Alethic Modal Logic Research Articles

Normative Consistency: An Xstit Account

In this paper we take inspiration from a couple of authors on how to think about normative (in)consistency, and then show how to conceive of normative inconsistency in an xstit framework. One view on normative inconsistency is from von Wright, and the other from Hamblin. These two accounts share a conception of normative inconsistency, but their formal frameworks are very different. We propose a way to get the best of both views on normative inconsistency by using an xstit framework, mixed with a version of Anderson’s reduction of deontic logic to alethic modal logic. We consider variations on those ideas and relate it to a work of Ruth Barcan Marcus.

Remarks on the logic of imagination. A step towards understanding doxastic control through imagination

Imagination has recently attracted considerable attention from epistemologists and is recognized as a source of belief and even knowledge. One remarkable feature of imagination is that it is often and typically agentive: agents decide to imagine. In cases in which imagination results in a belief, the agentiveness of imagination may be taken to give rise to indirect doxastic control and epistemic responsibility. This observation calls for a proper understanding of agentive imagination. In particular, it calls for the development of a semantics of imagination ascriptions. In the present paper an earlier suggestion by Ilkka Niiniluoto for a logic of imagination is considered. This proposal does not capture the agentive nature of imagination, and an alternative semantics is suggested. The new semantics combines the modal logic of agency with the neighbourhood semantics from alethic modal logic.

Truth Values, Neither-true-nor-false, and Supervaluations

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.

Three characterizability problems in deontic logic

We consider an infinite hierarchy of systems of Alethic Modal Logic with so-called Levels of Perfection, and add to them suitable definitions of such interesting deontic categories as those of supererogation, offence, conditional obligation and conditional permission. We then state three problems concerning the proper characterization of the resulting logic(s) for our defined notions, and discuss two of these problems in some detail.

Reply to Shogenji on relativism

In A Consistent Relativism (Hales 1997) 1 offered a formal logical system (RL) for truth relativism, defining two new operators for absolutely and relatively. This system was intended to be, as the title of the paper suggested, a consistent relativist logic, not the only possible one. I modeled the system on the alethic modal logic S5 for two reasons. The first is that the traditional self-refutation problem of global relativism can be beautifully and compellingly demonstrated in an S5 framework in a way that conforms to people's gut worries about relativism. The second reason is that S5 is, for better or worse, the default modal system philosophers seem to accept. Thus I hoped that the relativist logic I presented spoke a logical lingua franca. Shogenji (1997) argues that the self-refutation proof I gave fails if one uses an S4 relativist logic instead of S5. This should come as no surprise. The choice of logical system is crucial for all proofs. Even modus ponens fails if we abandon traditional logic for certain relevance logics. Two questions immediately arise: (1) should an S4 relativist logic be employed instead of S5? and (2) can everything is relative be proven false in an S4 relativist logic? Shogenji argues that the answer to the first question is yes. His argument proceeds by questioning the symmetry of the commensurability relation, a requirement of RL. To my mind, however, the rejection of symmetry is too hasty, and implicitly relies on a dubious and undefended concept of a perspective. Some concept of a perspective is needed by all parties, but Shogenji's realist vs. idealist example trades on perspectives as somehow being established by the intentions of individual persons, which seems suspicious. No matter. The answer to question (2) is also yes, as Shogenji admits in footnote 5. Since everything is relative is false in both an S5 relativist logic (like RL) and an S4 one, it is pretty hard to see the great advantage the defender of everything is relative gains by going the S4 route. His central thesis is false in both. In fact, I agree with Shogenji that everything is relative turns out false no matter which modal system is supplemented with absolutely and relatively operators. What more is needed to persuade us that everything is relative is false and to look about for which logical system will permit the strongest relativistic claims? Strangely, Shogenji is still not convinced. Instead, he concludes that this only means that a glo-

The Modal Ontological Argument

We know more today about the second, or so-called 'modal', version of St. Anselm's ontological argument than we did when Charles Hartshorne and Norman Malcolm brought it to the attention of philosophers some years ago. 1 But there is still much to be learned. Criticisms of the modal ontological argument focus on its premises, namely (i) the claim that perfection, or maximal greatness, implies necessary existence, and (ii) the claim that the existence of a perfect being is logically possible. Less attention has been given by the critics of the argument to the modal principles that allow one to get from these premises to the conclusion that a perfect being exists. The modal ontological argument is valid in some standard systems of alethic modal logic, e.g., S5 (which is assumed by advocates of the argument like Hartshorne and Plantinga), and in at least one weaker system. But it is not valid in all. For example, it is not valid in T (Von Wright's M) or any Lewis system weaker than S5, including S4. What should all this mean for our assessment of the argument? Is choosing a modal system in theological contexts as innocent as choosing hors d'oeuvres? Can it be so with a proof for the existence of God at stake? Reflecting on these questions leads to two important, though often neglected, objections to the modal ontological argument (hereafter the 'modal OA'). The narrower aim of this paper is to explore these two objections; the wider aim, continuous with the narrower one, is to describe the present status quaestionis concerning the argument. As I see it, there are three central, as yet unresolved, problems that stand in the way of accepting the modal OA. Two of these problems are associated with the two objections just mentioned. The third problem will be mentioned here, but not discussed at length, because it is better understood than the two problems I shall discuss and has been widely discussed in the literature. I am not myself sure how to respond to all of these

Consideraciones semánticas sobre lógica deóntica, con especial referencia a la jurisprudencia

Resumen

Alan Ross Anderson and Omar Khayyam Moore. The formal analysis of normative concepts. American sociological review, vol. 22 (1957), pp. 9–17. - Alan Ross Anderson. A reduction of deontic logic to alethic modal logic. Mind, n.s. vol. 67 (1958), pp. 100–103. - Alan Ross Anderson. The logic of norms. Logique et analyse (Louvain), n.s. vol. 1 (1958), pp. 84–91.

Alan Ross Anderson and Omar Khayyam Moore. The formal analysis of normative concepts. American sociological review, vol. 22 (1957), pp. 9–17. - Alan Ross Anderson. A reduction of deontic logic to alethic modal logic. Mind, n.s. vol. 67 (1958), pp. 100–103. - Alan Ross Anderson. The logic of norms. Logique et analyse (Louvain), n.s. vol. 1 (1958), pp. 84–91.