Truth-Bearers and Truth Value*

I. Introduction The purpose of this document is to explain the following concepts and the relationships between them: statements, propositions, and truth value. In what follows each of these will be discussed in turn.

II. Language and Truth-Bearers A. Statements 1. Introduction For present purposes, we will define the term “statement” as follows. Statement: A meaningful declarative sentence.1 It is useful to make sure that the definition of “statement” is clearly understood.

2. Sentences in General To begin with, a statement is a kind of sentence. Obviously, not every string of words is a sentence. Consider: “John store.” Here we have two nouns with a period after them—there is no verb. Grammatically, this is not a sentence—it is just a collection of words with a dot after them. Consider: “If I went to the store.” This isn’t a sentence either. “I went to the store.” is a sentence. However, using the word “if” transforms this string of words into a mere clause that requires another clause to complete it. For example, the following is a sentence: “If I went to the store, I would buy milk.” This issue is not merely one of conforming to arbitrary rules. Remember, a grammatically correct sentence expresses a complete thought.2 The construction “If I went to the store.” does not do this. One wants to

By Dr. Robert Tierney. This document is being used by Dr. Tierney for teaching purposes and is not intended for use or publication in any other manner. 1 More precisely, a statement is a meaningful declarative sentence-type. Sentence-types are to be distinguished from sentence-tokens. To see the difference, consider the items in Box A. Box A

The cat is on the mat. The cat is on the mat. The cat is on the mat. The cat is on the blue mat.

Box A contains two sentence-types, but four sentence-tokens. The first three items in Box A are all sentence- tokens of a single sentence-type. The last item in Box A is a single sentence-token of a sentence-type that is different from that of the first three items. Each individual utterance or inscription of a sentence is a sentence- token. The general form of the words being uttered or inscribed is the sentence-type. The same sentence—or, as we would now say, sentence-token—can uttered (spoken) or inscribed (written) on any number of occasions. Each such utterance or inscription is a sentence-token of the same sentence-type. Sentence-tokens of the same sentence type are all comprised of exactly the same words and punctuation in exactly the same order. (Of course, to be more precise, we could distinguish word-tokens from word-types; but we needn’t worry about that here.) For most purposes, we will be discussing sentence-types rather than sentence-tokens. Hence, to avoid unnecessarily complicated language, you can take the word “sentence” to mean “sentence-type” unless the contrary is specifically indicated. In other words, we will generally use the term “sentence” to be shorthand for “sentence-type.” 2 Strictly speaking, the point about sentences expressing a complete thought only applies to sentences that are both grammatically correct and meaningful. Some statements can be grammatically correct but not meaningful. See supra §II.A.4.

Page 1 of 4 Truth-Bearers and Truth Value v4.0 say: ‘“If you went to the store” what? You haven’t said anything. You haven’t completed the thought.’”

3. Declarative Sentences The definition of “statement” also tells us that a statement is a particular kind of sentence— a declarative sentence. Consider the following: (S1) Shut the door! This is an imperative sentence. Imperative sentences express a command or a request. Imperative sentences cannot be true or false. That is, it makes no sense to talk of truth or falsity here. Think about it. What would it mean to say S1 is true (or false)? It doesn’t make any fact claim. Suppose that the person to whom S1 is addressed, call him John, complies by shutting the door in question. Would that mean that S1 is true? No it wouldn’t. S1 does not say “John will shut the door.” It does not purport to report or represent a fact about the door being closed at some point in the future, or about who will close the door, or anything of the sort. It simply commands John to close the door.

Similarly, an interrogative sentence cannot be true or false. An interrogative sentence is a sentence that asks a question. Consider the following. (S2) Did John close the door? S2 is not a candidate for being true or false. It does not purport to express or represent any fact. It just asks whether something is the case. S2 doesn’t claim that something is the case; it asks whether it is the case. Hence, there is there is no claim being made that could be either true or false.

A declarative sentence, in contrast to an imperative sentence and an interrogative sentence, does make a fact claim.3 Roughly speaking, a declarative sentence is always a claim that something or other is the case or, to put it another way, that the world4 is thus-and-so. Another way to put this point is to say that a declarative sentence claims that a certain state-of-affairs obtains. Consider the following. (S3) The cat is on the mat. (S4) Is the cat on the mat? (S5) John, please put the cat on the mat. S3 is a declarative sentence. It claims that a certain cat is on a certain mat. By contrast, neither S4 (an interrogative sentence) nor S5 (an imperative sentence) makes a fact claim. Since S3 makes a fact claim, it makes sense to ask whether it is true or false.

3 After the discussion of “propositions” in what follows, we will see that we should revise what is said in this paragraph to the extent that it is not actually (meaningful) declarative sentences (i.e. statements) that are true or false, but, rather, the propositions that they express. See supra §§II.B. & II.C.

4 The term “world” in this sort of context does not mean “planet earth.” Rather, it means “the universe” in the broadest possible sense of the term or, if you like, “reality” or “all that is the case.”

Page 2 of 4 Truth-Bearers and Truth Value v4.0 4. Meaningful Sentences Finally, we come to the last term in our definition of “statement.” We said that statements are meaningful declarative sentences. Consider the following. (S6) Colorless green ideas sleep furiously.5 While, in grammatical terms, this is a properly formed declarative sentence, when we look at its content we see that it is nonsense, i.e. that it is meaningless. To say that it is meaningless is just to say that it fails to successfully articulate a fact claim. Thus, it does not succeed in expressing anything that could be either true or false.6

B. Propositions For present purposes, we will define the term “proposition” as follows. Proposition: The meaning of a statement. To get a clearer understanding of the definition, consider the following. (S7) It’s raining. (S8) Es Regnet. S7 and S8 mean precisely the same thing—they just express it in different languages (English and German, respectively). This is to say that S7 and S8 express the same proposition. To make sure that the point is clear, consider, also, the following. (S9) NY is larger than LA. (S10) LA is smaller than NY. Notice the differences between S9 and S10. S9 begins with the word “NY” and ends with the word “LA.” S10 begins with the word “LA” and ends with the word “NY.” S9 contains the word “larger” while S10 does not. S10 contains the word “smaller” while S9 does not. Clearly, S9 and S10 are different sentences. Nonetheless, S9 and S10 express the same proposition in that they make the same fact claim about the respective sizes of NY and LA. From what has been said we can see that propositions are expressed in declarative sentences, but that more than one declarative sentence can be used to express a single proposition.

C. Propositions as Truth Bearers For our purposes, the only things that can be accorded the status of true or false are propositions. In that a proposition is the meaning of a declarative sentence a proposition is a fact claim. In this regard, we can now see that what was said in §II.B.4. about (meaningful) declarative sentences making fact claims is more correctly attributed to the propositions that those declarative sentences express.

Finally, we should make the following terminological note. We will typically use the convenient term “statement” when we really mean “proposition expressed by the statement.” As long as we know the difference between propositions and statements, this convenient usage will not cause any problems. In contexts where clarity requires that we

5 This particular example was first used by the linguist, Noam Chomsky, and has since become a stock example in the relevant literature.

6 Further discussion of meaningless sentences can be found elsewhere. See John Hospers, An Introduction to Philosophical Analysis , Ch. 1: “Knowledge,” §1: “Truth” (3rd ed. 1990).

Page 3 of 4 Truth-Bearers and Truth Value v4.0 distinguish between a statement, properly speaking, and the proposition that it expresses, we will mark out this distinction.

III. Propositions and Truth Values At this point, we will introduce another technical term: “truth value.” We will not offer a formal definition of this term. To keep things simple, there are only two things that we need to know about the concept of “truth value” for this course. The first is that all and only propositions have a truth value. The second is that there are only two possible truth values, namely, “true” and “false.” Consider the following two statements: (S11) Houston is a city in Texas. (S12) Beijing is a city in Texas. The truth value of the proposition expressed by S11 is “true.” The truth value of the proposition expressed by S12 is “false.” That is all there is to say about the truth value of those propositions. Of course, we might encounter a statement expressing a proposition whose truth value is unknown to us. In fact that is often the case. Consider the following statement. (S13) At 1:15 PM, on January 1, 1905, there was at least one person named “Albert” standing in Red Square in Moscow, Russia. Is the proposition expressed by S13 true? I haven’t the slightest idea, and would guess that you don’t either. But there is one thing that we can say for sure about the truth value of the proposition expressed by S13: It is either true or it is false. That is to be understood so as to imply that the proposition expressed by S13 is not both true and false. It is logically impossible7 that it be both true and false.8 Moreover, it cannot be the case that the proposition expressed by S13 is neither true nor false.9 Again, it must be one or the other: either it is true or it is false.

7 The concept of logical impossibility is something that we may discuss later in the course. For present purposes, something is logically impossible if the statement that purports to express the idea is either self-contradictory or meaningless. 8 Philosophers and logicians refer to this general principle as “the law of non-contradiction.” Of course, there are some philosophers who disagree with this view. You can look up the issue on the Internet if you are interested. For our purposes, we will assume that the law of non-contradiction is correct. 9 Philosophers sometimes call this the “law of bivalence” or the “principle of bivalence.” Of course, there are philosophers who criticize this view. You can look up the issue on the Internet if you are interested. For our purposes, we will assume that the principle of bivalence is correct.

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