Two-dimensional Semantics Research Articles

Epistemic Two-Dimensionalism and the Epistemic Argument

One of Kripke's fundamental objections to descriptivism was that the theory misclassifies certain a posteriori propositions expressed by sentences involving names as a priori. Though nowadays very few philosophers would endorse a descriptivism of the sort that Kripke criticized, many find two-dimensional semantics attractive as a kind of successor theory. Because two-dimensionalism needn't be a form of descriptivism, it is not open to the epistemic argument as formulated by Kripke; but the most promising versions of two-dimensionalism are open to a close relative of that argument.

Vagueness, uncertainty and degrees of clarity

In this paper we compare different models of vagueness viewed as a specific form of subjective uncertainty in situations of imperfect discrimination. Our focus is on the logic of the operator “clearly” and on the problem of higher-order vagueness. We first examine the consequences of the notion of intransitivity of indiscriminability for higher-order vagueness, and compare several accounts of vagueness as inexact or imprecise knowledge, namely Williamson’s margin for error semantics, Halpern’s two-dimensional semantics, and the system we call Centered semantics. We then propose a semantics of degrees of clarity, inspired from the signal detection theory model, and outline a view of higher-order vagueness in which the notions of subjective clarity and unclarity are handled asymmetrically at higher orders, namely such that the clarity of clarity is compatible with the unclarity of unclarity.

Formalizing Medieval Logic: Suppositio, Consequentiae and Obligationes (review)

Reviewed by: Formalizing Medieval Logic: Suppositio, Consequentiae and Obligationes Mary Sirridge Catarina Dutilh-Novaes . Formalizing Medieval Logic: Suppositio, Consequentiae and Obligationes. Logic, Epistemology and the Unity of Science, 7. Dordrecht: Springer, 2007. Pp. xii + 316. Cloth, $169.00. The overarching aim of this excellent book is to demonstrate the common ground between medieval logic and logical theories of the twentieth century by analyzing some important medieval approaches to three important topics in medieval logic (suppositio, consequentiae, and obligationes) and then showing that in each case, once we determine what is really going on in the medieval theory, it can be formalized in such a way as to show how it resembles one or more developments in twentieth-century logical theory. Analysis in terms of modern logical theory has a lot to offer the study of medieval logical theories, the author claims, and twentieth-century theory can learn some interesting lessons from examining its medieval counterparts. Much of the material in these specific discussions has been presented in earlier versions in the author's published articles. "The Philosophy of Formalization" (part 4), however, is new. English translations are used in the text, though the Latin is supplied in the footnotes. Mature supposition theory (Ockham, Burley, and Buridan) is presented, not as a theory of reference or quantification, but as a unified theory of "algorithmic hermeneutics" (77); it allows us to calculate systematically the possible meanings of a proposition as a whole from the meanings of its component parts, with an eye to "distinguishing" the different contents that can be asserted (denotatur) by speaking, writing, or thinking the expression. Such multiple meanings may be due to allowing terms to be self-referential (material supposition) or to stand for species or the like (simple supposition), but also to the sorts of ambiguities of quantifier scope, modal operators, and the like that regularly turn up in discussions of fallacies and sophisms. Some results agree substantially with earlier analyses of Ockham's logic. Ockham's theory ultimately amounts to "an extensional theory of intensions" (31), and the distinction among personal, simple, and material supposition turns out to be more or less functionally independent of the modes of personal supposition. Still, it is a very interesting metatheoretical point that the two parts of the theory are co-ordinate strategies for achieving a single objective: the calculation of propositional meanings. Moreover, the understanding of the distinctive usage of denotatur is not only new and significant; it sounds right. Part 2, "Buridan's notion of consequentia," energetically pursues a search for twentieth-century near-relatives to Buridan's theory of entailment, an "intriguing combination of semantics and pragmatics" (113). Buridan's "token-based semantics" resembles the "two-dimensional semantics of Kaplan and Stalnaker," on which evaluation of the truth of a proposition involves both its content (abstract meaning) and its particular context of [End Page 469] utterance. Buridan's theory of consequence resembles the "hybrid" view of Shapiro (121); there is a broader notion of consequence (material consequence) similar to Etchemendy's "representational" approach to logical consequence that extends to entailments due to specific intensional content and to the inferability of one proposition from another due to pragmatic factors (inferences involving self-referential propositions or indexicals); a subset of these are valid on the basis of their logical form (formal consequence). One interesting side benefit of comparing Buridan's theory to its twentieth-century relatives is that it becomes clear that his token-based ontology, useful as it is for dramatizing self-reference, needs to be abandoned in favor of linguistic types and schemata if sense is be made of his theory of logical consequence, even if we allow him to avail himself of the notion of "equiform propositions." "Obligationes as Formal Games" (part 3) presents the theories of Burley, Swynshed, and Strode, in which the Respondent must accept or deny propositions put forward by the Opponent based on their relation to an initial supposition (positum), in modern game-theoretical terms as variants of a game of logical consistency, not as exercises for school boys, explorations of counterfactuals, or just a means of dealing with sophismata and Liar-like paradoxes. The game-theoretical analysis of Burley's theory...

Two-dimensional Truth

The paper identifies two major strands of truth theories, ontological and epistemological ones, and argues that both are of equal primacy and find their home within two-dimensional semantics. Contrary to received views, it argues further that epistemological truth theories operate on Lewisian possible worlds and ontological truth theories on Wittgensteinian possible worlds and that both are mediated by the so-called epistemic-ontic map the further specification of which is of utmost philosophical importance.

Two-dimensionalism and the “Knowing Which” Requirement

Two-dimensional semantics aims to eliminate the puzzle of necessary a posteriori and contingent a priori truths. Recently many argue that even assuming two-dimensional semantics we are left with the puzzle of necessary and a posteriori propositions. Stephen Yablo (Pacific Philosophical Quarterly, 81, 98–122, 2000) and Penelope Mackie (Analysis, 62(3), 225–236, 2002) argue that a plausible sense of “knowing which” lets us know the object of such a proposition, and yet its necessity is “hidden” and thus a posteriori. This paper answers this objection; I argue that given two-dimensional semantics you cannot know a necessary proposition without knowing that it is true.

Critical Notice of Scott Soames's Case against Two-Dimensionalism

This rich and stimulating book1 gives both a clear and cogent big picture of what is at stake in an important current philosophical debate, and some spirited and incisive arguments about the details. It is a defense of what its author regards as a major shift in our thinking about the sources of the intentionality of our speech and thought, and an attack on what he regards as a counterrevolution that is attempting to resurrect the view that was dominant prior to the shift in a new and more sophisticated form. The shift was “the Kripkean revolution,” led by Saul Kripke in Naming and Necessity, assisted by David Kaplan, Hilary Putnam, and Keith Donnellan. The counterrevolution exploits a piece of technical apparatus, two-dimensional modal semantics, which has its original application in the work of the revolutionaries themselves. The “ambitious two-dimensionalists,” who are the primary target of Soames’s critique, generalized the kind of two-dimensional semantics for indexical and demonstrative expressions developed by David Kaplan and used it to try “to reinstate descriptivism in the philosophy of language, internalism in the philosophy of mind, and some version of conceptualism in our understanding of modality in the face of the challenges that rocked these positions more than thirty years ago” (329). But, Soames argues, this attempt avoids “the deepest and most important philosophical consequences” (54) of Kripke’s work.

Buridan's Consequentia: Consequence and Inference Within a Token-Based Semantics

I examine the theory of consequentia of the medieval logician, John Buridan. Buridan advocates a strict commitment to what we now call proposition-tokens as the bearers of truth-value. The analysis of Buridan's theory shows that, within a token-based semantics, amendments to the usual notions of inference and consequence are made necessary, since pragmatic elements disrupt the semantic behaviour of propositions. In my reconstruction of Buridan's theory, I use some of the apparatus of modern two-dimensional semantics, such as two-dimensional matrices and the distinction between the context of formation and the context of evaluation of utterances.

Two-dimensional semantics and the articulation problem

David Chalmers's version of two-dimensional semantics is an attempt at setting up a unified semantic framework that would vindicate both the Fregean and the Kripkean semantic intuitions. I claim that there are three acceptable ways of carrying out such a project, and that Chalmers's theory does not coherently fit any of the three patterns. I suggest that the theory may be seen as pointing to the possibility of a double reading for many linguistic expressions (a double reading which, however, is not easily identified with straightforward semantic ambiguity).

Holisme, référence et irréductibilité du mental

ABSTRACTI examine in detail the argument vaguely suggested by Davidson to the effect that holism entails the irreducibility of the mental. I defend this argument against two objections often made against arguments that attempt to derive metaphysical theses from premises that concern our ordinary criteria for applying terms. I appeal to two-dimensional semantics to explain the links between these criteria and issues about reference and reduction. I show how the irreducibility of the mental follows from the holistic and flexible character of our criteria for attributing beliefs and intentional states.

The Rationalist Foundations of Chalmers's 2-D Semantics

In “Epistemic Two-Dimensional Semantics”, David Chalmers seeks to develop a version of 2-D semantics which can vindicate the rationalist claim that there are constitutive connections between meaning, possibility and a priority. Chalmers lays out different ways of filling in his preferred epistemic approach to 2-D semantics so as to avoid controversial philosophical assumptions. In these comments, however, I argue that there are some distinctively rationalist commitments in Chalmers's epistemic approach to 2-D semantics. I start by explaining why Chalmers's approach requires a canonical language that affords subjects accurate a priori access to the space of possibility. I then argue that traditional worries about rationalism will simply re-emerge as worries about whether there can be a canonical vocabulary and how we could come to recognize one if there were. The moral is that Chalmers's 2-D semantic framework builds in substantive metaphysical and epistemological commitments which stand in need of further defense.

Epistemic Two-Dimensional Semantics

Epistemic Two-Dimensional Semantics

Assertion Revisited: On the Interpretation of Two-Dimensional Modal Semantics

This paper concerns the applications of two-dimensional modal semantics to the explanation of the contents of speech and thought. Different interpretations and applications of the apparatus are contrasted. First, it is argued that David Kaplan's two-dimensional semantics for indexical expressions is different from the use that I made of a formally similar framework to represent the role of contingent information in the determination of what is said. But the two applications are complementary rather than conflicting. Second, my interpretation of the apparatus is contrasted with that of David Chalmers, Frank Jackson, and David Lewis. It is argued that this difference reflects a contrast between internalist and externa list approaches to the problem of intentionality.

Actual truth, possible knowledge

The well-known argument of Frederick Fitch, purporting to show that verificationism (= “Truth implies knowability”) entails the absurd conclusion that all the truths are known, has been disarmed by Dorothy Edgington's suggestion that the proper formulation of verificationism presupposes that we make use of anactuality operator along with the standardly invoked epistemic and modal operators. According to her interpretation of verificationism, the actual truth of a proposition implies that it could be known in some possible situation that the proposition holds in theactual situation. Thus, suppose that our object language contains the operatorA — “it is actually the case that ...” — with the following truth condition: ⊢ vA ⌽ iff ⊢w0⌽, wherew 0 stands for the designated world of the model — the actual world. Then we can formalize the verificationist claim as follows: $$A\phi \to \diamondsuit KA\phi .$$ However, while Edgington's introduction of the actuality operator dissolves Fitch's paradox, our troubles are not yet over. When we combine the truth-condition for the actuality operator with the standard truth-clauses for necessity and knowledge, formulated in terms of appropriate accessibility relations between worlds, it turns out that we once again have to accept the absurd claim: all actual truths must be known! Thus, the standard truth-conditions for the actuality operator and for the epistemic operator do not mix: when we try to combine them, they yield absurdities. To get a proper mix, we need a new semantics for actuality and knowledge. What is distinctive for our semantic proposal is that we give up the idea of afixed actual world (the designated point) and replace it with avariable perspective. The latter is contrasted with areference-world, which is being referred to, or described. We get what is sometimes called a two-dimensional semantics, in which a formula is being evaluated not just at one point (⊢ v ⌽) but at a pair of points (w⊢v⌽, wherew is a point of perspective, whilev is a point of reference). Intuitively, a formula says somethingabout the reference-world, butwhat it says is partially determined by the world of perspective. In particular, a formula such asA⌽ is true from a perspectivew at any reference-worldv iff it is true fromw's perspective atw itself. It turns out that, in a two-dimensional semantics, it is possible to formulate verificationism in a non-paradoxical way, provided we treat knowledge as a “variable-perspective” operator. The truth-condition of such an operator does not keep the perspective-world fixed. It does not involve an accessibility relation between worlds, but rather a relation between pairs of worlds:w⊢vK ⌽ iffw′⊢ν′⌽ everyw′ andν′ such that 〈w,ν〉E 〈w′, ν′〉. The relationE is meant to model epistemic uncertainty that originates from two different sources: The knower's information about the reference world may be more or less limited, and the same applies to his knowledge about the world that constitutes the point of perspective.