Nuel Belnap

(? - 2024)

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.).

The first section (§1) of this essay defends reliance on truth values against those who, on nominalistic grounds, would uniformly substitute a truth predicate. I rehearse some practical, Carnapian advantages of working with truth values in logic. In the second section (§2), after introducing the key idea of auxiliary parameters (§2.1), I look at several cases in which logics involve, as part of their semantics, an extra auxiliary parameter to which truth is relativized, a parameter that caters to special kinds of sentences. In many cases, this facility is said to produce truth values for sentences that on the face of it seem neither true nor false. Often enough, in this situation appeal is made to the method of supervaluations, which operate by “quantifying out” auxiliary parameters, and thereby produce something like a truth value. Logics of this kind exhibit striking differences. I first consider the role that Tarski gives to supervaluation in first order logic (§2.2), and then, after an interlude that asks whether neither-true-nor-false is itself a truth value (§2.3), I consider sentences with non-denoting terms (§2.4), vague sentences (§2.5), ambiguous sentences (§2.6), paradoxical sentences (§2.7), and future-tensed sentences in indeterministic tense logic (§2.8). I conclude my survey with a look at alethic modal logic considered as a cousin (§2.9), and finish with a few sentences of “advice to supervaluationists” (2.10), advice that is largely negative. The case for supervaluations as a road to truth is strong only when the auxiliary parameter that is “quantified out” is in fact irrelevant to the sentences of interest—as in Tarski’s definition of truth for classical logic. In all other cases, the best policy when reporting the results of supervaluation is to use only explicit phrases such as “settled true” or “determinately true,” never dropping the qualification.

permits a sound and rigorously definable notion of ‘originating cause’ or causa causans—a type of transition event—of an outcome event. Mackie has famously suggested that causes form a family of ‘inus’ conditions, where an inus condition is ‘an insufficient but non-redundant part of an unnecessary but sufficient condition’. In this essay the needed concepts of BST theory are developed in detail, and it is then proved that the causae causantes of a given outcome event have exactly the structure of a set of Mackie inus conditions. The proof requires the assumption that there is no EPR-like ‘funny business’. This seems enough to constitute a theory of ‘causation’ in at least one of its many senses. Introduction The cement of the universe Preliminaries 3.1 First definitions and postulates 3.2 Ontology: propositions 3.3 Ontology: initial events 3.4 Ontology: outcome events 3.5 Ontology: transition events 3.6 Propositional language applied to events Causae causantes 4.1 Causae causantes are basic primary transition events 4.2 Causae causantes of an outcome chain 4.3 No funny business Causae causantes and inns and inus conditions 5.1 Inns conditions of outcome chains: not quite 5.2 Inns conditions of outcome chains 5.3 Inns conditions of scattered outcome events 5.4 Inus conditions for disjunctive outcome events 5.5 Inns and inus conditions of transition events Counterfactual conditionals Appendix: Tense and modal connectives in BST.

Popper’s introduction of ‘‘propensity’’ was intended to provide a solid conceptual foundation for objective single-case probabilities. By considering the partly opposed contributions of Humphreys and Miller and Salmon, it is argued that when properly understood, propensities can in fact be understood as objective single-case causal probabilities of transitions between concrete events. The chief claim is that propensities are well-explicated by describing how they fit into the existing formal theory of branching space-times, which is simultaneously indeterministic and causal. Several problematic examples, some commonsense and some quantum-mechanical, are used to make clear the advantages of invoking branching space-times theory in coming to understand propensities.

“Flat pre-semantics” lets each parameter of truth be considered separately and equally, and without worrying about grammatical complications. This allows one to become a little clearer on a variety of philosophical-logical points, such as the usefulness of Carnapian tolerance and the deep relativity of truth. A more definite result of thinking in terms of flat pre-semantics lies in the articulation of some instructive ways of categorizing operations on meanings in purely logical terms in relation to various parameters of truth ; namely, closing vs. leaving open, local vs. translocal, and anchored vs. unanchored. Basic relations among these categories are established.

“Branching space-times” is intended as a representation of objective, event-based indeterminism. As such, BST exhibits both a spatio-temporal aspect and an indeterministic “modal” aspect of alternative possible historical courses of events. An essential feature of BST is that it can also represent spatial or space-like relationships as part of its relativistic theory of spatio-temporal relations; this ability is essential for the representation of local indeterminism. This essay indicates how BST might be seen to grow out of Newton ’s deterministic and non-relativistic theory by two independent moves: Taking account of indeterminism, and attending to spatio-temporal relationships in a spirit derived from Einstein’s theory of special relativity. Since and are independent, one can see that there is room for four theories: Newtonian determinism, branching time indeterminism, relativistic determinism, and branching space-times indeterminism.

Branching space-time is a simple blend of relativity and indeterminism. Postulates and definitions rigorously describe the causal order relation between possible point events. The key postulate is a version of everything has a causal origin; key defined terms include history and choice point. Some elementary but helpful facts are proved. Application is made to the status of causal contemporaries of indeterministic events, to how splitting of histories happens, to indeterminism without choice, and to Einstein-Podolsky-Rosen distant correlations.