Haim Gaifman

Vagueness, tolerance and contextual logic

The goal of this paper is a comprehensive analysis of basic reasoning patterns that are characteristic of vague predicates. The analysis leads to rigorous reconstructions of the phenomena within formal systems. Two basic features are dealt with. One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-increment of a walking distance is a walking distance). The other is the existence of borderline cases. The paper shows why these should be treated as different, though related phenomena. Tolerance is formally reconstructed within a proposed framework of contextual logic, leading to a solution of the Sorites paradox. Borderline-vagueness is reconstructed using certain modality operators; the set-up provides an analysis of higher order vagueness and a derivation of scales of degrees for the property in question.

Reasoning with Limited Resources and Assigning Probabilities to Arithmetical Statements

There are three sections in this paper. Thefirst is a philosophical discussion of the general problem of reasoning under limited deductive capacity. The second sketches a rigorous way of assigning probabilities to statements in pure arithmetic; motivated by the preceding discussion, it can nonetheless be read separately. The third is a philosophical discussion that highlights the shifting contextual character of subjective probabilities and beliefs.

Self-reference and the acyclicity of rational choice

Abstract Guided by an analogy between the logic of truth and the logic of a rationally choosing agent, I propose for the latter a principle of acyclicity, which blocks paradoxical self-referring reasoning. Certain decision-theoretic paradoxes are used to illustrate what can happen when acyclicity is violated. The principle, however, is argued for directly on grounds of coherence. Among its consequences are certain decision-theoretic rules, as well as a guiding line for setting Bayesian prior probabilities. From this perspective I discuss in the last two sections Prisoners Dilemma and Newcombs Paradox.

Deceptive updating and minimal information methods

The technique of minimizing information (infomin) has been commonly employed as a general method for both choosing and updating a subjective probability function. We argue that, in a wide class of cases, the use of infomin methods fails to cohere with our standard conception of rational degrees of belief. We introduce the notion of a deceptive updating method and argue that non-deceptiveness is a necessary condition for rational coherence. Infomin has been criticized on the grounds that there are no higher order probabilities that ‘support’ it, but the appeal to higher order probabilities is a substantial assumption that some might reject. Our elementary arguments from deceptiveness do not rely on this assumption. While deceptiveness implies lack of higher order support, the converse does not, in general, hold, which indicates that deceptiveness is a more objectionable property. We offer a new proof of the claim that infomin updating of any strictly-positive prior with respect to conditional-probability constraints is deceptive. In the case of expected-value constraints, infomin updating of the uniform prior is deceptive for some random variables but not for others. We establish both a necessary condition and a sufficient condition (which extends the scope of the phenomenon beyond cases previously considered) for deceptiveness in this setting. Along the way, we clarify the relation which obtains between the strong notion of higher order support, in which the higher order probability is defined over the full space of first order probabilities, and the apparently weaker notion, in which it is defined over some smaller parameter space. We show that under certain natural assumptions, the two are equivalent. Finally, we offer an interpretation of Jaynes, according to which his own appeal to infomin methods avoids the incoherencies discussed in this paper.

The sure thing principle, dilations, and objective probabilities☆

Abstract The common theme that unites the four sections is STP, the sure thing principle . But the paper can be divided neatly into two parts. The first, consisting of the first two sections, contains an analysis of STP as it figures in Savageʼs system and proposals of changes to that system. Also possibilities for partially ordered acts are considered. The second, consisting of the last two sections, is about imprecise probabilities, dilations and objective probabilities. Variants of STP are considered but this part is self-contained and can be read separately. The main claim there is that dilations, which can have extremely counterintuitive consequences, can be eliminated by a more careful analysis of the phenomenon. It outlines a proposal of how to do it. Here the concept of objective probabilities plays a crucial role.

ON ONTOLOGY AND REALISM IN MATHEMATICS

The paper is concerned with the way in which “ontology” and “realism” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections.

Context-Dependent Utilities

Savage’s framework of subjective preference among acts provides a paradigmatic derivation of rational subjective probabilities within a more general theory of rational decisions. The system is based on a set of possible states of the world, and on acts, which are functions that assign to each state a consequence. The representation theorem states that the given preference between acts is determined by their expected utilities, based on uniquely determined probabilities (assigned to sets of states), and numeric utilities assigned to consequences. Savage’s derivation, however, is based on a highly problematic well-known assumption not included among his postulates: for any consequence of an act in some state, there is a “constant act” which has that consequence in all states. This ability to transfer consequences from state to state is, in many cases, miraculous – including simple scenarios suggested by Savage as natural cases for applying his theory. We propose a simplification of the system, which yields the representation theorem without the constant act assumption. We need only postulates P1-P6. This is done at the cost of reducing the set of acts included in the setup. The reduction excludes certain theoretical infinitary scenarios, but includes the scenarios that should be handled by a system that models human decisions.

Erratum to: Vagueness, tolerance and contextual logic

During the production of the article, the author overlooked some errors. Below you will find the corrected text. Page 5, line 4-5 in the abstract should read: One is tolerance: the insensitivity of predicates to small changes in the objects of predication (a one-foot increment of a walking distance is a walking distance). Page 6, line 6-8 should read: A walking distance is still a walking distance if we increment it by one foot (but not by 5 miles); a child is still a child one hour later (but not 5 years later); and so on. As of page 7 line 3 the words borderline vagueness should read: borderline-vagueness Page 8, line 1, first paragraph of Sect. 2: the line should read: In ordinary usage, ‘vagueness’ is a broad term that covers an assortment of loosely connected linguistic phenomena: imprecision, fuzziness, ambiguity, obscurity, lack of specificity (hence the expression ‘vague generalities’), and their like. Page 8, the last line of the first paragraph of Sect. 2 should read: The indeterminateness, moreover, does not mean that the question is out of order (as would be a category mistake, like asking whether number 3 is happy); the question is appropriate, and unambiguous, but the semantics does not seem to decide it. Page 16, line 7-4 from the bottom should read: