Possible Worlds: A Neo-Fregean Alternative

Abstract

I outline a neo-Fregean strategy in the debate on the existence of possible worlds. The criterion of identity and the criterion of application are formulated. Special attention is paid to the fact that speakers do not possess proper names for worlds. A broadly Quinean solution is proposed in response to this difficulty.

Appendices

Appendix A

A general strategy for defining the category of singular terms was suggested by Dummett and later refined by Wright and Hale. The essence of their proposal consists in demarcating singular terms from other expressions in accordance with their inferential role. As the first step we segregate singular terms from substantival terms, such as ‘nothing’, ‘someone’, or ‘everything’, which can grammatically be put in name-positions. The tactic is to contrast directly the inferential behaviour of singular and substantival terms by applying the familiar rules of passage. For instance, if one says, ‘Jim is perfect’, we can infer that there is somebody who is perfect. But if one says, ‘Nobody is perfect’, we cannot infer that there is somebody who is perfect. This still leaves with the problem of higher generality where predicates can masquerade as singular terms. For example, if we say, ‘Jim is good at tennis’, you can still infer that there is something at which Jim is good. To deal with predicates we can use the Aristotelian intuition that qualities have their opposites, but substances do not. For example, the opposite of the predicate ‘① is white’ would be ‘① is black’, but there will not be any opposite for ‘Socrates’. The proposal can therefore be put as follows:²⁵ Syntactic criteria of singular termhood. An expression t functions as a singular term in a sentential context A(t) just in case:

• 1. The following conditions are satisfied:

  • (a) The inference is valid from A(t) to ‘Something is such that A(it)’.

  • (b) For some sentence B(t) the inference is valid from {A(t), B(t)} to ‘Something is such that A(it) and B(it)’.

  • (c) For some sentence B(t), the inference is valid from ‘It is true of t that A(it) or B(it)’ to the disjunction ‘A(t) or B(t)’.

• 2. There are no terms ‘opposite’ to t:

  $$ \neg\Sigma\alpha\Pi\beta((\alpha,\beta)\leftrightarrow \neg(t,\beta)), $$

  where the class β contains any expression which can be fitted into the sentential construct A(t) save those that fail the conditions of the first part.

The formalism of the second part demands some explanation. Suppose t is an expression that could be part of a sentence. Let ${\mathfrak{S}}()$ be a sentential function. We then use Σα and Πβ as substitutional quantifiers, aimed at replacing t and ${\mathfrak{S}}()$ respectively in the complete expression ${\mathfrak{S}}(t)$, where α and β comprise the classes of the grammatically legitimate substitutions of t and ${\mathfrak{S}}()$ respectively. The pair (α, β) designates the sentential construction containing one expression from α and one expression from β. The condition demarcates between singular terms and predicates: for every predicate it is possible to find an opposite predicate applied to the same quasi-singular term (i.e. the term certified by the three conditions of the first part of our definition). For genuine singular terms no such opposite term is to be found.

Let us see whether PW-terms qualify syntactically as singular terms. The first part of the test should not present difficulties. Consider, for example, condition (1b). Suppose that the following premisses hold:

• 1. Socrates is wise in w.

• 2. Socrates is fat in w.

There is no problem to infer:

• 3. Something is such that Socrates is fat and Socrates is wise in it.

Some doubts may persist about the second part of the test. Since worlds are the entities in which statements are true or false, could there be a world u in which every statement true in w is false? The answer is in the negative. We are guaranteed to have necessarily true statements, with the notion of necessity fixed appropriately, that will be true in every possible world.

Appendix B

For the axioms and rules of inference see Davies and Humberstone (1980).

Lemma 1

${\bf S5}{\mathcal{AF}}\vdash{\mathcal{FA}}\alpha\supset {\mathcal{FAFA}}\alpha$ (analogue of$\square\alpha \supset \square\square\alpha$).

Proof

$$ \begin{array}{ll} 1. {\mathcal{FA}}\alpha & \hbox{hyp}\\ 2. \square{\mathcal{FA}}\alpha &\hbox{1, Nec}\\ 3. {\mathcal{AFA}}\alpha &\hbox{2, A3, MP}\\ 4. {\mathcal{FAFA}}\alpha &\hbox{3, Fix}\\ 5. {\mathcal{FA}}\alpha\supset{\mathcal{FAFA}}\alpha & 1, 4, \supset {\bf I} \end{array} $$

Lemma 2

${\bf S5}{\mathcal{AF}}\vdash\neg{\mathcal{FA}}\alpha\supset {\mathcal{FA}}\neg{\mathcal{FA}}\alpha$ (analogue of$\lozenge\alpha \supset \square\lozenge\alpha$).

Proof

$$ \begin{array}{ll} 1. \neg{\mathcal{FA}}\alpha &\hbox{hyp}\\ 2. \square\neg{\mathcal{FA}}\alpha & \hbox{1, Nec}\\ 3. {\mathcal{A}}\neg{\mathcal{FA}}\alpha &\hbox{2, A3, MP}\\ 4. {\mathcal{FA}}\neg{\mathcal{FA}}\alpha &\hbox{3, Fix}\\ 5. \neg{\mathcal{FA}}\alpha\supset{\mathcal{FA}}\neg{\mathcal{FA}} \alpha & 1, 4, \supset {\bf I} \end{array} $$