3.4 Arguments for Propositional Necessitism
3.4.2 The Truth-Values Argument
The first argument for Propositional Necessitism that will be offered is theTruth-Values Argument. This argument is not the main argument for Propositional Necessitism to be presented. The reason is that, on its own, its cogency can be resisted. The reason for presenting it anyway is that it has strong similarities to the stronger arguments for Propositional Necessitism yet to be offered, and to an argument of Plantinga’s that will be discussed in §3.5.
I will begin by offering an argument for an instance of Propositional Necessitism. The argument’s conclusion is the claim that the proposition that Obama is a president is necessarily something. The proposition that Obama is a president thus takes the role of an arbitrary proposition that is possibly something (note that if necessarily there are no propositions, then Propositional Necessitism is true; a counterexample to the truth of Propositional Necessitism requires that there could have been some proposition that could have been nothing).
Afterwards the premises of the argument for the necessary being of the proposition that Obama is a president will be generalised to the premises of the Truth-Values Argument for Propositional Necessitism.
17The thesis that the members of a set are essential to it does not follows from the axioms of ZFC. Yet, those axioms do preclude some natural alternative conceptions of sets on which the members of a set are not essential to it.
One such conception is anintensionalconception of set. According to this conception, it is of the nature of sets to be the extension of (at least some) properties. For instance, according to this conception the set{x:xis a man}could have had more members than it actually has, since there could have been more men than the ones there actually are.
On the intensional conception of set the argument for the nonbeing of the set{x:xis Obama}in circumstances in which Obama is nothing would fail. Since in such circumstances nothing is Obama, the set{x:xis Obama}is empty at that world. Yet, the set is still something.
Let me offer an argument against the intensional conception. Suppose, absurdely, that the set{x:xis a man}could have had more members than the ones it actually has. Lethbe an enumeration of all the men, and suppose that the cardinality of{x:xis a man}isn. Thus,{x:xis a man}={x:x=h(1)orx=h(2)or. . .orx=h(n)}, by the axiom of extensionality. Consider now a circumstancewat which{x:xis a man}has more members than it actually has. Since
{x:xis a man}={x:x=h(1)orx=h(2)or. . .orx=h(n)}, it follows that, atw,{x:x=h(1)orx=h(2)or
. . .orx=h(n)}has more members than it actually has. Thus, atw, there is someo∈ {x:x=h(1)orx=h(2)or. . .
orx=h(n)}such thato6=h(i), for allisuch1≤i≤n. But in such case,odoes not satisfy the condition of being anx
such thatx=h(1)orx=h(2)or. . .orx=h(n). Soodoes not belong to the set{x:x=h(1)orx=h(2)or. . .
orx=h(n)}. Contradiction. Hence, it is not the case that{x:xis a man}could have had more members than the ones it actually has.
Letpstand for the proposition that Obama is a president, assuming that this proposition is possibly something. It will be argued that the following claim is true:
(NecPropObama)
1. Necessarily, the proposition that Obama is a president is something. 2. 2(∃q(p=q)).
The premises of the argument for the necessary being of the proposition that Obama is a president are the following:
(P1-TVAi)
1. Necessarily, Obama is a president or Obama is not a president.. 2. 2(p∨ ¬p).
(P2-TVAi)
1. Necessarily, if Obama is a president, then it is true that Obama is a president. 2. 2(p→T p).
(P3-TVAi)
1. Necessarily, if Obama is not a president, then it is false that Obama is a president. 2. 2(¬p→F p)
(P4-TVAi)
1. Necessarily, if it is true that Obama is a president, then the proposition that Obama is a president is something.
2. 2(T p→ ∃q(p=q))
(P5-TVAi)
1. Necessarily, if it is false that Obama is a president, then the proposition that Obama is a president is something.
2. 2(F p→ ∃q(p=q)).
One important remark is that ‘it is true that’ and ‘it is false that’ are here understood as properties of propositions, i.e., as standing for entities of typehhii. Thus, they do not stand for properties of sentences, nor of any other individuals. That is, they do not stand for entities of typehei, since on the typology of entities being presupposed, propositions are not individuals, but instead higher-order entities.18 The argument from (P1-TVAi)-(P5-TVAi) to (NecPropObama) goes as follows.
Premises (P1-TVAi), (P2-TVAi) and (P3-TVAi) together imply:
(10) 1. Necessarily, it is true that Obama is a president or it is false that Obama is a president. 2. 2(T p∨F p)
Moreover, (10), (P4-TVAi) and (P5-TVAi) together imply (NecPropObama). The premises of the Truth-Values Argument are the following:19
Premises of the Truth-Values Argument
(P1-TVA) Excluded Middle .
1. Necessarily, for everyp, necessarily,por¬p. 2. 2∀p2(p∨ ¬p)
(P2-TVA) Truth Introduction.
1. Necessarily, for everyp, necessarily, ifp, then it is true thatp. 2. 2∀p2(p→T p)
(P3-TVA) Falsity Introduction .
1. Necessarily, for everyp, necessarily, if¬pthenphas the property of being false. 2. 2∀p2(¬p→F p)
(P4-TVA) Thorough Serious Actualism.
Premise (P1-TVAi) is an instance of Excluded Middle, (P2-TVAi) is an instance of Truth Introduction, (P3-TVAi) is an instance of Falsity Introduction. Finally, (P4-TVAi) and (P5-TVAi) are both instances of Thorough Serious Actualism. It should be clear that the truth of Propositional Necessitism follows from (P1-TVA) - (P4-TVA), given the reasoning presented in the argument for the truth of (NecPropObama).
The weakest assumption of the Truth-Values Argument, in the sense of being the least controversial, is Excluded Middle. Every instance of the schemaϕ∨ ¬ϕis a propositional tautology, and thus every instance of the schema2(ϕ∨ ¬ϕ)is a theorem of the very weak propositional modal logicK. Moreover, the fact that every instance of2(ϕ∨ ¬ϕ)is a theorem ofKis no accident owing to the lack of expressive resources ofK. Instances of the schema are true no matter what possible proposition turns out to be the semantic value ofϕ. This means that necessarily, for everyp, necessarily,por¬p. That is, Excluded Middle is true.
Thorough Serious Actualism was defended in §3.3. Discussion of Truth Introduction and Falsity Introduction will be left for §3.5. Suffice it to say for now that I think that propositional contingentists 19The argument does not require the full strength of Thorough Serious Actualism. Rather, the following thesis would suffice:
Serious Actualismhhii
1. Necessarily, for every propertyXof propositions, necessarily, for every propositionp, necessarily, ifphasX, thenp
is something.
2. 2∀X2∀p2(Xp→ ∃q(p=q)).
The same remark applies to the remaining arguments to be considered in this section.
The reason for appealing to the full strength of Thorough Serious Actualism concerns the fact that later it will be queried what would the status be if modal expressions were seen as being analysable in terms of truth at a world, instead of constituting themselves predications of properties to propositions. It will be shown that the Possibility Argument, yet to be presented, is still valid under such understanding of modal expressions. Yet, this would not be so if the Possibility Argument were formulated in terms of Serious Actualismhhiiinstead of being formulated in terms of Thorough Serious Actualism.
have interesting objections to Truth Introduction and Falsity Introduction. Given the other arguments for Propositional Necessitism, these objections only have a local impact, since they are not applicable to the remaining arguments.