Tuesday, September 19, 2006

Tense Logic, Bivalence, and Open Theism

In previous blog posts I've discussed two opposing tense logics: (a) Ockhamist and (b) Peircean. Ockhamist tense logic takes as its characteristic axiom the claim that
(O) □(∀p)(∀t)(∀u: u<t)(IS(p,t) → WAS(WILL(p,t),u))

i.e., necessarily, for all propositions p, and for all times t and u such that u is prior to t, p is the case at t, then at u it was the case that p will be the case at t
In contrast, Peircean tense logic insists that (O) is a non sequitur, and replaces it with:
(P) ◊(∃p)(∃t)(∃u: u<t)(IS(p,t) ∧ ¬WAS(WILL(p,t),u))

i.e., possibly, for some proposition p and some times t and u such that u is prior to t, p is the case at t and it was not the case at u that p will be the case at t
The basis of the disagreement turns on whether the future tense operator, WILL, carries determinative force. In other words, if one is speaking strictly (not loosely) and predicts that something will happen, is it implied that the predicted event is necessitated by what obtains at the time the prediction is made? If so, then the Peircean is right. Otherwise, the Ockhamist is right.

Without trying to resolve that particular issue right now, I'd like to note that there are three possible results if the Peircean is correct, depending on how one answers two questions: (1) are there any propositions about the future that carry absolutely no determinative force whatsover, i.e., Ockhamist-style predictions? (2) If so, are these propositions bivalent, i.e., must they be either true or, if not true, then false?
  • Option 1: Yes on (1). Yes on (2). This means that the future is alethically settled, i.e., there is an unchanging set of true Ockhamist-style predictions that completely characterizes the future as of any given time.
  • Option 2: Yes on (1). No on (2). The future is alethically open (i.e., not alethically settled), but some Ockhamist-style prediction (those pertaining to future contingents) are neither true nor false.
  • Option 3: No on (1). The future is alethically open and bivalence is preserved because there simply are no Ockhamist-style predictions.
Now, suppose one is an open theist. That is, suppose that one affirms the conjunction of (a) monotheism, (b) future contingency, and (c) the incompatibility of future contingency with God's knowing the future as wholly determinate and settled. It turns out that there are three major versions of open theism, each corresponding to one of the three options above.
  • Version 1 - Involuntary Partial Nescience: God knows all that can be known, but there are truths about the contingent future that simply cannot be known, not even by God. (Richard Swinburne and William Hasker have espoused this view.)
  • Version 2 - Non-bivalentist Omniscience: God is fully omniscient, i.e., knowing all and only truths. Ockhamist-style propositions about the contingent future are neither true nor false. (J.R. Lucas, among others, has espoused this view.)
  • Version 3 - Bivalentist Omniscience: God is fully omniscience. There are no Ockhamist-style propositions about the contingent future. (Greg Boyd has espoused this view.)
Each view faces some challenges. The proponent of version 1 needs to explain how there can be truths that are in principle unknowable. The proponent of version 2 needs to motivate the denial of bivalence and the attendent rejection of standard logic. The proponent of version 3 needs to make a persuasive case that Ockhamist-style predictions are not really propositions.

Saturday, September 16, 2006

How to Respond to the Liar Paradox

One form of the infamous Liar Paradox asks us to consider a person uttering the phrase "I am lying right now" and to determine if they are or are not lying. The paradox is supposed to emerge once one notices that if the person is lying then it follows that they are not lying, and if the person is not lying, then it follows that the person is lying.

I believe this form of the paradox is bogus because it trades on the erroneous assumption that to lie one must say something false. Once one realizes the falsity of this assumption, the paradox can be dissolved as follows:
Person: I am lying right now.

Reply: To be lying you would have to be trying to get me to believe something that you do not believe by presenting yourself as though you believed it. Therefore, (a) if you are trying to get me to believe that you are lying and you do not believe that you are lying, then you are lying. Or, (b) if you are trying to get me to believe that you are lying and you do believe that you are lying, then you are not lying but simply confused about what 'lying' means. And, finally, (c) if you are not trying to get me to believe that you are lying then you are not lying whether you believe that you are lying or not.
In short, whether one lies or not is purely a matter of motive and not at all a matter of whether the normal meaning of what one says is true or false.

Another version of the Liar Paradox asks us to consider the sentence: "This sentence is false." (Why is this considered a version of the "Liar Paradox" when it has nothing to do with lying? Probably because it often comes up in connection with the "I am lying" version.) This is thought to be paradoxical because, allegedly, the sentence is true if false, and false if true, and it must either be true or false.

There are a couple problems with this. The first problem has to do with the sentence's self-referentiality. Now, self-reference is not necessarily problematic. For example, "This sentence has five words" is clearly meaningful and true, as we can see by replacing the indexical phrase "this sentence" with its referent:
"This sentence has five words" has five words.
This is unproblematic because having five words is an intrinsic property of sentences, so we can determine it's truth value simply by inspecting the properties of the quoted sentence.

But truth and falsity are not intrinsic properties of sentences. A sentence is true if and only if it expresses a true proposition and false if and only if it expresses a false proposition. So even though sentences can be said to be true or false, they can be so only in a secondary sense by virtue of expressing a proposition that bears the property in a primary sense. This point is significant because when we replace the indexical phrase in "This sentence is false" with its referent, we get:
"This sentence is false" is false.
which is equivalent to:
"This sentence is false" expresses a false proposition.
But now we run into the problem that no sentence-type, simply qua sentence-type, expresses anything at all. A sentence-type is a pattern of words that can be used to express a complete thought, but it does not actually express anything until it is so used, that is, until it is tokened by a speaker in a given context. It is the speaker's intent that gives a sentence-token its meaning, not the bare pattern of words by itself. For example, suppose that by pure chance the waves and wind formed on a beach a pattern in the sand having the form HELLO WORLD. Would that have a meaning? Of course not. Meaning, in the cognitive sense, comes from minds and only from minds. Thus, it does not follow simply from the fact that "This sentence is false" is a grammatically well-formed English sentence that it expresses any proposition at all, must less a false one.

And even if we assume that the sentence does express a proposition, we still need some way of determining what that proposition is. That this cannot be done in the present case the following dialogue is intended to illustrate:
A: "This sentence is false" expresses a false proposition.
B: What proposition?
A: That "this sentence is false" expresses a false proposition.
B: Yes, but what proposition is that?
A: I already told you, that "this sentence is false" expresses a false proposition.
B: No, you haven't. What exactly does "a false proposition" refer to?
A: To the proposition that "this sentence is false" expresses a false proposition.
B: That's unenlightening because circular. What you're saying is that "This sentence is false" expresses the false proposition that "this sentence is false" expresses a false proposition that "This sentence is false" expresses a false proposition that .... The meaning of the proposition can never be pinned down because the analysis continually gets pushed back another step.
A: So?
B: Not every verbal pattern of the sort that could be used to express a proposition actually expresses a proposition. Why should anyone think this one does? That the meaning of this alleged proposition cannot be articulated in a non-circular way shows that the expression has no definite meaning and thus does not express a proposition.
Another problem with this version of the Liar Paradox is the assumption that every sentence must be either true of false. The principle of bivalence - that every proposition is either true or, if not true, false - applies to propositions, not sentences. And there is no good reason to suppose that it can be extended to sentences. For example, suppose an actor on the stage says "Methinks it is like weasel" (a line from Hamlet). He has uttered a sentence, but has he expressed a proposition? I think not. If it did express a proposition it would have to be something like I, Hamlet, think that yonder cloud is shaped like a weasel. But there is and never has been any Hamlet, hence there is no autobiographical 'I' to give a definite meaning to the first-person indexical, and there is and never has been any particular cloud to give a definite meaning to "it". So here we have a sentence that does not actually express any proposition, even though it could, in other contexts, be used to express a proposition.

I conclude, then, that neither of these two forms of the Liar Paradox is particularly problematic. There are other, more complex versions, such as those involving pairs of sentences with first saying of the second that it is true and the second saying of the first that it is false. But I don't see that any reason to think that these versions will prove any harder to defang provided we (1) maintain a sharp distinction between sentences and propositions as I have indicated, and (2) insist on pinning down the cognitive meaning of alleged propositions instead of resting content with mere verbal formulas.

Concepts and Propositions - I

Sorry, folks, about being absent from my blog for so long. I'm currently under a crunch-time of sorts regarding several different projects. Blogging for me is a spare-time thing, not an obsession, so when things get too hectic I have to take a break. And with this year's philosophy job search fast approaching, posting will continue to be sparse for the next several weeks.

Anyway, I am, yet again, rethinking my views on propositions. I have already written four posts on this topic (I, II, III, and IV). After the last post I thought I had everything settled in my mind, but recent reflections on tense logic (which I'll discuss in a later post) have brought a new issue to my attention: the distinction between concepts and propositions.

First, I take it to be a definitional truth that a proposition is a unit of meaning that is capable of bearing a truth value. One issue that arises concerning this is how to tell when we have such a unit of meaning. In particular, what I'd like is a test that will enable us to distinguish sharply between concepts and propositions. As I see it, (a) both concepts and propositions are units of meaning, and (b) necessarily, every proposition contains at least two concepts - a subject and a predicate. But then is a proposition merely a type of compound or complex concept? If so, then how can a proposition meaningfully be said to possess a truth value? If not, then what is the extra ingredient required to transform a complex concept into a proposition?

It seems to me that for a unit of meaning to qualify as a proposition it must be something that can intelligibly be either asserted or denied. This requirement can be understood in two different ways, depends on what we mean by "assert" and "deny". In the weaker sense, to "assert" or "deny" 'p' is just to utter a phrase like "'p' is true" or "'p' is false". In other words, take any putative proposition-expressing phrase, prefix it with "It is true that ..." or "It is false that ..." and you should get a verbally meaningful result. I'll call this the grammatical assertibility-deniability test (GAD test, for short). To illustrate, here are a few examples. In each case, the italicized phrase is the one being tested for propositionality.
  • It is true/false that dog. (Nonsensible. Not a proposition, but a concept.)
  • It is true/false that red roses and a bluejay. (Nonsensible. Not a proposition, but a compound concept.)
  • It is true/false that being about to sing the national anthem. (Nonsensible. Not a proposition, but a complex concept.)
  • It is true/false that 2+2=4. (Makes sense. A necessarily true proposition.)
  • It is true/false that a cat is on the mat. (Makes sense. A contingently true proposition.)
  • It is true/false that square circles exist. (Makes sense. A necessarily false proposition.)
I regard the GAD test as a necessary condition for propositionality, but I doubt that it is a sufficient condition. The reason is that there are various expressions that satisfy the test but that also are, in some sense, "pathological". For example:
  • It is true/false that this sentence is false.
  • It is true/false that I am lying right now.
Both of these are grammatically well-formed sentences, but they are also problematic.

The first is problematic because it is self-referential in a way that makes it impossible to assign a fully determinate meaning. I'll try to spell this out more in a follow-up post, but it suffices for now to point out that sentences aren't true or false in themselves but only by virtue of expressing a true or false proposition. So to say "this sentence is false" is to say that this sentence expresses a false proposition. But we are left no way to specify which proposition is false other than the proposition expressed by "this sentence is false", which brings us right back to where we started. The lack of any independent means of expressing the proposition in question indicates that the sentence is cognitively incomplete.

The reason why the second is problematic is that the proposition that I am lying right now creates a performative contradiction when it is genuinely (as opposed to merely verbally) asserted. After all, a genuine assertion is a sincere reflection of one's beliefs. A lie, however, is a deliberate misrepresentation of one's belief. Hence, a lie is by definition insincere, and thus not a genuine assertion.

What I want to suggest, then, is that the sort of assertibility and deniability that is relevant to propositionality is stronger than what the GAD test gives us. In the stronger sense, to "assert" or to "deny" are partially epistemic (or better, doxastic) notions. To "assert" p in this sense is not essentially to utter a phrase like "'p' is true", but rather to cognitively assent to p. Thus, a assertion or denial (in the verbal sense) is only a genuine assertion or denial if it is a sincere expression of a person's beliefs. For a unit of meaning to be assertible or deniable in this stronger sense it must be potentially believable. In other words, there must be some possible being in some possible world that could be in a position to give it either cognitive assent or dissent. I call this the cognitive assertibility-deniability test (CAD, for short).

CAD rules out as propositions anything that is either cognitively incomplete (as is the case with the self-referential example) or that is in principle unassertible in the cognitive sense (as is the case with the liar example). It is important to note that CAD does not require that a proposition be both cognitively assertible and cognitively deniable. It only has to satisfy one of those disjuncts to quality. The reason for this stipulation is that even if a necessarily false proposition like I am a married bachelor is not cognitively assertible in any possible world, it most certainly is cognitively deniable. Likewise, even if a necessarily true proposition like all triangles have three sides is not cognitively deniable in any possible world, it most certainly is cognitively assertible.

I believe that both the GAD and CAD tests for propositionality are useful, but the stronger CAD test may also be able to help resolving various sematic paradoxes, such as those stemming from self-reference and liar-type paradoxes. In short, if CAD is right, then such paradoxes arise from falsely supposing that all grammatically well-formed declarative sentence-tokens express propositions. Rather, such sentence-tokens express propositions if and only if they reflect the actual beliefs of the speaker.

To be continued ...