Abstract
The article looks at the structure of impossible worlds, and their deployment in the analysis of some intentional notions. In particular, it is argued that one can, in fact, conceive anything, whether or not it is impossible. Thus a semantics of conceivability requires impossible worlds.
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Notes
Cohen and Cohen (1992, p. 450).
See Smith (2011).
See Knuuttila (2013).
See Garson (2014).
See Berto (2013).
See Priest (2008, chs. 2, 3).
What follows applies equally to a first-order languages, but their specificities are not relevant to the following considerations.
In some standard presentations, there is no designated world, @, and validity is defined as truth preservation over all possible worlds. As long as no special constraints are put on @, this is, of course, equivalent. However, it will be useful in what follows to have @ at our disposal.
See Carnielli and Pizzi (2008).
I note that if we make all the propositional parmeters true and false at w, then every formula is true there; and if we make all the propositional parameters neither true nor false at w, all formulas are neither true nor false there. Hence, the primary directive can, in fact, be satisfied with just these two worlds. However, the two, on their own, hardly do justice to the diversity of impossible situations.
Any such world is obviously closed under the consequence relation. Conversely, if w does not satisfy Exh and Exc, it is clearly not closed under the relation. If it accesses a world, \(w'\), that is not so closed, then for some, p, p is either both true and false at \(w'\) or neither true nor false there. In the first case, \(\diamondsuit (p\wedge \lnot p)\) is true at w; but in S5, \(\diamondsuit (p\wedge \lnot p)\models A\), so the world is either not closed under the consequence relation, or is trivial. In the second case, it is not true that \(\Box (p\vee \lnot p)\) at w, so the world is not closed under Necessitation.
For these non-classical modal logics, see Priest (2008, ch. 11a).
As argued by Mortensen (1989). Even if X is not P, this may still be the case. In FDE and LP everything is logically possible, because of the trivial possible world. The thought that every situation is logically possible may initially seem an odd one. But it should be remembered that logical possibility is a very weak constraint. Even if one is of a classical persuasion, it is a logical possibility that I can jump a kilometre into the air, that the moon is made of blue cheese, etc. Usually, when we are concerned with possibility, we are concerned with much more restricted notions, especially physical possibility.
See Priest (1987, ch. 13).
See Priest (2008, ch. 5).
One notable exception: logics where the inference \(A\vdash A\) fails.
In truth, it is only \(\nu ^{+}\) that is required to deliver the Secondary Directive. We cannot give up \(\nu ^{-}\), though, since it may be involved in the falsity conditions of modal formulas at @; though this leaves us free to impose constraints on \(\nu ^{-}\) if required for any reason.
Suggested to me by Hartry Field.
In fact, we do not need to consider what is accessed by non-@ worlds under \(R_{\kappa }\), since the truth/falsity of modal sentences at such worlds is taken care of by \(\nu\). We may therefore take \(R_{\kappa }\) simply to be of the form \(\{\left\langle @,y\right\rangle :y\in Y\}\) for some \(Y\subseteq W\).
And the new context may suggest some new members of K, such as ‘It is intuitionistically possible that...’.
See Priest (2008, ch. 6).
See Priest (2008, ch. 10).
OED, to conceive: ‘to take or admit into the mind, to form in the mind, to grasp with the mind’.
OED, to imagine: ‘to form a mental image of, to represent to oneself in imagination, to create as a mental conception, to conceive’. There is one sense of the word according to which what is imagined ‘should not be known with certainty’ (OED, again). This is not the sense at issue here.
Selby-Bigge and Nidditch (1978, p. 32).
For Hume, for something to be absolutley impossible is for it to imply a contradiction. (See Lightner (1997, p. 115)). I take it that he holds that the negation of any “relation of ideas” would do this.
See Yablo (1993). Yablo’s own account of conceivability (in Section 10) is that A is conceivable if one can imagine a world that verifies A. In fact, I agree with this, since I take everything to be conceivable/imaginable. This is not what Yablo intends, however. For, by ‘world’, he means ‘(classically) possible world’. Yablo tells us (p. 30) that one cannot imagine, e.g., tigers that lick all and only those tigers that do not lick themselves. I find this no harder to imagine than a set that contains all those sets which are not members of themselves. (And I could imagine this even before I became a dialetheist.)
For discussion and references, see Priest (2006, 3.3).
Priest (1997b).
There is a somewhat thorny issue here about what it is, exactly, to understand. Can a congenitally blind person understand the predicate ‘is red’, for example? I am inclined to the view that they can, if they can use the word—by whatever means—in a roughly normal way. When they imagine something red, the phenomenological content may, however, be quite different from that of a sighted person who imagines something red.
Semantics for a logic of imagination can be found in Niiniluoto (1985), Costa-Leite (2010), and Wansing (2015). These are all variations on possible-world semantics, and hence do not allow for imagining the impossible. Even worse, they all require imagination to satisfy certain logical closure conditions. Thus, they all validate the principle that if A is imagined, and A is logically equivalent to B, then B is imagined. This is clearly incorrect. A is logically equivalent to \((A\wedge C)\vee A\), but I can imagine that Sherlock Holmes lived in Baker St without imagining that (Sherlock Holmes lived in Baker St and \(E=mc^{2}\), or Holmes lived in Baker St). Nothing about Special Relativity need have crossed my mind at all. It is precisely this to which the Secondary Directive caters. Berto (2012, ch. 7) has a semantics for conception/representation which uses impossible worlds. He does not requre that everything be conceivable, but the semanatics does allow for that possibility.
One might also doubt that a person understands indefinitely long sentences of such a language. By the same token, one might doubt that such sentences are really grammatical. One might therefore be inclinded to put the same bounds of finitude on both both.
See Berto (2012, 6.3.2.1) for references and discussion.
Chalmers (2002) constructs an eightfold taxonomy of notions of conceivability, and argues that at least one of these entails possibility: ideal primary positive conceivability. This may well be different from the notion of conceivability I am discussing here—though the circularity in his glosses of these notions make me less than certain. But in any case, one thing is clear: the ideality involved is that of some infinite and infallible a priori reasoner—not a very useful notion for mere mortals.
OED, to suppose: ‘to think or assume that something is true or probable but lack proof or certain knowledge’, ‘used to introduce a hypothesis and imagine its development’.
See Priest (1995, 4.8). Again, I am assuming that the agent understands the term ‘t’.
Here, \(\varepsilon\) is the indefinite description operator: a (particular) object such that.
As the old saying goes: be careful of what you wish for; you might just get it.
Cohen and Cohen (1992, p. 291).
Versions of this paper were given at the Fordham University Metaphysics and Mind Group, the Departments of Philosophy at the University of Amsterdam and the Australian National University, and the conference Thinking the Impossible, at the University of Turin. Thanks go to those present for their helpful comments. Thanks, too, go to Hartry Field for comments on an earlier draft.
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Priest, G. Thinking the impossible. Philos Stud 173, 2649–2662 (2016). https://doi.org/10.1007/s11098-016-0668-5
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DOI: https://doi.org/10.1007/s11098-016-0668-5