Nuel Belnap

(? - 2024)

This paper offers a lighthearted presentation of some of the chief ideas about truth that are shared by theories similar to those of Kripke, Herzberger, and Gupta. The problem is to explain the concept of truth for a language that contains its own truth predicate. The proposal of these theories is that one can unwind the tangles that threaten by invoking a transfinite series of stages of semantic reflection as one ascends the ordinals. The presentation emphasizes how each stage begins, to the extent that it can, with the fruits of semantic reflection at earlier stages. Special attention is given to clarifying how these theories about truth treat the three chief kinds of ordinal numbers: the initial ordinal, successor ordinals, and limit ordinals.

We are grateful to Professor Robert Koons for his excellent, and generous, review (henceforth KR) of our book The Revision Theory of Truth (henceforth RTT). Koons provides in KR a welcome guide to our RTT, and he puts forward objections that deserve serious consideration. In this note we shall respond only to his principal objection.' This objection, which is developed on pp. 625 — 628 of KR, calls into question our main thesis. As we argue below, however, the objection is not successful. We should forewarn the reader that this note is not self-contained. It presupposes familiarity with RTT (primarily, Chapter 4) and with KR. The main thesis of RTT is that truth is a circular concept. We argued that the Tarski biconditionals, read as partial definitions, constitute an intensionally adequate definition of truth. In other words, if T is a predicate defined by the Tarski-style infinitistic definition (I).

There is no EPR-like funny business if (contrary to apparent fact)our world is as indeterministic as you wish, but is free from theEPR-like quantum mechanical phenomena such as is sometimes described interms of superluminal causation or correlation between distant events.The theory of branching space-times can be used to sharpen thetheoretical dichotomy between EPR-like funny business and noEPR-like funny business. Belnap (2002) offered two analyses of thedichotomy, and proved them equivalent. This essay adds two more, bothconnected with Reichenbachs principle of the common cause, theprinciple that sends us hunting for a common-causal explanation ofdistant correlations. The two previous ideas of funny business and thetwo ideas introduced in this essay are proved to be all equivalent,which increases ones confidence in the stability of (and helpfulnessof) the BST analysis of the dichotomy between EPR-like funny businessand its absence.

Using Aristotle's well-known sea battle as our example, we offer a precise, intelligible analysis of future contingent assertions in the presence of indeterminism. After explaining our view of the problem, we present a picture of indeterminism in the context of a tree ofbranching histories. There follows a brief description ofthe semantic bases for our double-time-reference theory of future contingents. We then set out our account. Before concluding, we discuss some ramifications of, and alternatives to, a double-time-reference approach to the problem of future contingents. There are some technical ideas at the foundation of our analysis, ideas of which most philosophers are largely ignorant; on our view, in the absence of mastery ofthese ideas it is quite impossible to speak responsibly about either indeterminism or free will

Tableau formulations are given for the relevance logics E (Entailment), R (Relevant implication) and RM (Mingle). Proofs of equivalence to modus-ponens-based formulations are vialeft-handed Gentzen sequenzen-kalküle. The tableau formulations depend on a detailed analysis of the structure of tableau rules, leading to certain global requirements. Relevance is caught by the requirement that each node must be used; modality is caught by the requirement that only certain rules can cross a barrier. Open problems are discussed.

A structure is described that can serve as a foundation for a semantics for a modal agentive construction such as sees to it that Q ([ stit: Q]). The primitives are Tree,,Instant, Agent, choice. Eleven simple postulates governing this structure are set forth and motivated. Tree and encode a picture of branching time consisting of moments gathered into maximal chains called histories. Instant imposes a time-like ordering. Agent consists of agents, and choice assigns to each agent and each moment in Tree a set of possible choices, where each possible choice is a set of histories. All of these ingredients are referred to in the semantics suggested for [ stit: Q]. The most complex part of the discussion is the motivation for the definition of what it means for a typically non-terminating chain of moments jointly to witness the truth of [ stit: Q] at a moment.The paper begins with an informal account of the Refref conjecture, which says that the only way to refrain from refraining from seeing to something is to see to it. The paper ends with a consideration of an argument of Prior's that in a certain sense contemplation and action are inconsistent.

1. Rescher 1964 — henceforth HR — proposes a way of reasoning from a set of hypotheses which may include both some of our beliefs and also hypotheses contradicting those beliefs. The aim of this paper is to point out what I take to be a fault in Rescher’s proposal, and to suggest a modification of it, using a nonclassical logic, which avoids that fault. The paper neither attacks nor defends the broader aspects of Rescher’s proposal, but merely assumes that it is at least prima facie worthwhile and therefore worthy of amendment; consequently, I shall try to tinker as little as possible. In particular, the use of a nonclassical logic which I propose does not replace any use by HR of classical logic — in those places where Rescher is classical, I shall be classical, too. (Instead, the amendment introduces a nonclassical logic at a point where HR uses no logic at all.).