Online research in philosophy

I analyze here some insights of Naming and Necessity that have become especially influential in the subsequent philosophy of language and metaphysics. Kripke's distinction between necessary truth and a priori truth plays a central role in two respects: it is fundamental to understand how essentialism has received support from Kripke's book; and it can be seen as closely related to another distinction between two independently plausible but confronted conceptions about what is the proposition expressed by a sentence.

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination.

One approach to the paradoxes of self-referential languages is to allow some sentences to lack a truth value (or to have more than one). Then assigning truth values where possible becomes a fixpoint construction and, following Kripke, this is usually carried out over a partially ordered family of three-valued truth-value assignments. Some years ago Matt Ginsberg introduced the notion of bilattice, with applications to artificial intelligence in mind. Bilattices generalize the structure Kripke used in a very natural way, while making the mathematical machinery simpler and more perspicuous. In addition, work such as that of Yablo fits naturally into the bilattice setting. What I do here is present the general background of bilattices, discuss why they are natural, and show how fixpoint approaches to truth in languages that allow self-reference can be applied. This is not new work, but rather is a summary of research I have done over many years.

Machine generated contents note: Introduction Alan Berger; Part I. Naming, Necessity, Identity, and A Priority: 1. Kripke on proper and general names Bernard Linsky; 2. Kripke on vacuous names and names in fiction Nathan Salmon; 3. Kripke on epistemic and modal possibility: two routes to the necessary a posteriori Scott Soames; 4. Possible world semantics and its philosophic foundations Robert Stalnaker; Part II. Formal Semantics, Truth, Philosophy of Math, and Philosophy of Logic: 5. Kripke models for modal logic and intuitionism John Burgess; 6. Kripke's theory of truth John Burgess; 7. Kripke on logicism, Wittgenstein, and de re beliefs about numbers Mark Steiner; 8. Kripke on the incoherency of adopting a logic Alan Berger; Part III. Language and Mind: 9. Kripke's new puzzle about belief and our principles of belief attribution Mark Richard; 10.; A note on Kripke's puzzle about belief Nathan Salmon; 11. Kripke's version of Wittgenstein: some conceptions and misconceptions George Wilson; 12. Kripke on color words and the primary, secondary quality distinction Mario Gomez-Torrente; Part IV. Philosophy of Mind and Philosophical Psychology: 13. Kripke's views on carteisianism and naturalism Sydney Shoemaker; 14. Kripke's critique of functionalism Jeff Buechner.

In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. (Martin and Woodruff independently developed this semantics, but not to the same extent as Kripke.) Kripke’s work suggests a number of related fixed-point theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixed-point theories. The current paper considers three natural ways to compare the various resulting theories of truth, and establishes the resulting relationships among these theories. The point is to get a sense of the lay of the land amid a variety of options. Our results will also provide technical fodder for the methodological remarks of the companion paper to this one.

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.

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀.

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀.

What is truth? -- Varieties of deflationism -- A defense of minimalism -- The value of truth -- A minimalist critique of Tarski -- Kripke's paradox of meaning -- Regularities, rules, meanings, truth conditions, and epistemic norms -- Semantics : what's truth got to do with it? -- The motive power of evaluative concepts -- Ungrounded reason -- The nature of paradox -- A world without 'isms' -- The quest for reality -- Being and truth -- Provenance of chapters.