The Bearers of Truth-Values
Professor Norman Swartz
Simon Fraser University
Here is an outline of the major points of the dialectical argument, on pp. 68-86 in Possible Worlds, that addresses the question "What (sorts of) things have truth-values (i.e. bear truth-values)?" These notes are intended to supplement and to elucidate the argument in the text; they are not intended to substitute for that (more detailed) argument.
Objection: Thesis #1 is unclear because the terms "belief", "statement", etc. are ambiguous. We could be referring to acts of belief (or acts of stating, asserting, etc.), or we could be referring to the objects of those acts of belief (stating, asserting, etc.) We will examine each of these alternatives in turn (Theses #2 and #3).
Thesis: Acts of believing (stating, asserting, etc.) are those things that have truth-values.
Objection A: There are not enough acts of belief for them to constitute all truths and falsehoods. There will have been (when this universe finally expires) a large, but still only a finite, number of acts of belief; but there are an infinite number of truths and falsehoods (e.g. in arithmetic).
Objection B: Someone may believe some specific truth, let's say , while no one ever believes ~. Thus we would have a case of a proposition which had no contradictory. This would make a shambles of logic. We need, as our bearers of truth and falsity, some 'things' that exist in infinite numbers and are such that for every truth, there is (at least) one corresponding falsehood (i.e. its contradictory). Thus, we examine next the suggestion that the bearers of truth-values are not acts of believing (asserting, hypothesizing, etc.) but the objects of those beliefs (assertions, hypotheses, etc).
Thesis: The objects of belief (assertion, hypothesizing, theorizing, etc.) are the 'things' which bear truth-values.
Objection: It is unclear what sorts of 'things' the objects of belief are. Are 'objects of belief' to be regarded as sentences or as something else, viz. propositions? We need to examine these latter two alternatives. (We look at the first of these two latter proposals [viz. that the bearers of truth-values are sentences] in Theses #4-#9; and at the second [viz. that the bearers of truth-values are propositions] in Theses #10-#13.)
Thesis: Sentences are the 'things' that bear truth-values.
Objection A: Only certain kinds of sentences, namely indicative (very roughly "fact stating") sentences are plausible candidates for having truth-values. Interrogative sentences (e.g. "Where are the scissors?") and commands (e.g. "Wash the dishes!") do not bear truth-values – they are neither true nor false.
Objection B: Thesis #4 is unclear because the term "sentence" is ambiguous. We could be referring to sentence-tokens or we could be referring to sentence-types. We will examine each of these alternatives in Theses #5 through #9.
Why will we need five theses (#5-#9) rather than just two? Well, as we shall see, even distinguishing between sentence-tokens and sentence-types is not the end of the matter. We shall have to distinguish between those (indicative [see above]) sentences that are context-free and those that are not. So we next turn to consider the thesis that it is sentence-tokens that are the 'things' that bear truth-values.
Thesis: (Indicative) sentence-tokens are the bearers of truth-values.
Objection: The same difficulty arises as we saw earlier in Objection A to Thesis #2, namely there are more truths than there are sentence-tokens. (The number of sentence-tokens will always remain finite; the number of truths and the number of falsehoods are infinite.)
But if sentence-tokens do not exist in the requisite numbers, sentence-types (which are abstract objects) do exist in the requisite numbers: there are an infinite number of sentence-types. So, let's next examine the proposal that it is sentence-types which are the 'things' that bear truth-values.
Thesis: (Indicative) sentence-types are the 'things' which bear truth-values.
Objection: Thesis #6 will lead to a logic in which the law of noncontradiction is violated. Suppose Jack is ill and says "I am ill." What he says is true. Suppose Jill is perfectly healthy, but also says, "I am ill." Clearly, what Jill says is false. Now, if we identify the bearers of truth and falsity with sentence-types, then insofar as both Jack and Jill, while uttering different sentence-tokens, have uttered tokens of one and the same type, then that (one) type would turn out to be both true and false. But this latter consequence is intolerable. Is there any way to persevere with the theory that sentences are the bearers of truth and falsity?
One way that has been suggested to 'repair' the sentence-theory so as to avoid violating the law of noncontradiction is to introduce a further distinction: that between those (indicative) sentences that are context-free and those that are not (i.e. are not context-free).
Thesis: Context-free sentences are the bearers of truth-values.
Consider again the sentence-token "I am ill" uttered by Jack and another token (of that same type) uttered by Jill. Clearly these two persons, although using tokens of the same type, are making different claims: Jack is taking about himself (i.e. Jack), while Jill is talking about herself (i.e. Jill). There is a sense in which these two different tokens (of the same type) are being used to say different things (to make different claims - one about Jack, the other about Jill). So, perhaps, we should be looking, not at the literal sentences themselves, but at sentences which 'capture' the relevant differences we have just discussed. Let's paraphrase (i.e. 'translate within the very same language') these two sentence-tokens so as to make explicit all the tacit references (to the persons being referred to, to the time, to the place, etc.). If we do this, we might then have these context-free sentences:
'Repairing' (or elaborating) the two original sentences in this manner, now removes the objection that we will violate the law of noncontradiction. We can perfectly well maintain – without violating the law of noncontradiction – that the first of these two context-free sentences is true and that the second one is false.
So at this stage we seem to have arrived at the thesis that context-free sentences are those 'things' that bear truth-values.
Unfortunately, we cannot let the matter rest here, because, once again, there is an unclarity, namely, which is it to be: context-free sentence-tokens are the 'things' that bear truth-values, or context-free sentence-types are the 'things' that bear truth-values? We will need to examine each of these latter two possibilities (Theses #8 and #9).
Thesis: Context-free sentence-tokens are those 'things' that bear truth-values.
Objection: We run into the same problem with Thesis #8 that we earlier encountered with Theses #2 and #5: there simply are not enough of these 'things' (in this case context-free sentence-tokens) for them to be identified as the bearers of truth-values. The class of context-free sentence-tokens is a proper subset of the class of sentence-tokens, and since (we have already decided that) there there are insufficient numbers of sentence-tokens for them to be identified with the members of the class of truths and falsehoods, a fortiori, context-free sentence-tokens cannot be identified with the members of the class of truths and falsehoods. In short, whatever sorts of things truths and falsehoods turn out to be, they must 'exist' in infinite numbers. So – obviously – the next thesis to try is that of context-free sentence-types.
Thesis: Context-free sentence-types are those 'things' that bear truth-values.
Objection A: (We'll skip this first objection although you are welcome to read it in the textbook. We will jump ahead to the second paragraph on p. 78.)
Objection B: Even context-free sentences (whether tokens or types) can still be ambiguous, e.g. "Flying planes exhilarated Janice". The sentence is context-free: it contains no pronouns, and (presumably) does not depend on who says it or when. Even so, it is ambiguous. If it is taken to mean (something like) "Seeing (or hearing) planes flying overhead exhilarated Janice" we may suppose that that is true; but if it is taken to mean "Janice was exhilarated whenever she personally piloted a plane", we might suppose that that is false. Thus one sentence (depending on its 'interpretation') would turn out to be both true and false. Again, this is intolerable: it constitutes a violation of the law of noncontradiction. We must find a thesis that avoids this consequence.
Objection C: No theory that makes sentences – whether tokens or types, context-free or not – the 'things' that bear truth-values can be reconciled with the fact that (at least) some creatures (animals) that lack a language can, nonetheless, have (some true) beliefs.
What we are looking for is a theory that accommodates a number of disparate facts:
But if it is not (indicative) sentences (type /token /context-free /non-context-free) that are the 'things' that bear truth-values, what other candidates are there for that role?
Thesis: Propositions are those 'things' that bear truth-values.
Objection: OK, so if propositions are not any kind of (indicative) sentence, if they are not linguistic items, what, then, are they?!
Thesis: Propositions are the meanings of sentences.
Three objections are offered in the textbook. I will here offer just one (an elaboration of the paragraph that begins "Secondly, and more importantly, ..." on page 81.)
For there to be meanings of sentences there must be (on one account of what the meanings of sentences are) sentences. But if the theory that the 'things' that are the bearers of truth-values presupposes (or requires) that there exist sentences, then all the objections that have been levelled against Theses #4-#9 come flooding back. To avoid this latter consequence, we need an account of propositions which does not presuppose the existence of sentences, i.e. an account of propositions that does not make them 'depend' on the existence of sentences..
Thesis: Propositions are to be identified with sets of possible worlds.
This thesis may be stated this way: rather than saying that propositions are true in some possible worlds, we will simply identify each proposition with the set of possible worlds in which it is true. Sets of possible worlds are not linguistic entities; unlike sentences, they do not presuppose the existence of language-users, nor even, for that matter, conscious creatures. In addition, sets of possible worlds exist in infinite numbers. The concept of ambiguity does not even apply to sets of possible worlds.
On this account, the proposition that Canada is north of Mexico would (just) be the set of possible worlds in which (it is true that) Canada is North of Mexico. And the proposition that Mexico is South of Canada would (just) be the (very same) set of possible worlds as just described. I.e. the latter proposition is identical to the former. But a different proposition, e.g. that some stars explode, is true in a different set of possible worlds. (The worlds-diagram depicting the modal relation between the proposition that Canada is north of Mexico and the proposition that some stars explode is, of course, WD15.)
Objection A: We'll skip this objection.
Objection B: This account (i.e. Thesis #12) seems alright if we confine our examples (as in the paragraph "On this account..." above) to contingent propositions: nonequivalent contingent propositions will be identified with different sets of possible worlds. There will be as many different such sets as there are nonequivalent contingent propositions. But this 12th account gets into difficulty with noncontingent propositions. If propositions just are the sets of possible worlds in which they are true, then all necessarily true propositions would turn out to be identical (not just equivalent to one another – a tolerable oddity) and all necessarily false propositions would turn out to be identical. Put another way: on this 12th account, although there would be an infinite number of nonequivalent contingent propositions, there would be only one necessarily true proposition and only one necessarily false proposition. For example, the (necessarily true) proposition that
So, what's left?
Thesis: Propositions are not to be identified with sentences (tokens /types /context-free /non-context-free), not with the meanings of sentences, indeed not with any linguistic 'thing' at all, nor are they to be identified with possible worlds. Propositions are not to be identified with anything else. They are sui generis, a unique kind of thing. They are abstract entities, but they are a distinct kind of abstract entity.
Objection: Some philosophers, nominalists, object to (i.e. are uncomfortable with) positing any kind of abstract entity. For these philosophers, propositions are suspect or abhorrent entities. For other philosophers, abstract entities (including propositions) seem to be required – other sorts of things (e.g. sentences, meanings, etc.) just don't have the right sorts of properties to be the bearers of truth-values. The debate over the need to posit abstract entities has raged throughout all of the history of philosophy. Understand that the theory, concerning the bearers of truth-values, presented here is not universally accepted among philosophers. Even so, anyone who wants to go on in philosophy needs to know the major 'moves' in this series of dialectical steps whatever one's ultimate opinion may be as to the cogency of some of them.