Alternative Arguments: The Possibility Argument

The only difference between the Possibility and the Possibility Or Necessity Argument is that the Possibility Argument has as a premise the thesis ofSerialityinstead of the thesis of Possibility Or Necessity.

Seriality.

1. Necessarily, for everyp, necessarily, it is possible thatpor it is possible that¬p. 2. ₂∀p₂(₃p∨₃¬p).

The premises of the Possibility Argument are thus the following:

Premises of the Possibility Argument

(P1-PA) Seriality.

(P2-PA) Serious Actualism. (P3-PA) Contradictoriness.

The Possibility Argument proceeds in a fashion somewhat similar to the Truth Argument, and so the details will be left out. Contrary to the theses of Possibility Or Impossibility and Possibility Or Necessity, the thesis of Seriality is not supported by systemK. Rather, it is supported by the very weak normal modal propositional logicKD. This logic results from adding toKall instances of the following schema:

Axiom schema D. ₂ϕ→₃ϕ

In the context ofK, each instance₂ψ→₃ψof axiom schemaDturns out to be equivalent to an instance of the following schema:

(19) ₃ϕ∨₃¬ϕ

This means thatKDis also the system that results from adding toKthe schema (19).

The truth of each instance of (19) is extremely plausible given the interpretation of ‘₃’ as meta- physical possibility. All that is required for each instance of (19) to be true is that necessarily, things could have been some way or another. Take any possible circumstancew. If, atw, things could had been some wayw0, thenϕwould have been true atw0or¬ϕwould have been true atw0, by Excluded Middle. Thus, atw, it is possible thatϕor it is possible that¬ϕ.

Is it the case that, for each possible circumstancew, things could have been some way or another? Yes, since at each possibilityw, things could at least have been as they are inw(this just consists in the observation that, necessarily,p→₃p). Let me now turn to the justification for the assumption of Contradictoriness.

(20) If you are consistent and believe thatϕ, then there is something that you do not believe, namely, that¬ϕ.

Arguably, the truth of every instance of this schema is supported by unreflective common sense. The truth of every instance of this schema presupposes the truth of its universal generalisation:

(21) For every propositionp, if you are consistent and believe thatp, then there is something that you do not believe, namely, you do not believe that¬p.

If (21) codifies a constraint on what it is to be consistent, as it appears to do, then its necessitation is true. Since the necessitation of (21) implies Contradictoriness, then, arguably, Contradictoriness is one of the commitments of unreflective common sense.

This argument presupposes that propositional quantification is appropriate in this context, rather than quantification over propositions understood as individuals, i.e., as entities of typee. Such understanding of propositions is a presupposition of this chapter. As mentioned in §3.1, propositions are here understood as entities of typehi, and as the objects of the attitudes. If it turns out that there could be no propositions as these are here understood, then Propositional Necessitism is vacuously true.

I suspect that there are many other commitments of unreflective common sense supporting Contradictoriness. Anyway, the commitment to Contradictoriness is, on its own, independent of Propositional Necessitism. For instance, the Russellian theory of propositions, one of the theories falling under the Classical Conception, appears to imply Contradictoriness, given natural auxiliary assumptions.

According to the Russellian theory, propositions are structured entities, containing other entities as their constituents. For instance, according to standard Russellians, the proposition that Obama is a president is composed of Obama and the property of being a president.

An argument from the Russellian theory to Contradictoriness appeals to the plausible assumption that if the constituents of a structured proposition are all something, then the proposition itself is something. It is also plausible to think that the operation of negation is necessarily something. So, if a propositionpis something, then any proposition that haspand the operation of negation as its only constituents is also something. On the structured accounts of propositions, one such proposition is the proposition that¬p. Thus, necessarily, for every propositionp, ifpis something then¬pis something.

Thus, arguably, Contradictoriness is supported by unreflective common sense, and it is implied by some of the theories committed to the falsity of the Classical Conception of propositions. Contradic- toriness is thus an assumption shared with at least some propositional contingentists.

Finally, independently of what positive support there is for Contradictoriness, the fact that the thesis is independent of Propositional Necessitism, would make it a surprising result if the best line of reply to an argument for Propositional Necessitism consisted in rejecting the truth of Contradictoriness.

This concludes the presentation of the arguments for Propositional Necessitism. In the next section objections to the arguments are considered, and replies to these objections are offered. I will be referring to the Possibility Or Impossibility Argument, Possibility Or Necessity Argument and Possibility Argument as themodal argumentsfor Propositional Necessitism.