Charles B. Cross

I present and discuss two logical results. The first shows that a non-trivial counterfactual analysis exists for any contingent proposition that is false in at least two possible worlds. The second result identifies a set of conditions that are individually necessary and jointly sufficient for the success of a counterfactual analysis. I use these results to shed light on the question whether disposition ascribing propositions can be analyzed as Stalnaker-Lewis conditional propositions. The answer is that they can, but, in order for a counterfactual analysis to work, the antecedent and consequent must be related in a particular way, and David Lewis’s Time’s Arrow constraints on comparative world similarity must be relaxed. The upshot is that counterfactual analyses are easy to come by, in principle, even if not in practice. In that sense, it’s easy to be iffy.

Possible worlds talk is, in my view, a metaphor, and what makes it a good metaphor is its capacity to be extended and elaborated in fruitful ways. The essays in this dissertation all concern ways of adding structure to the basic apparatus of possible worlds semantics--the Kripke frame--so as to make it bear more fruit. ;One way of adding structure is to think of possible worlds as histories. In "A Theory of Conditionals in the Context of Branching Time" Richmond Thomason and Anil Gupta use this approach to place Stalnaker's theory of the conditional into a temporal context. They formulate two principles about the relation between historical necessity and the conditional--Past Predominance and the Edelberg Inference--and investigate how these principles place constraints on branching time models of conditional tense logic. In the second chapter of this dissertation I show that Past Predominance can be articulated using Kripke frames, and I prove a completeness theorem for such structures. ;Another way of elaborating the possible worlds metaphor is to introduce apparatus for describing relationships between or among possible worlds. Using this strategy Robert Stalnaker devised a semantic theory of conditional logic that involves a selection function, and in the third and fourth chapters of this dissertation I show how to use selection functions to organize intuitions about causation and ability, respectively. ;A third way of elaborating the possible worlds metaphor is to generalize the notion of a Kripke frame. Bas van Fraassen has shown that the most rational way of characterizing semantic entailment in probabilistic semantics involves replacing individual Popper functions with well-behaved sets of them: probabilistic model sets. The upshot of this is that Popper functions in probabilistic model sets are analogous to possible worlds in Kripke frames, and probabilistic model sets are themselves analogous to Kripke frames. In the final chapter of my dissertation I carry the analogy further by giving a probabilistic semantics for modal logic in which modal operators function as quantifiers over Popper functions in probabilistic model sets, thereby generalizing Kripke's semantics for modal logic along the lines van Fraassen sets out

In The Scientific Image B. C. van Fraassen argues that a theory of explanation ought to take the form of a theory of why-questions, and a theory of this form is what he provides. Van Fraassen's account of explanation is good, as far as it goes. In particular, van Fraassen's theory of why-questions adds considerable illumination to the problem of alternative explanations in psychodynamics. But van Fraassen's theory is incomplete because it ignores those classes of explanations that are answers not to why-questions but to how-questions. In this article I provide a unified theory of explanatory questions that comprehends both how-questions and why-questions, and I show that a question-theoretic approach to explanation can be defended independently of van Fraassen's programme of Constructive Empiricism.

I compare the failure of counterfactual dependence as a criterion of event causation to the failure of stochastic dependence as a criterion of causal law. Counterexamples to the stochastic analysis arise from cases of Simpson's Paradox, and Nancy Cartwright has suggested a way of transforming the stochastic analysis into something that avoids these counterexample. There is an analogical relationship between cases of Simpson's Paradox and cases of causal overdetermination. I exploit this analogical relationship to motivate my own view about the connection between counterfactuals and event causation. According to my view, counterfactuals and event causation are intimately connected by a principle which falls short of being an analysis but which, as I show, correctly handles not only overdetermination but common causes as well.

In A World of States of Affairs(Cambridge University Press, 1997) David Armstrong offers acomprehensive metaphysics based on the thesis that the world consistsof states of affairs. Among the entities postulated by Armstrong's theory are relations, including non-symmetrical relations, and whileArmstrong does not agree with Russell that all relations have adirection or definite order among their places, he does explicitlyacknowledge that the slots of a non-symmetrical relation have adefinite order or direction. I first show that non-symmetricalrelations pose a problem for Armstrong's theory by raising TheProblem of Converse Relations. I then argue that the bestresolution of this problem in the context of Armstrong's theoryinvolves adopting an analysis of the order or direction of a relationthat differs from the analysis that Russell assumes. I conclude bydiscussing a further problem facing Armstrong's ontology: TheProblem of Converse Relational Properties.

Discrepancies between an agent's goals and beliefs play an important, if implicit, role in determining what a rational agent is motivated to do. This is most obvious in cases where an agent achieves a complex goal incrementally and must deliberate anew as each milestone is reached. In such cases the concept of goal/belief discrepancy defines an appropriate space to which a degree-of-achievement yardstick can be applied. This paper presents soundness and completeness results concerning a logic for reasoning about goal/belief discrepancy, and it is suggested that a certain species of goal/belief discrepancy captures the concept of desire

I give a critique of the argument against the Identity of Indiscernibles found in Max Black's dialogue "The Identity of Indiscernibles". I begin by postulating and giving existence and individuation conditions for actually existent thought experiment characters on analogy with fictional characters as postulated in Peter van Inwagen's "Creatures of Fiction". I then show that Black's two-spheres thought experiment raises not one but two discernibility questions: 1) Is it true in the two-spheres thought experiment that there exist two indiscernible spheres? NO. 2) Is it true in the actual world that there are two indiscernible sphere-characters? YES.

In his well-known 1952 dialogue Max Black describes a counterexample to the Principle of the Identity of Indiscernibles (PII). The counterexample is a world containing nothing but two purportedly indiscernible iron spheres. Reflecting on Black's example, Robert Adams uses the possibility of a world containing two almost indiscernible spheres to argue for the possibility of the indiscernible spheres world. One of Adams's almost indiscernible spheres has a small impurity, and, Adams writes, "Surely... the absence of the impurity would not make such a universe impossible." The appeal to "surely" constitutes a gap in Adams's argument. This paper bridges the gap with a premise that exploits the counterfactual conditional and the related notion of causal independence. The paper then argues that causal independence bears in a similar way on an issue Nathan Salmon raises concerning Kripke's argument for the Essentiality of Origins.

[IMPORTANT CORRECTION - See end of abstract.] In Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability, Acta Philosophica Fennica 16 (1963), 153–167, Richard Montague shows that the use of a single syntactic predicate (with a context-independent semantic value) to represent modalities of alethic necessity and idealized knowledge leads to inconsistency. In A Note on Syntactical Treatments of Modality, Synthese 44 (1980), 391–395, Richmond Thomason obtains a similar impossibility result for idealized belief: under a syntactical treatment of belief, the assumption that idealized belief is deductively closed, together with certain other plausible conditions on idealized belief, imply that an ideal believer with consistent beliefs cannot believe the truth of Robinson's Arithmetic. In this essay I assert an impossibility result similar to Thomason's but which does not assume that belief is deductively closed or ideal in any other way. There are technical mistakes in the formulations of Lemmas 1-4 and Theorem 2, however. Corrections can be found in the Erratum section of the author's website.

Nonmonotonic consequence is the subject of a vast literature, but the idea of a nonmonotonic counterpart of logical inconsistency—the idea of a defeasible property representing internal conflict of an inductive or evidential nature—has been entirely neglected. After considering and dismissing two possible analyses relating nonmonotonic consequence and a nonmonotonic counterpart of logical inconsistency, this paper offers a set of postulates for nonmonotonic inconsistency, an analysis of nonmonotonic inconsistency in terms of nonmonotonic consequence, and a series of results showing that nonmonotonic inconsistency conforms to these postulates given the analysis of nonmonotonic inconsistency presented here and certain postulates for nonmonotonic consequence. The results presented here establish the interest of certain previously undiscussed postulates of nonmonotonic consequence. These results also show that nonmonotonicity, which has never seemed useful in the formulation of general principles governing nonmonotonic reasoning, is relevant to the positive characterization of nonmonotonic inference after all.

Stephen Barker argues that a possible worlds semantics for the counterfactual conditional of the sort proposed by Stalnaker and Lewis cannot accommodate certain examples in which determinism is true and a counterfactual Q > R is false, but where, for some P, the compound counterfactual P > (Q > R) is true. I argue that the completeness theorem for Lewis’s system VC of counterfactual logic shows that Stalnaker–Lewis semantics does accommodate Barker’s example, and I argue that its doing so should be understood as showing that the example is an exception to Lewis’s Time’s Arrow requirements

Despite the results of David Lewis, Peter Gärdenfors, and others, showing that imaging and classical conditionalization coincide only in the most trivial probabilistic models of belief revision, it turns out that imaging on a proposition A can always be described via Popper function conditionalization on a proposition that entails A. This result generalizes to any method of belief revision meeting certain minimal requirements. The proof is illustrated by an application of imaging in the context of the Monty Hall Problem.

Temporal necessity and the subjunctive conditional appear to be related by the principle of Past Predominance, according to which past similarities and differences take priority over future similarities and differences in determining the comparative similarity of alternative possible histories with respect to the present moment. R. H. Thomason and Anil Gupta have formalized Past Predominance in a semantics that combines selection functions with branching time; in this paper I show that Past Predominance can be formalized and axiomatized using ordinary possible worlds semantics (without branching time).

In “The Paradox of the Knower without Epistemic Closure”, MIND 110:319-33, 2001, I develop a version of the Knower Paradox which does not assume epistemic closure, and I use it to argue that the original Knower Paradox does not support an argument against epistemic closure. In “The Paradox of the Knower without Epistemic Closure?”, MIND 113:95-107, 2004, Gabriel Uzquiano, using his own result, argues that my rebuttal to the anti-closure argument is not successful. I respond here by arguing that in order to use Uzquiano’s result in an argument against closure, one must assume an implausible skepticism about arithmetic.

In ‘Two Spheres, Twenty Spheres, and the Identity of Indiscernibles,’ Della Rocca argues that any counterexample to the PII would involve ‘a brute fact of non-identity [. . .] not grounded in any qualitative difference.’ I respond that Adams's so-called Continuity Argument against the PII does not postulate qualitatively inexplicable brute facts of identity or non-identity if understood in the context of Kripkean modality. One upshot is that if the PII is understood to quantify over modal as well as non-modal properties, the qualitative explicability of numerical distinctness requires not the PII but a principle of the identity of necessary indiscernibles

The coherence of the whole truth is a presupposition of any holistic coherence theory of justification that postulates a positive connection between justification and truth, for unless the whole truth is itself systemically coherent there is no reason to look for systemic coherence when deciding whether one is justified in accepting a given body of beliefs as true. This paper develops a formal model of holistic evidential coherence and uses this model to formalize and defend the claim that the whole truth must be coherent in an evidential sense.

This essay corrects an error in the presentation of the Paradox of the Knowledge-Plus Knower, which is the variant of Kaplan and Montague’s Knower Paradox presented in C. Cross 2001: ‘The Paradox of the Knower without Epistemic Closure,’ MIND, 110, pp. 319–33. The correction adds a universally quantified transitivity principle for derivability as an additional assumption leading to paradox. This correction does not affect the status of the Knowledge-Plus paradox as a rebuttal to an argument against epistemic closure, since the quantified transitivity principle is true in the standard model of arithmetic and therefore innocuous.

I show the incompatibility of two theses: (a) to desire the truth of p amounts to believing a certain proposition about the value of p’s truth; (b) one cannot be said to desire the truth of p if one believes that p is true. Thesis (a), the Desire-As-Belief Thesis, has received much attention since the late 1980s. Thesis (b) is an epistemic variant of Socrates’ remark in the Symposium that one cannot desire what one already has. It turns out that (a) and (b) cannot both be true if it is possible for there to exist an agent who has a desire initially, say the desire for the truth of p, and then expands the corpus of propositions she believes to include p. This result provides a new route to the denial of (a).