There is no consensus as to whether a Liar sentence is meaningful or not. Still, a widespread conviction with respect to Liar sentences (and other ungrounded sentences) is that, whether or not they are meaningful, they are useless . The philosophical contribution of this paper is to put this conviction into question. Using the framework of assertoric semantics , which is a semantic valuation method for languages of self-referential truth that has been developed by the author, we show that certain (...) computational problems, called query structures , can be solved more efficiently by an agent who has self-referential resources (amongst which are Liar sentences) than by an agent who has only classical resources; we establish the computational power of self-referential truth . The paper concludes with some thoughts on the implications of the established result for deflationary accounts of truth. (shrink)

If we want to retain classical logic and standard syntax in light of the liar, we are forced to restrict the T-schema. The traditional philosophical justification for this is sentential – liar sentences somehow malfunction. But the standard formal way of implementing this is conditional, our T-sentences tell us that if “p” does not malfunction, then “p” is true if and only if p. Recently Bacon and others have pointed out that conditional T-restrictions like this flirt with incoherence. If we (...) want to keep the “malfunction” motivation, our only other option is to non-conditionally restrict the T-schema, but Field and others have given powerful philosophical and technical arguments against this kind of approach. Here I argue that if we really take the philosophical motivation for restricting T-sentences seriously, we can explain why conditional restrictions fail, answer Field's argument, and reason in a coherent way about truth using a non-conditional restriction strategy. This cracks the door open for a fully classical response to the liar and related paradoxes. In closing, I argue that, when properly understood, this kind of “restriction” is not really a restriction at all. If this is right, then the holy grail of liar studies (classical logic, naïve truth, and standard syntax) may yet be attainable. (Note for readers: when this paper was first put online, the typesetters had changed some instances of corners to brackets, leading to confusion. The paper has now been corrected online by PPR, so please make sure you have the corrected version before reading or citing.). (shrink)

In Hard Truths Elijah Millgram argues that ‘partial truth’ is pervasive and for that reason bivalence 1 may rarely be assumed. As classical logic assumes bivalence, and inferences from partial truths to partial truths are central to our thinking, classical logic rarely has application, the argument continues. And so we must develop an account of inference from partial truths to partial truths; as well, our metaphysics must be revised. While the novel positions of Hard Truths 2 raise many interesting challenges, (...) my focus is limited to offering an alternative to Millgram’s account of predication and to exploring some issues raised by his appeal to ‘fit’ in his definitions of truth and partial truth. 1. Introduction to partial truth Millgram defines truth and partial truth in terms of fit, as follows: Advancing a claim as fully true demands complete fit; assessing a claim as partially true allows for slack … but only so much of it. Claims are said to completely fit, slackly fit or not fit, the world, in §6.3. But, the word ‘fit’ is first introduced in §1.2, and it is experience or proto-experience, that fits into moulds provided by claims, thoughts, sentences and predicates. I won’t dwell on which fits what, 3 but on the notion of a ‘complete fit’ and the contrast with a ‘slack fit’; this discussion leads me to ponder the clarity and implications of appealing to these notions in definitions like the above. Millgram’s explanation of fit comes in the context of his account of predication, according to which, most predicates do not quite apply. It’s this that is primarily responsible for his promotion of partial truth. I sketch an alternative account of language whereby predicates do apply because language has come to life as a constituent of the things we do: …. (shrink)

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ?external? negation not-R; and the assertion of R in the pair of propositions ?it is (un)true that R? belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended. (shrink)