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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.

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Notes

  1. 1.

    This may be Stalnaker’s ultimate view. See Stalnaker (1996).

  2. 2.

    For some doubts about the transition from #3 to #4 of this argument see Girle (2003: 160–163).

  3. 3.

    See Frege (1980: §60) and Dummett (1981: 369).

  4. 4.

    See Dummett (1981: 380).

  5. 5.

    See Wright (1983: 52).

  6. 6.

    See Forbes (1985: 80ff). I believe that pursuing the modalist strategy further will result in a version of fictionalism. The details cannot be elaborated here.

  7. 7.

    The full story is spelled out in the works of Kripke, Dummett, and Evans.

  8. 8.

    See Wright (1983: 139).

  9. 9.

    Compare e.g. Wright (1990: 154) and Hale and Wright (2001a: 12).

  10. 10.

    See Kripke (1980: 19, 44).

  11. 11.

    See Kripke (1980: 18).

  12. 12.

    Unless, that is, we are allowed to treat them as postulates useful for all kinds of theoretical purposes. This is the main strategy adopted in Lewis (1986).

  13. 13.

    Other versions may include:

    ‘Possibly’\(^{\frown}S\) is true iff there is a possible circumstance w such that S is true in w.

    ‘Possibly, …’ is true iff there is a possible circumstance w such that ‘…’ is true in w.

  14. 14.

    As shown by the Kaplan-Peacocke’s paradox. See Lewis (1986) and Davies (1981).

  15. 15.

    See Humberstone (1981).

  16. 16.

    See Humberstone (1981) and Forbes (1985).

  17. 17.

    See Quine (1939: 707).

  18. 18.

    See Quine (1951: 205).

  19. 19.

    For an account that assigns referential role to variables see Fine (1985).

  20. 20.

    See Quine (1969: 95).

  21. 21.

    Clearly laptop is not a stand-alone sortal concept. It is rather an impure sortal concept in Wiggins’ terminology. See Hale and Wright (2001b: 387) and references therein. I also ignore the view on which people are sophisticated computers. The concept computer in the text must be understood as factory-built computer, not as Turing machine.

  22. 22.

    See Hale and Wright (2001b: 368) and Wright (1983: 116–117).

  23. 23.

    See Stalnaker (1998: 99) with a nod to Wittgenstein’s Tractatus.

  24. 24.

    See Crossley and Humberstone (1977: 19) and Davies and Humberstone (1980: 2–3).

  25. 25.

    I omit various qualifications made by Hale in response to criticisms. For the latest version see Hale (1994: 68).

References

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  2. Davies MK (1981) Meaning, quantification, necessity. Blackwell, Oxford

  3. Davies MK, Humberstone IL (1980) Two notions of necessity. Philos Stud 38:1–30

  4. Dummett MAE (1981) The interpretation of Frege’s philosophy. Duckworth, London

  5. Fine K (1985) Reasoning with arbitrary objects. Basil Blackwell, Oxford

  6. Forbes G (1985) The metaphysics of modality. Clarendon Press, Oxford

  7. Frege G (1980) The foundations of arithmetic. Northwestern University Press, Evanston

  8. Girle RA (2003) Possible worlds. McGill–Queen’s University Press, Montreal

  9. Hale B (1994) Singular terms (2). In: Hale B, Wright CJG (eds) The reason’s proper study. Clarendon Press, Oxford

  10. Hale B, Wright CJG (2001a) The reason’s proper study. Clarendon Press, Oxford

  11. Hale B, Wright CJG (2001b) To bury Caesar… In: Hale B, Wright CJG (eds) The reason’s proper study. Clarendon Press, Oxford

  12. Humberstone IL (1981) From worlds to possibilities. J Philos Log 10:313–339

  13. Kripke SA (1980) Naming and necessity. Basil Blackwell, Oxford

  14. Lewis DK (1986) On the plurality of worlds. Blackwell, Oxford

  15. Quine WVO (1939) Designation and existence. J Philos 36:701–709

  16. Quine WVO (1951) On Carnap’s views on ontology. In: The ways of paradox and other essays. Random House, New York

  17. Quine WVO (1969) Existence and quantification. In: Ontological relativity and other essays. Columbia University Press, New York, pp 91–113

  18. Stalnaker RC (1984) Inquiry. The MIT Press, Cambridge

  19. Stalnaker RC (1996) On what possible worlds could not be. In: Morton A, Stich S (eds) Benacerraf and his critics. Blackwell, Oxford

  20. Stalnaker RC (1998) On the representation of context. In: Context and content. Oxford Univeristy Press, Oxford

  21. Wright CJG (1983) Frege’s conception of numbers as objects. Aberdeen University Press, Aberdeen

  22. Wright CJG (1990) Field and Fregean platonism. In: Hale B, Wright CJG (eds) The reason’s proper study. Clarendon Press, Oxford

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Acknowledgement

I am grateful to Dorothy Edgington for many helpful comments on the earlier version of this material.

Author information

Correspondence to Sandy Berkovski.

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:Footnote 25 Syntactic criteria of singular termhood. An expression t functions as a singular term in a sentential context A(t) just in case:

  1. 1.

    The following conditions are satisfied:

    1. (a)

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

    2. (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)’.

    3. (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. 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. 1.

    Socrates is wise in w.

  2. 2.

    Socrates is fat in w.

There is no problem to infer:

  1. 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

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

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Berkovski, S. Possible Worlds: A Neo-Fregean Alternative. Axiomathes 21, 531–551 (2011). https://doi.org/10.1007/s10516-010-9096-x

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Keywords

  • Possible worlds
  • Realism
  • Context principle
  • Possibility semantics
  • Wright
  • Dummett
  • Stalnaker