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The account of ‘possibility’ and ‘necessity’ that will be investigated, theActual Truth At Account, is closely related to the Truth At Account. The account is the following:
Actual Truth At Account
1. Possibility
• Jx3ϕyKis the proposition thatactually,JϕKis true at some worldw.
• Jx3ϕyKis true at a worldwif and only ifJϕKis true at some worldw’ accessible from the
actual world. 2. Necessity
• Jx2ϕyKis the proposition thatactually,JϕKis true at every worldw. • Jx2ϕyKis true at a worldwif and only ifJϕKis true at every worldw
0 accessible from the
actual world.
Note that the expression ‘actually’ is used in its rigid sense, not in an indexical sense. Letαname the actual world. Then, it is the case that actually,pif and only if it is true atαthatp. The Possibility Argument turns out not to be cogent given the assumption that the Actual Truth At Account is true, as dialectically required by propositional contingentists.
Consider the reading of (P2-PAi) according to the account:
(P2-PAi-AcTrAt)
1. It is necessary that if actually some possible worldwis such that it is true atwthat Obama is a president, then the proposition that Obama is a president is something;
2. 2(@∃w(T wp)→ ∃q(p=q))
Contrary to what was the case with the reading of (P2-PAi-TrAt), (P2-PAi-AcTrAt) is not an instance of Thorough Serious Actualism. Suppose that it is possible that Obama is a president at an actual or counterfactual circumstancew. According to the Actual Truth At Account this means that actually, the proposition that Obama is a president is true at some worldw0. If actually, the proposition that Obama is a president is true at some worldw0, then actually, the proposition that Obama is a president bears the being true at relation tow0. So, by Thorough Serious Actualism, it follows thatactually,w0
and the proposition that Obama is a president are both something. However, from the fact that, atw, actually, the proposition that Obama is a president is something it does not follow that ifwhad been realised, then the proposition that Obama is a president would have been something. It only follows that the proposition that Obama is a president is somethingatthe actual world.
So, once possibility is understood according to the Actual Truth At Account, there is no longer reason to hold that (P2-PAi) is true. And similarly with respect to (P3-PAi). Moreover, the objections
to Plantinga’s Argument and the Truth-Values argument will still go through. In particular, truth in a world may still be defined in terms of truth at a world. Thus, the Actual Truth At Account promises to provide propositional contingentists with a way out of the Truth At Dilemma.
In what follows I will show that this is not so. But first I will quickly show that propositional contingentists have the resources required to reject the claim that the Actual Truth At Account has as an unwanted consequence the claim that it is necessary that every proposition is actually something. To see how this worry arises, recall that from the assumption that at some worldw, it is possible that Obama is a president, it follows that actually, the proposition that Obama is a president is something. Similarly, from the assumption that, at some worldw, it is possible that Obama is not a president it follows that the proposition that Obama is not a president is actually something. So, the proposition that Obama is a president is actually something, or the proposition that Obama is not a president is actually something. In such case it follows, by Boolean Structure, that actually, the proposition that Obama is a president is something.
Since the proposition that Obama is a president plays, in the above argument, the role of an arbitrary proposition, it would appear that from the claim that actually, the proposition that Obama is a president is something it could be legitimately inferred that it is necessary that, for every proposition
p, actuallypis something. But the conclusion that it is necessary that, for every propositionp, actually,
pis something is problematic for propositional contingentists.
On the one hand, the claim that it is necessary that, for every propositionp, actuallypis something conflicts with the conjunction of the following claims: i) there could have been some things that actually are nothing; ii) there could have been propositions directly about those things; iii) the thesis of Existentialism. Suppose that Noman is a merely possible individual, and that the proposition that Noman is a human is about him. Since it is necessary that every proposition actually is something, it follows that the proposition that Noman is a human is actually something. But then, by Existentialism, Noman is also something, contrary to the assumption.
On the other hand, suppose that it is indeed true that it is necessary that every proposition actually is something. Then, it seems plausible to think that it is necessary that every proposition is necessarily something. Why should the actual world be special in this respect? For these reasons, if the Actual Truth At Account implies that it is necessary that every proposition actually is something, then propositional contingentists may prefer to avoid a commitment to its truth.
Proponents of the Actual Truth At Account have the resources to reject the legitimacy of the inference from the claim that actuallypis something, for an arbitrary propositionp, to the claim that it is necessary that, for every propositionp, actuallypis something.
The legitimacy of the inference can be resisted by adopting the view that the functions that are the semantic values of ‘3’ and ‘2’ have an empty extension at other worlds. They relate nothing whatsoever in worlds other than the actual world.
Once this view is adopted, the inference is illegitimate. Since actuallypis something, for an arbitrary propositionp, it may be legitimately inferred that for every propositionp, actuallyp is
something. Butnecessitationis illegitimate. The inference of the claim that for every propositionp, actuallypis something was legitimate due to a fact that holds only of actual propositions: only these are arguments of the semantic values of the modal expressions.25
Call this objection to the Possibility Argument theTruth At Objection. I turn now to the problems with the Truth At Objection. An important initial observation is that the Truth At and the Actual Truth At accounts are mutuallyconsistent. Thus, the Actual Truth At Objection is successful only if the following claim is true:
(31) There is some possible propositionpand possible worldwsuch that i) it is true atwthat there is some worldw0accessible from theactual worldsuch thatpis true atw0(and so it is true atwthat3paccording to the Actual Truth At Account) and yet ii) it is not true atwthat there is some worldw0accessible fromwsuch thatpis true atw0.
The truth of (31) is presupposed by the Actual Truth At Objection because otherwise it would follow that if it is true at a worldwthat3p, thenpis something atw, and so the Possibility Argument would indeed be valid.
The main problem with the Actual Truth At Objection is simply that (31) is false. Assuming that propositions are true at worlds, (31) is false because:
(32) Any case in whichpis not true at a worldw0accessible fromwis a case in which it is not true atwthat3p.
My argument for (32) will be based on showing that a certain debate on the correct logic for meta- physical modality presupposes that3pis judged to be false atwon those scenarios in whichpis not true at a worldw0accessible fromw, regardless of whetherpis true at a worldw0accessible from the actual world. The example should make it obvious that this is a general feature, and so that (32) is true.
This will reveal that the Actual Truth At Account is based on a erroneous view of the semantics of modal expressions. I will focus on one such scenario, inspired by the arguments for the claim that axiom schema4of propositional modal logic, according to which if it is possible that it is possible thatϕ, then it is possible thatϕ(i.e.,33ϕ→3ϕ) has false instances.
25I ultimately think that such move is unsuccessful, for reasons related to the case against the partial functions account of the semantics of first-order modal languages presented in §2.5.
In general, the problem with this strategy is that it does not have the resources to make sense of the semantic values of open formulas and what their contribution is to the semantic values of the sentences in which they occur is. For instance, what is the semantic value of3pin the context of the formula3∃p(3p∧@¬∃q(p=q))?
The view that the semantic value of ‘3’ has no extension in worlds other than the actual world leaves it mysterious what the semantic value of ‘3p’ is. The reason is that whatever possible proposition witnesses the truth of ‘3∃p(3p∧@¬∃q(p=q))’, this proposition is nothing in the actual world. The problem, from the standpoint of many propositional contingentists, is that they accept the truth of the sentence. That is, they accept that there could have been true propositions that are actually nothing.
These propositional contingentists appear to be left with no satisfactory account of the semantic value of ‘3p’ if they indeed endorse the view that the semantic value of ‘3’ has an extension only in the actual world.
A putative counterexample to axiom schema4starts by assuming the truth of the following two principles about bicycles:26 i)Tolerance, according to which necessarily, any bicycle could have been constituted by any two thirds of its original constitution; ii)Restriction, according to which necessarily, no bicycle could have been constituted by one third of its original constitution.
Let me describe a scenario, theBicla Scenario, of the sort envisaged by proponents of the claim that axiom schema4has false instances. In this scenario,Biclais a bicycle. At the actual world,α, Bicla is constituted by frameα-Frame and wheelsα-FrontWheel andα-BackWheel.
Suppose thatw1-FrontWheel is one of the wheels of which Bicla could have been constituted,
according to Tolerance. Letw1be a possible world witnessing the fact that Bicla could have been just
as it is, except for havingw1-FrontWheel as its front wheel.
Letw2-Frame be a frame that is something atw1. According to proponents of the view that
axiom schema4has false instances there is a possible worldw2accessible fromw1such that Bicla is
constituted atw2byw2-Frame,w1-FrontWheel andα-BackWheel. Worldw2is taken to witness the
truth of Tolerance.
LetBiclalternatestand for the claim that Bicla is constituted byw2-Frame,w1-FrontWheel and
α-BackWheel.Biclalternateis true atw2. The fact thatw2is accessible fromw1andBiclalternate
is true atw2is taken by proponents of the view that axiom schema4has false instances to suffice for
it to be true atw1that3Biclalternate.
Worldw2 is not accessible fromα. If it were, then Bicla would have been constituted by one
third of its original constitution, which would violate Restriction. Thus, according to Restriction,
3Biclalternateis false atα. Sincew1 is accessible from αand3Biclalternate is true atw1,
33Biclalternateis true atα. Thus, proponents of the view that axiom schema 4 has false instances take the Bicla Scenario to show that33Biclalternate→3Biclalternateis false atα.
If the Actual Truth At Objection were successful, then the thought that scenarios such as the Bicla Scenario constitute counterexamples to axiom schema4 would be deeply misguided. For instance, according to the Actual Truth At Account the fact thatBiclalternateit is true atw2and
w2 is accessible fromw1is irrelevant to whether3Biclalternateis true atw1. Moreover, since
3Biclalternateis false atα, there is no possible worldwaccessible fromαsuch thatBiclalternate
is true atw. Hence, according to the Actual Truth At Account,3Biclalternateis false atw1, and
indeed at every worldswsuch thatwis accessible toα. A fortiori,33Biclalternateis false atw0.
Thus, if the Actual Truth At Objection were true, then even if the Bicla Scenario were metaphysically possible, this would provide no counterexample to the truth of every instance axiom schema4.
Importantly for the present purposes, proponents and opponents of the view that there are false instances of axiom schema4agreethat if scenarios such as the Bicla Scenario are metaphysically possible, then axiom schema4has false instances. To repeat, such widespread agreement makes no sense from the standpoint of proponents of the Actual Truth At Objection. After all, the Actual Truth 26The example in the text is close the one offered in (Chandler,1976). Besides Chandler,Salmon(2005) has defended forcefully the failure of axiom-schema4.
At Account implies the truth of every instance of axiom-schema4.
Suppose that it is true that it is possible that it is possible thatp, for an arbitrary propositionp. In such case, according to the Actual Truth At Account, there is some possible worldwsuch thatwis accessible fromαand3pis true atw. So, there is some possible worldwsuch thatwis accessible fromαand there is some possible worldw0such thatw0is accessible fromαandpis true atw0. This is so if and only if there is some possible worldwaccessible fromαsuch thatpis true atw. So, if
33p, then3p.27
The discussion concerning the Bicla Scenario shows that the Actual Truth At Account yields the wrong semantics for ‘3’. Ifpis not true at a worldw0 accessible fromw, then3pis not true atw, independently of what happens in the actual world. That is, the Actual Truth At Objection is wrongly committed to the truth of (31).
I find the predicament of proponents of the Actual Truth At Objection to be similar to that of a biologist, say, Bio, that endorses the view that all living things are composed of carbon partly on the grounds that he takes ‘all’ to mean all things on Earth. Just as it may be that all living thingstout courtare composed of carbon, it may very well be that all instances of axiom schema4are true. But just as Bio supports the claim that all livings things are composed of carbon on the basis of a faulty semantics for ‘all’, proponents of the Actual Truth At Account support the truth of every instance of axiom schema4on the basis of a faulty semantics for ‘3’.
Read(2005, p. 321) captures the present point when referring to the contrapositive of axiom schema 4, saying that ‘even when ... there is equivalence, it is misleading to say that nothing is added by prefixing ‘it is necessary that’. It is a substantive thesis that necessity is idempotent (that22pis equivalent to2p)’. If the Actual Truth At Objection were successful, then the truth of every instance of axiom schema4would turn out to be non-substantive. The truth of every instance of axiom schema 4 would be consistent with scenarios like the Bicla Scenario. This reveals that the Actual Truth At Objection is unsuccessful.