On Propositions As Sets Of Possible Worlds

In a recent paper Aviv Hoffmann argues that analyses of propositions as sets of possible worlds are to be rejected because the costs of such analyses are too high. I aim to protect some such analyses from this charge. I will first review Hoffmann's arguments before considering responses to them. My claim is that while Hoffmann's premises may be plausible under some disambiguations of 'proposition' and 'possible world', not all are plausible under the conception of these terms relevant to the analyses he attacks. I will focus on a particular premise of Hoffmann's and show that it is implausible once the ambiguity in the key terms has been resolved.

Hoffman's Charge I will begin by presenting a concise form of Hoffmann's reductio argument against the thesis that propositions are sets of possible worlds.

Assuming that propositions are sets of possible worlds, let Somewise be the proposition that someone is wise and Wisebert be the proposition that Russell is wise. Then this is Hoffmann's argument, with some auxiliary premises made explicit:

"A phrase of the form 'the proposition that S' designates the same object with respect to every possible world where the object exists, and it never designates anything else." Necessarily, if the proposition that someone is wise exists, then, Somewise exists. Wisebert entails Somewise. "If a proposition p entails a proposition q, then, necessarily, if q exists, then p exists." Necessarily, if Somewise exists, then Wisebert exists. Wisebert is directly about Russell. (Object Dependence) "If a proposition p is directly about an object o, then, necessarily, if p exists, then o exists." Necessarily, if Wisebert exists, then Russell exists. (By Object Dependence). (F) Necessarily, if the proposition that someone is wise exists, then Russell exists. (By 1-8).

According to Hoffmann, (F) is obviously false; hence one can reject analyses of propositions in terms of possible worlds. Hoffmann concludes this because he believes his premises to be on firm ground, so what should be rejected is the assumption that propositions are sets of possible worlds.

I agree that (F) is false, but disagree that we should reject possible-worlds-analyses of propositions. I reject premise 7, Object Dependence. Most of the remainder is a di(SC)ussion of this premise and why I think it should be abandoned. I will concede the other premises as true and not di(SC)uss them further. Hoffmann says that the cost of rejecting Object Dependence is too high. I will show that there is no cost in rejecting Object Dependence for an analysis of propositions as sets of possible worlds.

In Support of Object Dependence To reject Object Dependence I must fault Hoffmann's argument for it, which I will now examine.

Let the proposition that S be directly about an object o. Then the following is reasonable:

"The proposition that S is a proposition p such that necessarily, if p exists, then p is directly about o." (SC) Necessarily, if p is directly about o, then o exists. "The proposition that S is a proposition p such that, necessarily, if p exists, then o exists."

I will not contest that Object Dependence follows from the conclusion of this argument; it is only the premise (SC) that I repudiate. I will show that under the disambiguations of 'proposition' and 'possible world' relevant to analyses of propositions as sets of possible worlds, (SC) is implausible. To do this, I will explain exactly what analyses of propositions as sets of possible worlds are, and what the relevant terms mean according to such an analysis.

Possible World Analyses of Propositions To show why I reject Hoffmann's premise (SC), I will formulate a general account of analyses of propositions in terms of possible worlds and show that these analyses have no room for such a premise. To do this, I will follow Bricker and what he calls the Standard Theory of Propositions. The Standard Theory defines precisely what propositions and possible worlds are. Therefore, given that one has in mind the same conceptions of proposition and possible world as Bricker, the Standard Theory is not really up for debate.

Take as primitive the notion of one proposition implying another. Here we should understand implication as something akin to strength, so that one proposition implies another if it says everything the first says and may say more. The implied proposition is weaker than the proposition implying it; and its factual content, so to speak, is contained within the factual content of the stronger proposition. Implication forms a primitive partial ordering on the propositions. Where needed, I will symbolise implication by "?". The five theses of the Standard Theory are as follows:

Propositions form a Boolean algebra under "?". This Boolean algebra is complete. (i.e. Every set of propositions has a greatest lower bound. This amounts to stipulating that for any set of propositions, there is a conjunctive proposition that is equivalent to the set as a whole.) Propositions are truth functionally standard. (i.e. The expected truth conditions of propositional operations (in classical logic) hold. For example p&q (the conjunction of the propositions) is true iff p and q are both true, etc.) If two worlds w and v are distinct, then the set of propositions true at w is distinct from the set of propositions true at v. Every non-null (non-contradictory) proposition is true at some world. (i.e. There is a unique 'impossible proposition'. This implies that distinct propositions always play distinct world characterising roles.)

There is a lot of unintroduced terminology in these theses and they carry a lot of information. It would take more space than I have to do justice to the Standard Theory and Bricker's arguments for each of these theses. I attempt only to facilitate an understanding of the Standard Theory, since the possible-worlds-analyses I will be looking at abide by it.

From the Standard Theory, the natures of 'propositions' and 'possible worlds' follow. It is useful to give an informal characterisation of these terms so we can distinguish Bricker's use of them from other connotations the terms carry. Propositions, as the Standard Theory defines them, are what Bricker calls 'metaphysical propositions'. These are to be contrasted for example with semantic propositions, which are more fine grained. As an example, consider the purported propositions that Jane is a vixen and that Jane is a female fox. These are distinct qua semantic propositions, but identical qua metaphysical propositions. The difference between metaphysical and semantic propositions can be thought of as the difference between 'factual content' and 'meaning'; the examples above have the same factual content, but different meanings. Analyses of propositions as sets of possible worlds are about metaphysical propositions not semantic propositions. There are times in Hoffmann's paper where what he says is prima facie plausible in the case of semantic propositions, but incorrect for metaphysical propositions. The premise (SC) is an example of this.

Possible worlds also obtain their character from the Standard Theory. Bricker is employing a realist conception of possible worlds; I will follow him in this regard. This does not mean they must be the concrete possible worlds of Lewis, although these are a good example; they might instead be understood in Stalnaker's moderate-realist sense. What is common to these views is the close relationship between metaphysical propositions and possible worlds. Propositions are "world characterisers" as Bricker puts it. In our present vernacular, they might be thought of as something like maximal consistent factual contents.

I won't argue in favour of the Standard Theory; I will argue that some versions of it that offer an analysis of propositions as sets of possible worlds are immune to Hoffmann's criticism because they render the premise (SC) implausible. Given this plan, I will rely heavily upon taking the Standard Theory for granted; I take metaphysical propositions and possible worlds to be defined by the Standard Theory.

Now I can say what I am taking an analysis of propositions in terms of possible worlds to be. Such an analysis is what Bricker dubs a "world-based-theory" (WBT). A WBT adds to the Standard Theory the following 3 definitions:

Something is a proposition iff it is a set of possible worlds. One thing implies another iff the first is included in the second and the second is a set of possible worlds. (i.e. "?" is equivalent to "?") One thing is true at another iff the first is a set of worlds and the second a member of that set. (i.e. p is true at w iff w?p)

There are then different WBTs that disagree on points such as the ontological status of worlds, but I will use 'WBT' as an umbrella term for all theories that accept the Standard Theory and these definitions 1-3.

According to WBT, propositions exist at every possible world. If it were otherwise then some proposition must fail to exist at some world w. But then some set of worlds must fail to exist at w. This can only be the case if some member of the set fails to exist at w which is to deny that some member-world is possible with respect to w; contradiction. This will play an important role later on.

For Hoffmann's argument to have force against WBT it must be about the same notions of propositions and possible worlds as WBTs discuss. I will show that Hoffmann's argument involves the wrong notion of proposition. Furthermore, I will show that no matter how one tries to reconstruct the premise (SC), so that it does discuss metaphysical propositions, it will remain implausible.