)