Counterfactual Logic and the Necessity of Mathematics

Journal of Philosophical Logic - This paper is concerned with counterfactual logic and its implications for the modal status of mathematical claims. It is most directly a response to an ambitious...     

The Broadest Necessity

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.     

The Necessity of Conflict

No abstract available for this article.     

The Powerlessness of Necessity

This paper concerns anti‐Humean intuitions about connections in nature. It argues for the existence of a de re link that is not necessity. Some anti‐Humeans tacitly assume that metaphysical necessity can be used for all sorts of anti‐Humean desires. Metaphysical necessity is thought to stick together whatever would be loose and separate in a Hume world, as if it were a kind of universal superglue. I argue that this is not feasible. Metaphysical necessity might connect synchronically co‐existent properties—kinds and their essential features, for example—but it is difficult to see how it could also serve as the binding force for successions of events. That is, metaphysical necessity seems not to be fit for diachronic, causal affairs in which causal laws, causation, or dispositions are involved. A different anti‐Humean connection in nature has to do that job. My arguments focus mainly on a debate which has been the battleground for Humean vs. anti‐Humean intuitions for many decades—namely, the analysis of dispositional predicates—yet I believe (but do not argue here) that the arguments generalise to causation and causal laws straightforwardly.     

The Transcendental Necessity of Morality

David Gauthier tries to defend morality by showing that rational agents would choose to adopt a fundamental choice disposition that permits them to cooperate in prisoner's dilemmas. In this paper, I argue that Gauthier, rather than trying to work out a prudential justification for his favored choice disposition, should opt for a transcendental justification. I argue that the disposition in question is the product of socialization, not rational choice. However, only agents who are socialized in such a way that they acquire a disposition of this type could acquire the capacity to use language. Given the internal connection between language and thought, this means that no agent endowed with such a disposition could rationally choose to adopt another. Thus rational reflection by moral agents upon their own fundamental choice disposition will have no tendency to destabilize it.
"It is a necessary truth that people tend to do what they think they ought to do, for it is a necessary truth that people who occupy a linguistic position which means / ought to do A now , tend to do A. If they did not, the position they occupy could not mean I ought to do A now."
Wilfrid Sellars, "Some Reflections on Language Games."     

The Objectivity of Mathematics

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.     

The Miracle of Applied Mathematics

Synthese - Mathematics has a great variety ofapplications in the physical sciences.This simple, undeniable fact, however,gives rise to an interestingphilosophical problem:why should physical...     

The Role of Notations in Mathematics

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways.

     

A Posteriori Necessity as Restricted Necessity

I argue that conventionalists should construe a posteriori necessity as restricted necessity. I take Sidelle’s defence of the conventionalist explanation of a posteriori necessity against the contingency problem as the starting point. Sidelle construes a posteriori necessity as unrestricted necessity and then argues that a posteriori necessity is to be considered under a fixed convention (meaning) and is thus irrelevant to the contingent nature of our linguistic conventions. I offer a different solution to the contingency problem. I argue that conventionalists should construe a posteriori necessity as restricted necessity based upon a use-based meta-semantic picture. Restricted necessity is contingent in a broader sense and does not conflict with the contingency of conventions. My solution allows conventionalists to explain a posteriori necessity as restricted necessity. I show that the realist’s arguments for a posteriori necessity are unconvincing in regard to real modal features and a posteriori necessity as unrestricted necessity. We thus have good reason to continue to accept a posteriori necessity as restricted necessity under the conventionalist explanation.

     

The Necessity of a Non-Reductionist Science of Politics

The major tendency within the discipline of political science has been to try to achieve a science modeled on the natural sciences and mathematics, following the pattern of other social sciences. This tendency has led to many reductionistic efforts to explain political behavior in terms of one or more functions, such as power, linguistic, psychical, or the economic. The institutional community of government and citizens—the political community or state—is thus overlooked or reduced to one or more functions. In critique of this tendency, this paper shows how the work on entities and functions by the Dutch legal philosopher Herman Dooyeweerd illumines the errors of reductionism and points the way to a multi-functional entity science of the political community.