Abstract
The standard approach to the modalities necessity and possibility is given by modal logic together with possible worlds semantics. In the view of some philosophers, this semantics requires a realism which a fortiori is presupposed for the understanding of modalities. In this paper we will discuss some philosophical problems of this approach, and propose an alternative view which overcome some of these problems; in particular, it dispense with realism for possible worlds.
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Notes
- 1.
As standard reference we can refer to the introduction to modal logic by Bull and Segerberg in the Handbook of Philosophical Logic, Volume II, [3].
- 2.
For a brief history of modal logic, see the first part of [3].
- 3.
In fact, Carnap’s approach to modalities makes already extensive use of possible worlds; however, he did not provide a semantics for modal logic, since the accessibility relation was still missing.
- 4.
- 5.
Therefore, this axiom is sometimes also designated by E.
- 6.
- 7.
This relation is not accidental; compare the German words Wirklichkeit and bewirken.
- 8.
They are dubbed ersatzers by David Lewis and his followers.
- 9.
Of course, syntactically one can speak about expressions like \(\neg \Box \phi\). However, it seems to be the case that the standard systems of modal logic do not derive any interesting \(\neg \Box \phi\) theorems. We heard that this criticism was repeatedly put forward by Quine, but we are not aware of any reference.
- 10.
And both, Leibniz and Carnap, make extensive use of the actual world in their approaches to possible worlds.
- 11.
- 12.
This was observed, for instance, by Føllesdal in [8, p. 572f.]. Stalnaker [25, p. 333] is half on this way when saying: ‘The possible worlds representation of content and modality should be regarded, not as a proposed solution to the metaphysical problem of the nature of modal truth, but as a framework for articulating and sharpening the problem.’ We think that the contribution of possible worlds to articulate and sharpening the problem of modalities is rather limited.
- 13.
There are some attempts in the literature to ‘define’ worlds which are not closed under logical consequence, cf. e.g., [19]; however, we consider these approaches as rather premature. In particular, they are usually discussed only informally.
- 14.
We are not sure whether in this analysis necessity and possibility can still be classified as modi of sentences. If not, it would be odd to call them modalities. For this reason, it might be more correct to put quotes around modalities in the title: ‘Modalities’ without worlds.
- 15.
The restriction to a first-order language has the only purpose to fix a logical framework; the given analysis works for any other formal language of arbitrary complexity.
- 16.
However, the concrete formal calculus—a Hilbert-style calculus, a sequent calculus, natural deduction, or any other calculus—is not important.
- 17.
We have an ally of highest authority here: Aristotle derives this property from (one of) his definitions of possibility in the Prior Analytics, cf. [24, §5.6.3].
- 18.
We will come back the temporal aspect later, when we argue that possibility statements are normally statements about the future. A further specification of our principle could even be:
-
What is provable is no longer considered as possible.
The subjective aspect of ‘considered’ is in particular of relevance for the epistemic modalities discussed below.
-
- 19.
As far as we see, it does not follow from any of the original maxims of Grice [12]. In the form: ‘Do not make your contribution less informative than necessary’ it would be somehow a dual to the Maxim of Quantity: ‘Do not make your contribution more informative than is required.’
- 20.
This matches, in an epistemic context, with the first reading of possibility by Frege in his Begriffsschrift: ‘Wenn ein Satz als möglich hingestellt wird, so enthält sich der Sprechende entweder des Urtheils, indem er andeutet, dass ihm keine Gesetze bekannt seien, aus denen die Verneinung folgen würde; …’ [10, p. 5] (‘Is a proposition is advanced as possible, either the speaker is suspending judgment by suggesting that he knows no laws from wich the negation of the proposition would follow …’, [11, p. 13]).
- 21.
A similar point was discussed in [13] in relation with contradictory belief sets; it might well be that we have contradictory beliefs, but as long as a contradiction was not derived, we can live happily with them.
- 22.
In fact, if you believe in a complete determination of the future, you will not have other possibilities than epistemic ones.
- 23.
This is a separate discussion, which, we believe, will help to resolve a lot of the philosophical problems concerning the understanding and status of ‘laws’.
- 24.
Since we do not consider nested modalities, we do not need an accessibility relation.
- 25.
See the section on the normative character of unary necessity statements below.
- 26.
In fact, at some occasions necessity is used in a redundant way with respect to the truth of a sentences (or argument). In these cases it should just serve as special emphasize. The most prominent example for such a use is probably the use of \(\upalpha\upnu\acute{\upalpha}\upgamma\upkappa\upeta\) by Aristotle for the conclusion of syllogisms, cf. Łukasiewicz [18, §5].
- 27.
The problem starts already with the question what happens if we replace ϕ by a logically equivalent formula ϕ′; it might be that in this way we get a proof of ψ which does not use ϕ literally, but somehow implicitly.
- 28.
See the example in the following subsection.
- 29.
However, there are limit cases: The question whether the tertium-non-datur is necessary for the proof of a certain mathematical theorem, is an example where we like to consider a logical axiom in this context. And, in fact, it is a prime example for our reading of necessity.
- 30.
We would have non-normal circumstances if the soccer federation already announced an investigation of the doings of B.
- 31.
Thus, if you still prefer possible worlds semantics over our reading, you should at least consider to use the varieties of alternatives to determine your variety of possible worlds.
- 32.
If you object to proofs as formal objects in this analysis, you can replace it by the (less well defined) notion of argument; in particular for a notion of epistemic necessity, constructed in analogy to the epistemic possibility above, arguments—known to the defender of a necessity statements—could be more adequate than proofs.
- 33.
This is a more colloquial form of ‘To raise the taxes is necessary’ which would be (very close to) an instance of (3).
- 34.
In particular, in a possible worlds semantics, every tax payer could without any doubt imagine a world where the taxes do not raise.
- 35.
To go even a step further, we could suggest that the communicative content of a necessity statement is often just the exclusion of certain—perfectly reasonable—alternatives. In terms of possible worlds semantics: A necessity statement is not to be analyzed in terms of possible worlds; in contrast, it is meant to define the possible worlds which the utterer considers as ‘possible’. This latter ‘possible’ might be better called ‘preferable’ or the like.
- 36.
This is in line with Frege when he writes: ‘Das apodiktische Urtheil unterscheidet sich vom assertorischen dadurch, dass das Bestehen allgemeiner Urtheile angedeutet wird, aus denen der Satz geschlossen werden kann, während bei den assertorischen eine solche Andeutung fehlt. Wenn ich einen Satz als nothwendig bezeichne, so gebe ich dadurch einen Wink über meine Urtheilsgründe.’ [10, p. 4] (‘The apodictic judgment differs from the assertory in that it suggests the existence of universal judgments from which the proposition can be inferred, while in the case of the assertory one such a suggestion is lacking. By saying that a proposition is necessary I give a hint about the grounds for my judgment.’ [11, p. 13].)
- 37.
If one allows the symbol 4 in this group, of course, 4 equals 1 and one could try to save this particular necessity statement; but it is clear that a statement like ‘2+2 is necessarily different from 1’ would break down.
- 38.
Obviously, this view is closely related to Carnap’s meaning postulates.
- 39.
It would be probably more honest if the politician would say ‘We will raise the taxes’ instead of ‘We must raise the taxes’; but it is obvious that it is politically smarter to use the second statement.
- 40.
This does not hold for epistemic possibility for the case we did not check the truth or falsity.
- 41.
Such a dynamics should, of course, not apply to the logical and mathematical examples; however, it might apply to the ‘definitional’ examples: ‘civil marriage is a contract between a man and a woman’ might have been considered as a necessary truth due to the definition of marriage. Today, several countries have legislation redefining this notion.
- 42.
Defenders of possible worlds semantics may point out that different forms of modalities require also different varieties of possible worlds. We are not aware of any approach which combines such different modalities in a single framework; in fact, such a framework would probably have quite drastic implications for the underlying realism.
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Acknowledgements
I thank Jesse Alama, Volker Halbach, Greg Wheeler, Bartosz Więckowski, and John Woods for helpful comments on earlier versions of this paper. Research partially supported by the ESF research project Dialogical Foundations of Semantics within the ESF Eurocores program LogICCC (funded by the Portuguese Science Foundation, FCT LogICCC/0001/2007).
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Kahle, R. (2012). Modalities Without Worlds. In: Rahman, S., Primiero, G., Marion, M. (eds) The Realism-Antirealism Debate in the Age of Alternative Logics. Logic, Epistemology, and the Unity of Science, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1923-1_6
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