Infectious logics are systems that have a truth-value that is assigned to a compound formula whenever it is assigned to one of its components. This paper studies four-valued infectious logics as the basis of transparent theories of truth. This take is motivated as a way to treat different pathological sentences differently, namely, by allowing some of them to be truth-value gluts and some others to be truth-value gaps and as a way to treat the semantic pathology suffered by at least (...) some of these sentences as infectious. This leads us to consider four distinct four-valued logics: one where truth-value gaps are infectious, but gluts are not; one where truth-value gluts are infectious, but gaps are not; and two logics where both gluts and gaps are infectious, in some sense. Additionally, we focus on the proof theory of these systems, by offering a discussion of two related topics. On the one hand, we prove some limitations regarding the possibility of providing standard Gentzen sequent calculi for these systems, by dualizing and extending some recent results for infectious logics. On the other hand, we provide sound and complete four-sided sequent calculi, arguing that the most important technical and philosophical features taken into account to usually prefer standard calculi are, indeed, enjoyed by the four-sided systems. (shrink)

In Kripke’s classic paper on truth it is argued that by adding a new semantic category different from truth and falsity it is possible to have a language with its own truth predicate. A substantial problem with this approach is that it lacks the expressive resources to characterize those sentences which fall under the new category. The main goal of this paper is to offer a refinement of Kripke’s approach in which this difficulty does not arise. We tackle this characterization (...) problem by letting certain sentences belong to more than one semantic category. We also consider the prospect of generalizing this framework to deal with languages containing vague predicates. (shrink)

In Saving Truth from Paradox, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. I (...) close with some general considerations on the use of revision-theoretic methods in theories of truth. (shrink)

Kripke’s theory of truth succeeded in providing a trivalent semantics for a language that contains its own truth predicate and means of self-reference; but it did so by radically restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g. This very sentence is not true) receives the indeterminate truth value; but the logic cannot express the fact that the Liar is something other than true: in order to do so, a weak negation not* would be needed, but (...) it would also make the logic inconsistent (because the ‘Super Liar’ This very sentence is not* true could not be assigned any truth value). Taking a hint from the quantificational form of the problematic sentences (… is something other than true), we define a hierarchy of negations which each quantifies over a domain of truth values, assimilated to ordinals. The resulting logic has as many negations and truth values as there are ordinals. Unlike Kripke’s logic, it enjoys a form of expressive completeness. And although the logic is not monotonic, we show that under broad conditions we can construct a variety of fixed points; one of them emulates Kripke’s ‘least fixed point’, while another one assigns a different truth value to each Super Liar. (shrink)

We show how to construct certain L M, T -type interpreted languages, with each such language containing meaningfulness and truth predicates which apply to itself. These languages are comparable in expressive power to the L T -type, truth-theoretic languages first considered by Kripke, yet each of our L M, T -type languages possesses the additional advantage that, within it, the meaninglessness of any given meaningless expression can itself be meaningfully expressed. One therefore has, for example, the object level truth (and (...) meaningfulness) of the claim that the strengthened Liar is meaningless. (shrink)

We show how to construct certain " $[Unrepresented Character]_{M,T}$ -type" interpreted languages, with each such language containing meaningfulness and truth predicates which apply to itself. These languages are comparable in expressive power to the $[Unrepresented Character]_{T}$ -type, truth-theoretic languages first considered by. Kripke, yet each of our $[Unrepresented Character]_{M,T}$ -type languages possesses the additional advantage that, within it, the meaninglessness of any given meaningless expression can itself be meaningfully expressed. One therefore has, for example, the object level truth (and meaningfulness) (...) of the claim that the strengthened Liar is meaningless. (shrink)

Nonclassical theories of truth that take truth to be transparent have some obvious advantages over any classical theory of truth. But several authors have recently argued that there’s also a big disadvantage of nonclassical theories as compared to their “external” classical counterparts: proof-theoretic strength. While conceding the relevance of this, the paper argues that there is a natural way to beef up extant internal theories so as to remove their proof-theoretic disadvantage. It is suggested that the resulting internal theories are (...) preferable to their external counterparts. (shrink)

Bounds consequence provides an interpretation of a multiple-conclusion consequence relation in which the derivability of a sequent is understood as the claim that it is conversationally out-of-bounds to take a position in which each member of Γ is asserted while each member of Δ is denied. Two of the foremost champions of bounds consequence—Greg Restall and David Ripley—have independently indicated that the shape of the bounds in question is determined by conversational practice. In this paper, I suggest that the standard (...) treatments of bounds consequence have focused heavily on the matter of veridicality at the expense of ignoring other features by which conversational bounds are set, prime among them being the matter of content or subject-matter. Furthermore, I argue that the semantic behavior of propositions containing “monstrous” content—content whose introduction is inappropriate to a context independently of veridical considerations—leads to a weak Kleene account of bounds consequence. (shrink)

Systems ofillative logicare logical calculi formulated in the untypedλ-calculus supplemented with certain logical constants.1In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn’t so, by providing a nonempty class (...) of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox. (shrink)