Andrea Iacona

(Università del Piemonte Orientale)

Rethinking Bivalence

 

1. The philosophical controversy

Classical logic rests on the assumption that there are two truth values, truth and falsehood, which are mutually exclusive and jointly exhaustive. Truth and falsehood are mutually exclusive in that if something has one of them, then it does not have the other. They are jointly exhaustive in that there is no third way for something to be between having one of them and having the other. Although this simple and apparently unproblematic assumption is tacitly accepted by the great majority of logicians, it has always been surrounded by philosophical controversy. Doubts have been raised about its legitimacy, and hence about the legitimacy of classical logic. Usually, the assumption is explicitely stated in the form of a general principle, called the principle of bivalence. Accordingly, the philosophical controversy about its legitimacy is mostly framed in terms of the question whether or not the principle of bivalence holds.

The principle of bivalence is commonly phrased as the principle that every proposition is either true or false. Sometimes the term ‘statement’ is preferred to ‘proposition’ in phrasing the principle. But no significant difference seems to hinge on this preference, as the two terms are normally used as synonymous in this context. Way less common is to find the principle of bivalence phrased in terms of sentences or utterances, at least when it is presented as a general principle without restrictions. It could hardly make sense to claim that every sentence is either true or false. Sentences are not true of false simpliciter, but true or false relatively to the meaning they have and the way they are used by the speakers of the language they belong to. For example, the sentence ‘quadruplicity drinks procrastination’ is not evaluable as true or false, in that it doesn’t even have a meaning. Neither would it make sense to claim that every meaningful sentence is either true or false, as questions or commands are obviously not evaluable as true or false. Restricting the principle to meaningful declarative sentences wouldn’t do, for at least in some cases meaningful declarative sentences are evaluable as true or false only relatively to the circumstances of their use. For example the sentence ‘he is tall’ is evaluable as true or false only relatively to a context in which it is used to refer to this or that person. Utterances fare no better than sentences. Not many philosophers find natural to regard utterances as truth bearers. Besides, those who regard them as truth bearers tend to assume that ascription of truth to them is to be restricted to occasions in which something has been said to be the case. This obviously rules out the possibility that every utterance is either true or false. Rather, utterances turns out to be true or false to the extent to which they express propositions, as it normally taken for granted that what it is said to be the case is the proposition expressed by the utterance. Then, again, if the principle of bivalence is taken to hold, it is taken to hold for propositions.

As the principle of bivalence is commonly phrased as the principle that every proposition is either true or false, the philosophical controversy about its legitimacy is commonly framed in terms of the question whether or not every proposition is either true or false. Many logicians and philosophers seem to doubt that an affirmative answer may be returned to this question. The considerations to which they appeal are of various sorts. According to a line of argument that goes back to the ancient times, the rejection of the principle that every proposition is either true or false rests on considerations about the fatalist implications of ascription of truth or falsehood to future tense contingent propositions. Another line of argument against the principle hinges on considerations about vagueness. The idea is that propositions expressed by sentences containing vague words may have no determinate truth value. A third sort of considerations comes from Fregean semantics, and rests on the hypothesis that reference failure gives rise to truth value gaps, i.e., that sentences containing empty singular terms express propositions which lack a truth value. Lastly, considerations to the effect that there are propositions which are neither true nor false come from the intuitionistic reconstruction of mathematics and its underlying conception of truth as provability. As there is no guarantee that we can find either a proof or a disproof of certain mathematical propositions, we have no right to assume that those propositions are either true or false. Considerations such as those just seen naturally lead to revisionism about classical logic. That is, if classical logic rests on the principle that every proposition is either true or false, and the principle does not hold, then classical logic is not to be regarded as "correct" in some absolute sense. Accordingly, those who appeal to such considerations often construct or advertise systems of logic which are assumed not to embody the principle that every proposition is either true or false.

On the other hand, there are logicians and philosophers who seem not to be impressed by the purported arguments against the principle that every proposition is either true or false. Presumably, some of them simply think that those arguments are bad arguments. Others may concede that they have some degree of cogency or plausibility, but think that nonetheless there are stronger reasons to assume that the principle holds. Usually, the logicians and philosophers of this family think that classical logic is "correct" in some absolute sense, or that the systems of logic constructed or advertised by the logicians and philosophers of the other family give rise to theoretical troubles that are at least as thorny as those arising in connection with systems of classical logic. The assumption common to both families is that there is a philosophical question to be addressed whether the principle that every proposition is either true or false holds, and that the solution to this question depends on the weight that is to be assigned to the purported arguments in support of or against the principle. In what follows I argue that there is something wrong in this way of putting things. My point is that the usual way of understanding the principle of bivalence is misconceived, where the misconception lies in the underlying picture of propositions.

 

2. What is usually taken for granted about propositions

There is no widespread agreement among philosophers on the nature of propositions. But one thing that seems certain about them is that they are truth bearers, that is, entities to which truth or falsehood can be ascribed. We ordinarily speak of things said, stated, thought, and so on, where the things in question are what we identify by means of clauses of the form ‘that so-and-so’, in short, that-clauses. For example, in ‘Tom said that the sea is blue’ the expression ‘that the sea is blue’ seems to stand for what Tom said, namely, that the sea is blue. In accordance with this way of speaking, propositions are usually defined as the things we say, state, think, and so on, or what that-clauses stand for. As we normally use the adjectives ‘true’ and ‘false’ as predicates which can be attached to that-clauses, we seem to normally ascribe truth and falsehood to the things we say, state, think, and so on. For example, if one says ‘it is true that the sea is blue’, one seems to ascribe truth to what the expression ‘that the sea is blue’ stands for.

Another thing that seems certain about propositions is that they are other than sentences. Sentences are normally defined as sequences of words formed according to the syntactic rules of this or that language. By contrast, propositions are defined as what sentences express, and hence as things which are not themselves made of words. A familiar distinction in terms of which the difference is explained is that between the sentence one utters and the statement or assertion one makes by uttering it. Usually, the latter distinction is supported by considerations such as the following. A certain speaker may assert that so-and-so by uttering a certain sentence. But asserting that so-and-so is not the same thing as uttering that sentence. On the one hand, in order to assert that so-and-so it is not sufficient to utter that sentence, for one may utter that sentence and assert something else or nothing at all. On the other, in order to assert that so-and-so it is not necessary to utter that very sentence, for other sentences would do as well. The stronger conclusion is usually drawn that statements or assertions are other than sentences.

The assumption that propositions are truth bearers and the assumption that propositions are other than sentences are usually accompanied by a very influential and deep-rooted idea. The idea is that propositions are true or false in a sense that is different from that in which sentences can be said to be true or false. Sentences can be said to be true or false relatively to the meaning they have and the way they are used on certain occasions by the speakers of the language they belong to. By contrast, propositions are true or false independently of linguistic facts concerning the sentences that express them, and hence absolutely. In accordance with this idea, it is usually taken for granted that truth or falsehood are possessed by propositions as intrinsic or essential properties, namely, that for a certain proposition having a certain truth value is part of being that very proposition. The influence of the idea that propositions have a truth value intrinsically or essentially is clear in the standard treatment of the phenomenon of context-dependence. For example, it is usually taken for granted that the proposition expressed by

1. it is raining

as uttered on a certain occasion is something that cannot be individuated by means of (1) alone. We need, so to speak, more information than that contained in (1). For example, the proposition expressed by (1) as it is uttered in Barcelona the 16^th of October 2002 at 4 p.m. is something that cannot be individuated by means of (1) alone. One needs know that (1) is uttered in Barcelona the 16^th of October 2002 at 4 p.m. Thus, it is normally taken for granted that if there is a sentence by means of which the proposition in question can be individuated, it is the eternal sentence

(2) it rains in Barcelona the 16^th of October 2002 at 4 p.m.

It is easy to see the connection between the assumption that the proposition expressed by (1) on a given occasion cannot be individuated by means of (1) alone and the assumption that the proposition in question has its truth value intrinsically or essentially. On the one hand, if one starts from the assumption that the proposition expressed by (1) as it is uttered in Barcelona the 16^th of October 2002 at 4 p.m. has its truth value intrinsically or essentially, one is forced to allow that the proposition in question cannot be the same as that expressed by (1) as it is uttered in Barcelona the 16^th of October at 4.30 p.m. For it may be the case that in Barcelona it rains at 4 p.m. but it is sunny at 4.30 p.m. This leads to the conclusion that the proposition expressed by (1) on the first occasion must be something that cannot be individuated by means of (1) alone. On the other hand, the assumption that the proposition expressed by (1) on a given occasion is not individuated by means of (1) alone but it is individuated by means of an eternal sentence such as (2) leads us to think that the proposition in question has its truth value intrinsically or essentially. If it rains in Barcelona the 16^th of October 2002 at 4 p.m., then the proposition that it rains in Barcelona the 16^th of October 2002 at 4 p.m is intrinsically or essentially true.

More generally, the standard view is that if we take a sentence and we add all the relevant information about the context of its utterance, we get one proposition which is the proposition expressed by that sentence in that context. The proposition in question has a truth value absolutely, and hence intrinsically or essentially. To put things another way, the proposition expressed is complete with respect to the determination of its truth condition, in the sense that it embodies all the information needed to specify what has to be the case in order for it to be true. Consequently, the determination of its truth value does not depend on facts concerning the utterance of the sentence. On the contrary, the sentence uttered is not complete with respect to the determination of its truth condition, in that part of the information needed to specify what has to be the case in order for it to be true depends on facts concerning its utterance. Consequently, the determination of its truth value partly depends on such facts.

The standard treatment of the phenomenon of ambiguity is analogous. It is commonly assumed that ambiguous sentences are not complete with respect to the determination of their truth condition, in that they are capable of being interpreted in different ways, while the propositions they express are complete with respect to their truth condition, in that they are not themselves capable of being interpreted in different ways. For example, the sentence

(3) visiting relatives can be boring

is ambiguous, in that the expression ‘visiting relatives’ occurring in it may be interpreted in more than one way. Accordingly, the truth condition of (3) is relative to the interpretation of that expression. But the proposition expressed by (3) under this or that interpretation is complete with respect to the determination of its truth condition, in that it is not itself capable of being interpreted in different ways.

The influential and deep-rooted idea that propositions are true or false in a sense that is different from that in which sentences can be said to be true or false leads to the equally influential and deep-rooted idea that propositions are the bearers of truth, or that they are the primary bearers of truth. Sometimes the latter idea is accompanied by the claim that sentences are not bearers of truth. At other times it is accompanied by the weaker claim that sentences are true or false "derivatively". Accordingly, it is maintained that for a sentence to be true or false is for the proposition it expresses to be true or false. The claim that propositions are the (primary) bearers of truth goes together with the view that propositions are the things that (in the primary sense) function as the terms of logical relations like entailment, contradiction, and so on. To say that one thing entails another is to say that it cannot be the case that the first is true and the second is false. Similarly, to say that two things contradict each other is to say that it is impossible for both to be true. Then, it is argued, the items that constitute the (primary) terms of these relations are the things that are the (primary) bearers of truth and falsehood, namely, propositions.

 

3. What cannot be taken for granted about propositions

The justification of the notion of proposition outlined in §2, call it traditional notion of proposition, is seldom addressed in explicit terms. Most philosophers seem to assume that the notion is justified in that it rests on a solid intuitive basis. But this assumption is at least in part misguided. It is certainly plausible to say that to some extent the traditional notion of proposition finds support in our ordinary way of speaking. If we define propositions as the things we say, state, think, and so on, or the referents of that-clauses, our definition may be regarded as intuitive, in that it does not require any sort of theorization but the intuitive distinction between saying, stating, thinking, that so-and-so, and the thing said, stated, thought, that so-and-so. It may also be in accordance with our ordinary way of speaking to claim that propositions so intuitively defined are truth bearers, and that they are other than sentences. However, the idea that propositions have a true value absolutely, and hence intrinsically or essentially, can hardly be regarded as intuitive. Our ordinary way of identifying the things we say, state, think, and so on, by means of that-clauses does not provide reasons in support of that idea. Suppose Tom utters the sentence

(1) it is raining

at 4 p.m. One may report Tom as saying that it rains at 4 p.m. Otherwise, one may report Tom as saying that it is raining. In both cases the thing said by Tom is identified by means of a that-clause. Suppose now that at 4.30 p.m. Tom utters (1) again. One may report Tom as saying that it rains at 4.30 p.m. Otherwise, one may report Tom as saying that it is raining. In both cases the thing said by Tom is identified by means of a that-clause. Given our intuitive definition of proposition, it seems correct to say that on the two occasions Tom stated different propositions, namely, the propostition that it rains at 4 p.m. and the proposition that it rains at 4.30 p.m. But it seems equally correct to say that on the two occasions Tom stated the same proposition, namely, the proposition that it is raining. Our way of identifying propositions by means of that-clauses leaves indeterminate whether the proposition stated by Tom on the first occasion is the same as the proposition stated by Tom on the second occasion. Accordingly, it leaves indeterminate whether the proposition stated by Tom on the first occasion is capable of changing its truth value. If Tom stated two different propositions on the two occasions, then the first proposition is not affected by truth value changes: if it rains at 4 p.m., it remains true that it rains at 4 p.m. The same goes for the second proposition. Instead, if Tom stated the same proposition on both occasions, then the proposition in question can change its truth value. For it may be the case that it rains at 4 p.m. but it is sunny at 4.30 p.m. Therefore, nothing forces us to conclude that the proposition stated by Tom has a truth value intrinsically or essentially. In order to draw that conclusion it has to be assumed that the proposition stated by Tom at 4 p.m. is different from the proposition stated by Tom at 4.30 p.m., although both propositions can be identified by means of a that-clause embedding (1). This amounts to assuming that the proposition stated by Tom at 4 p.m. is something that cannot be individuated by means of (1) alone. The same goes for the proposition stated by Tom at 4.30 p.m. But there seems to be no reason to make this assumption unless it is taken for granted just what is at issue, namely, that those propositions have their truth value intrinsically or essentially.

More generally, nothing in our ordinary way of speaking leads us to think that the amount of information required by the individuation of a certain proposition must go beyond that encompassed by the identification of that proposition by means of a certain that-clause. Therefore, it is in accordance with our ordinary way of speaking to assume that insofar as different that-clauses may equally be used to identify what is said by uttering a certain sentence in a certain context, different propositions may equally be individuated in that context. This is to say that given any sentence and any context, there is no such thing as the proposition expressed by that sentence in that context. The case of ambiguity is analogous in this respect. For example, given the sentence

(3) visiting relatives can be boring

there may be circumstances in which it is equally correct to identify the thing said by uttering it as that it can be boring to visit relatives and to identify it as that visiting relatives can be boring. On the assumption that propositions can be individuated by means of that-clauses embedding ambiguous sentences just as by means of that-clauses embedding unambiguos sentences, this means that there may be circumstances in which (3) can be said to express the proposition that visiting relatives can be boring just as it can be said to express the proposition that it can be boring to visit relatives. Clearly, propositions so individuated are not guaranteed to have a truth value intrinsically or essentially, and hence they are not guaranteed to have a truth value absolutely.

To put things another way, nothing in our ordinary way of speaking leads us to think that propositions are complete with respect to the determination of their truth conditions. Take the proposition that it is raining. This proposition is obviously not complete with respect to the determination of its truth condition. The same goes for the proposition that visiting relatives can be boring. In substance, our ordinary way of speaking does not support the influential and deep-rooted idea that propositions are true or false in a sense that is different from that in which sentences can be said to be true or false. Accordingly, it does not support the claim that propositions are the bearers of truth, or that they are the primary bearers of truth. For the usual considerations to the effect that propositions are the bearers of truth, or that they are the primary bearers of truth, essentially rest on that idea.

So far I argued that part of the traditional notion of proposition is not intuitive, or that the traditional notion of proposition is not as intuitive as it is commonly taken for granted. Certainly, this does not entail that we have no reason to adopt that notion. It might be the case that we have a justification of it as a technical notion which plays some "theoretical role" in philosophical explanations. At least, this is what some of its advocates say. However, I doubt that such a justification may be provided. The fact is that the non-intuitive part of the traditional notion of proposition is to a good extent explanatorily idle. Even if we assume that some technical notion of proposition is to be adopted for explanatory purposes, it is simply not obvious that the explanatory purposes require the notion in question to be one according to which propositions enjoy properties such as that of having a truth value intrinsically or essentially. If a theoretical entity is postulated in order to explain certain phenomena, then the properties that can rightfully be attributed to the entity are the properties that play some role in the explanation of those phenomena. That is, we have reason to attribute a certain property to the entity insofar as some entity with that property is needed in order to explain those phenomena. However, it is not clear that properties such as that of having a truth value intrinsically or essentially are needed in this sense. Arguably, most of the things philosophers may want to explain about the things we say, state, think, and so on can be explained without resorting to them. For example, one of the things philosophers may want to explain is that if Tom and John utter the sentence

(4) my house is red

on a certain occasion, there is a sense in which they say different things on that occasion. The same goes for similar cases. Let us suppose that some technical notion of proposition is to be adopted in order to capture the sense in which Tom and John say different things. It seems that the property required by the "explanatory role" of the notion is that of saying the same thing about the same thing, or something like that. But saying the same thing about the same thing does not amount to having a truth value intrinsically or essentially. Take the following case. On Monday morning Tom utters (4) and his house is red. On Tuesday morning Tom utters (4) but his house is no longer red because it was painted on Monday night.

Obviously, in the latter case another difference may be taken into account, namely, that between the circumstances which make true (4) as it is uttered on Monday morning and the circumstances which make true (4) as it is uttered on Tuesday morning. In similar fashion, one may distinguish between the circumstances which make true (1) as it is uttered at 4 p.m. and the circumstances which make true (1) as it is uttered at 4.30 p.m. More generally, some technical notion of proposition might be adopted in order to account for the circumstances which make true sentences as we utter them on this or that occasion. But such a notion would collapse into that of truth condition, namely, that of the way the world is to be arranged in order for something to be true. Then, propositions in the supposed technical sense could hardly be regarded as the things we say, state, think, and so on, at least in the ordinary sense in which the things we say, state, think, and so on are the kind of things which can be true or false. For truth conditions are certainly not the kind of things which can be true or false. Therefore, the question whether or not propositions are to be postulated in this supposed technical sense is utterly irrelevant to the question we are dealing with, i.e. what notion of proposition is to be adopted in order to rightly understand the principle of bivalence.

The non-intuitive part of the traditional notion of proposition is not only explanatorily idle. It is explanatorily harmful, in that it surreptitiously introduces a theoretically biased reform of our intuitions. Take the case of (1). On the assumption that the traditional notion of proposition is to be adopted, the situation may be described by saying that on the two occasions Tom utters the same sentence but states two different propositions. That is, the sense in which Tom says the same thing on the two occasions is accounted for in terms of uttering the same sentence, while the sense in which Tom says different things is accounted for in terms of expressing different propositions, where propositions are understood in the traditional way as the bearers of truth or the primary bearers of truth. But this way of describing things leads to neglect that on the two occasions Tom states the same proposition in the intuitive sense, which seems to be in accordance with our ordinary way of speaking. On the one hand, the sense in which Tom says the same thing on the two occasion is not just a matter of uttering the same sentence. We are willing to admit that Tom said something on both occasions, namely, that it is raining. The something in question is not the sentence (1) itself. Even if Tom uttered the Italian sentence ‘piove’ instead of (1) on one of the two occasions, we could still report him as saying the same thing on both occasions, namely, that it is raining. On the other, the sense in which Tom says different things on the two occasions is not the only sense in which truth or falsehood can be ascribed to what Tom says. We are willing to admit that the thing which is the same on both occasions may be true on the first occasion and false on the second occasion. Thus, it seems that in this case the traditional notion of proposition affects our description of what is to be explained more than it helps us to explain it.

 

4. Interpretation

As the non-intuitive part of the traditional notion of proposition can hardly be justified, it seems right to accept at most its intuitive part. This amounts to saying that nothing but the intuitive notion of proposition considered above is to be accepted. According to that notion propositions are like sentences, in that they are true or false relatively to linguistic facts, and hence may fail to be complete with respect to the determination of their truth conditions. The crucial feature of propositions so understood is that they can be interpreted just like sentences. For example, just like the sentence (1) can be interpreted in more than one way, the proposition that it is raining can be interpreted in more than one way, where different interpretations induce different truth conditions on the proposition. More generally, insofar as the truth condition of the sentence ‘so-and-so’ may be relative to a certain interpretation, the truth condition of the proposition that so-and-so may be relative to that interpretation.

In §2 and §3 we saw how the truth condition of a sentence may depend on our way of interpreting it in standard cases of context-dependence and ambiguity. But independently of the phenomena of context-dependence and ambiguity, there seems to be another sense in which the truth condition of a sentence may depend on our way of interpreting it. It is the sense in which the sentence provides a description of reality which is capable of being specified in more than one way. Sentences provide descriptions of reality, that is, they provide descriptions that speak of ways for things to be. A sentence ‘so-and-so’, as it is uttered on a given occasion, describes things as being in a certain way, namely, as being such that so-and-so. But the question whether things are such that so-and-so may fail to be settled for every possible occasion. There may be occasions in which it is equally compatible with the correct use of the sentence ‘so-and-so’ to describe things as being such that so-and-so and to describe things as not being such that so-and-so. Take the sentence

(5) it is blue

This sentence may be used to describe some ink as being blue. However, there are different things to be said about what would count as being blue. Consider some ink which is black in fluid form but writes blue. There is a sense in which it seems right to say that it is blue. But there is also a sense in which it seems right to say that it is not blue. This means that it is equally compatible with the correct use of (5) to describe the ink in question as being blue and to describe it as not being blue. Things may be put another way by saying that the description provided by (5) admits of two specifications. One is that according to which blue things include ink which is black in fluid form but writes blue. The other is that according to which blue things do not include such ink. More generally, the descriptions of reality provided by our sentences may admit of specifications. A specification of a description provided by a certain sentence is a way of sharpening the description compatibly with the linguistic meaning of the sentence by taking into account possible ways for things to be that are not taken into account by the description. Thus, different specification of the description provided by a certain sentence may be equally compatible with the linguistic meaning of that sentence. This means that the linguistic meaning of a certain sentence leaves indeterminate which of the possible specifications of the description provided by that sentence is to be adopted on this or that occasion. The choice between specifications is occasion-relative and depends on what we take speakers to have in mind.

On the hypothesis that propositions are like sentences, the point may be extended to propositions. This is to say that the truth condition of a proposition may depend on our interpretation of it in the sense that it provides a description of reality which is itself capable of being specified in more than one way. This leads to the conclusion that interpreted propositions are not to be conceived as analogous to propositions in the traditional sense. Unlike propositions in the traditional sense, interpreted propositions are are not complete with respect to the determination of their truth conditions. The disanalogy between interpreted propositions and propositions in the traditional sense has important philosophical implications. There is a picture about our way of representing reality that has dominated a good part of the philosophy of language of the last century. The picture is that our way of representing reality is such that the representation has the same grain or the same format as the thing represented, in the sense that the thing represented is completely "mirrored" in the representation. According to this picture, all that is needed to determine whether the reresentation is correct or not is contained, so to speak, in the representation itself and in the relevant part of reality. Propositions in the traditional sense are intended to be representations of this kind. Then, saying that interpreted propositions are not like propositions in that sense amounts to saying that there are no such representations, and hence that the whole picture is misguided.

 

5. How all this matters to our initial question

The principle of bivalence is usually understood as the principle that every proposition is either true or false. On that understanding, truth and falsehood are properties that are possessed by propositions absolutely, in that each proposition is taken to have one of them simpliciter. But we saw that truth or falsehood can be ascribed to propositions only relatively to our way of interpreting them. Then, it simply makes no sense to say of a given proposition that it has one of them simpliciter, and hence it makes no sense to say that every proposition is either true or false. This suggests that the philosophical controversy about the legitimacy of the principle of bivalence rests on a misconception. As it makes no sense to say that every proposition is either true or false, there is no interesting question to be addressed whether or not every proposition is either true or false. That is, there is no interesting question to be addressed whether or not the principle of bivalence, understood in the usual way, holds. This is what I take to be the negative moral of the paper.

The positive moral goes as follows. There seems to be one understanding of the principle of bivalence on which it does make sense to ask whether or not the principle holds, namely, that on which every proposition is such that given a certain interpretation of it, either truth or falsehood are to be ascribed to it relatively to that interpretation. Assuming that this is the question to be addressed, there is at least one plausible sense in which an affirmative answer can be returned to it. It is the sense in which according to each interpretation either the proposition describes things as they are or it describes things as they are not. Suppose that a certain proposition describes a certain portion of ink as being blue. Since there are different understandings of being blue, there are different ways of interpreting that proposition. But given each understanding of being blue, either that portion of ink is blue on that understanding or it is not blue on that understanding. This means that according to each interpretation of that proposition, either the proposition describes things as they are or it describes things as they are not. It is quite natural to assume that a true proposition is a proposition that describes things as they are, and that a false proposition is a proposition that describes things as they are not. Therefore, it is quite natural to assume that, according to each interpretation of that proposition, either the proposition is true or it is false. More generally, for any given interpretation of the proposition that p, either the world is such that p or it is such that not-p. On the assumption that the proposition that p is true just in case p and false just in case not-p, this entails that the proposition that p is either true or false.

To say that the principle of bivalence holds in the sense considered is not to say that every proposition as it is actually interpreted on this or that occasion is either true or false. It may happen that our interpretation of a certain proposition on a certain occasion is less than adequately specified for the purpose of ascribing truth or falsehood to it on that occasion. The following example is drawn from Austin and Travis. Zoe unexpectedly expires in her chair before the fire. Some moments after her last breath, Max and Pia are discussing whether to pay a visit. Pia suspects that Zoe is out, but Max says that she will be at home. The question is whether things are as Max said. There is an understanding of being at home on which that is where Zoe is, dead in her chair. There is another understanding of being at home on which Zoe is no longer at home since, as the euphemism suggests, the departed are no longer with us. But Max is not to be understood to speak of her being at home on one of these understandings. Therefore, it seems that the way the world is arranged fails to decide whether or not things are the way described. To put things in our terms, the way the world is arranged provides no reason to say that the proposition that Zoe is at home as it is actually interpreted on that occasion is true, and, equally, no reason to say that it is false.

The fact that propositions as they are actually interpreted on this or that occasion are not guaranteed to be either true or false depends on some form of underspecification of interpretations essentially due to the fact that interpretations take into account what speakers have in mind. In the case considered, the interpretation of the proposition that Zoe is at home is less then adequately specified for the purpose of ascribing truth or falsehood to it in that it takes into account what Max has in mind. There obviously are specifications of that interpretation which make the proposition either true or false. But they are arbitrary with respect to what Max has in mind. Cases of underspecification such as this are compatible with the hypothesis of bivalence in that ascription of truth or falsehood to propositions does not entirely depend on what speakers have in mind. For example, ascription of truth or falsehood to the proposition that Zoe is at home does not entirely depend on what Max has in mind. The arbitrary specifications of its interpretation which make it either true or false may be legitimate for the purpose of ascribing truth or falsehood to it. Therefore, it seems correct to say that the hypothesis of bivalence holds for that proposition.

As the fact that propositions as they are actually interpreted on this or that occasion are not guaranteed to be either true or false has little relevance to the question whether or not the principle of bivalence holds, it has little relevance to the question whether or not classical logic is "correct" in some absolute sense. Classical logic rests on the assumption that there are two mutually exclusive and jointly exhaustive truth values, truth and falsehood, in that it deals with logical relations involving things which are truth valued in this sense. But it does not rest on the assumption that propositions as they are actually interpreted on this or that occasion are such things. For it is not part of classical logic to say what things are truth valued in the sense it presupposes. It may well happen that our interpretation of a certain proposition on a certain occasion is less than adequately specified for the purpose of ascribing truth or falsehood to it on that occasion. But this simply means that so interpreted that proposition is not one of the things classical logic speaks of. In other words, propositions may be regarded as the kind of things classical logic speaks of to the extent to which their interpretation is assumed to be adequately specified for the purpose of ascribing truth or falsehood to them.

 

References

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Dummett, M. 1978. Truth and Other Enigmas, Harvard University Press, Cambridge, Massachusetts.

Fine, K. 1975. Vagueness, Truth and Logic, Synthese 30, 265-300, reprinted in Keefe and Smith 1997.

Goble, L. 2001. (ed.), The Blackwell Guide to Philosophical Logic, Blackwell, Oxford.

Grayling, A. C. 1982. An Introduction to Philosophical Logic, Blackwell, Oxford.

Haack, S. 1978. Philosophy of Logics, Cambridge University Press, Cambridge.

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since december 2000