The Relationship between the Truth Value of Premises and the Truth Value of Conclusions in Deductive Arguments I. The Issue in Question This document addresses one single question: What are the relationships, if any, between the truth or falsity of the premises of a deductive argument and the truth or falsity of the conclusion of that argument? II. Framing the Issue: Statements and Their Truth Values The first thing to note is that we are talking about deductive arguments only. This document has nothing to say about non-deductive arguments. The second thing that we need to take note of is a bit of terminology. The title of this document makes use of a technical term, namely, truth value. To keep things simple, there are only two things that you need to know about the concept of truth value for this course. The first is that all and only meaningful statements have a truth value. The second is that there are only two possible truth values, namely, true and false. Consider the following two statements: (S1) Houston is a city in Texas. (S2) Beijing is a city in Texas. The truth value of S1 is true. The truth value of S2 is false. That is all there is to say about the truth value of those statements. Of course, we might encounter a meaningful statement whose truth value is unknown to us. In fact that is often the case. Consider the following statement. (S3) At 1:15 PM, on January 1, 1905, there was at least one person named Albert standing in Red Square in Moscow, Russia. Is S3 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 S3: S3 is either true or it is false. That is to be understood so as to imply that S3 is not both true and false. It is logically impossible 1 that it be both true and false. 2 Moreover, it cannot be the case that S3 is neither true nor false. 3 Again, it must be one or the other: either it is true or it is false. Note that we did restrict our rule about truth value to statements, 4 and by statements we meant, in effect, declarative sentences. 5 Consider the following: 1 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. (The idea of meaningless statements is discussed further in what follows.) 2 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. 3 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 the issue up on the Internet if you are interested. For our purposes, we will assume that the principle of bivalence is correct. 4 There are important philosophical disputes about whether the entities having truth value are best understood as sentences, propositions, or something else. ( Proposition is a special technical concept in philosophy.) We won t worry about these disputes. We will simply assume that it is best to talk about declarative sentences, i.e. statements, as being true or false. Page 1 of 5
(S4) Shut the door! This is an imperative sentence. Imperative sentences express a command or a request. It makes no sense to talk of truth or falsity here. Think about it. What would it mean to say S4 is true (or false)? It doesn t make any fact claim. Suppose that the person to whom S4 is addressed, call him John, complies by shutting the door in question. Would that mean that S4 is true? No it wouldn t. S4 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 does not have a truth value. An interrogative sentence is a sentence that asks a question. Consider the following. (S5) Did John close the door? This sentence 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. It doesn t claim that something is the case; it asks whether it is the case. So there is there is no claim being made that could be either true or false. We also restricted the application of our rule about truth value to meaningful statements not just any old statement will do. Consider the following. (S6) Colorless green ideas sleep furiously. 6 While, in grammatical terms, this is a properly formed declarative sentence, when we look at its content we see that it is nonsense. To say that it is nonsense 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. III. The Relationship between the Truth Value of Premises and the Truth Value of Conclusions Having clarified the framework of our discussion, we can now answer our central question about the relationship between the truth or falsity of the premises of a deductive argument and the truth or falsity of the conclusion of that argument. The answer is this. There is only one law-like relationship between the truth value of premises and the truth value of a conclusion in a deductive argument. That relationship is expressed in the definition of deductive validity. 5 Note that 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. The construction If I went to the store. does not do this. One wants to say: If you went to the store what? You haven t said anything. You haven t completed the thought. (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 the discussion of this issue below.) 6 This particular example was first used by the linguist, Noam Chomsky, and has since become a stock example in the relevant literature. Page 2 of 5
What does the concept of deductive validity tell us? As you may recall, it tells us that an argument that is deductively valid is an argument that exhibits the following relationship between its premises and its conclusion: If all the premises were true, then the conclusion would have to be true. It is vital that you notice the function of the word if. We have not said anything about whether the premises are, as a matter of fact, true. The actual truth or falsity of the premises is completely irrelevant to the issue of whether an argument is valid or invalid. Consider the following argument. 1. Obama is President. Argument #1 2. Any person who is the president is a government employee. Obama is a government employee. This argument is valid because it is the case that if the premises were true (and in this case they are) then it must be true that Obama is a government employee. There is just no other possible conclusion. But this does not depend upon the truth of the premises. Consider the following argument. 1. Romney is President. Argument #2 2. Any person who is the president is a government employee. Romney is a government employee. Just like Argument #1, Argument #2 is valid. This is just to say that if it were the case that Romney were president he would be a government employee. There is just no other way it could be. It would be impossible (assuming the truth of the second premise) for him to be president without also being a government employee. 7 Thus, an argument can be valid regardless of whether or not all its premises are true. Always remember that the issue of whether each of the premises of an argument is actually, i.e. as a matter of fact, true is an issue that is entirely separate and distinct from the issue of whether an argument is valid or invalid. As we saw above, Argument #2 is valid, but its conclusion is, nonetheless, false. How is this possible? It is possible because one of the premises turned out to be false i.e. the first premise. If that premise had been true, then the conclusion would have been true. We have established that. But since the premise turned out to be false, the guaranty as to the truth of the conclusion doesn t, so to speak, come into effect. Of course, the fact that a valid argument has a false premise does not guaranty that its conclusion will be false. Consider the following argument. 7 Of course, this does not prove that Romney is president. Remember, as we have discussed elsewhere, a sound deductive argument, i.e. a successful deductive argument, is one that is both logically valid and has only true premises (i.e. all of its premises are true). Page 3 of 5
Argument #3 1. Obama s last name begins with the letter O. 2. Any person whose last name begins with O is a government employee. Obama is a government employee. This argument is logically valid it has exactly the same structure as Argument #1 and Argument #2. As it turns out, the conclusion of Argument #3 is true. Yet, Argument #3 has a false premise, i.e. the second premise. We can see that the conclusion is true, but that Argument #3 does not offer proof that the conclusion is true. After all, you can t prove anything with false statements. If it helps, you might think of the conclusion of Argument #3 as being accidentally true. Its truth is unrelated to anything Argument #3 has to offer us. The upshot of this is that if the premises of a valid argument are false, the fact that they are false tells us nothing about whether the conclusion of the argument is true or false. Remember, the fact that an argument is valid only tells us something about the truth of the conclusion if it turns out that the premises are, as a matter of fact, true. Perhaps it will help you to think about this if you remember one of the advantages of knowing whether an argument is valid even if you are not sure whether the premises are true or not. That advantage is as follows: If you can safely conclude that an argument is valid, then you know that if you later find out that the premises are true then you also know that the conclusion must be true. A second practical advantage to using the concept of validity will become clear in what follows. Note that we can also take the relationship between the premises and conclusion of a valid argument as it was expressed above and, so to speak, turn it the other way around. Suppose that we know that an argument is valid and that its conclusion is false. In that case, we know that at least one of the premises is false. After all, since the argument is valid it is the case that if the premises were true then the conclusion would have to be true. But since the conclusion is false, it follows that something must have gone wrong with the premises at least one of the premises must be false. Consider the following argument. Argument #4 1. There are exactly iki planets orbiting our sun. 2. The word iki means two in Ottoman Turkish. There are exactly two planets orbiting our sun. This, as you can see, is a valid argument. If it were the case that the premises were both true, it would have to be the case that the conclusion is true. But I expect that when you read the first premise of this argument for the first time you had no idea whether it was true or false because you probably didn t know what it meant. However, you probably knew that the conclusion was false as soon as you read it. Thus, you know that since the argument is valid and the conclusion is false, something must have gone wrong with one of the premises. In fact, the first premise is false. In this particular case, you might check on the second premise and see that it is true. Thus, you would know that the first premise must be false. This is because, as we noted above: If the conclusion of a deductive argument is false then at least one of the premises of that argument must be false. As noted above, this just follows from the concept of validity. Page 4 of 5
This may leave us wondering whether the fact that the conclusion of a valid argument is true is an indication that all of its premises must be true. But we have already seen that this is not the case. Look back at Argument #3. There we had a valid argument with a true conclusion, but one of the premises was false. Of course, it might turn out that a valid argument with a true conclusion has all true premises. After all, that s what we are aiming for as our ideal argument that is the very definition of a sound argument. Perhaps you are wondering: What about invalid deductive arguments? Is there any relationship between the truth value of the premises and the truth value of the conclusion in an invalid deductive argument? The answer is, quite simply, no. The only relationship between the truth value of the premises and the truth value of the conclusion of a deductive argument is that which is expressed in the definition of validity. The results of this discussion of truth value relationships are summarized in the table below. Deductive Arguments All True Premises At Least One False All True Premises At Least One False True Possible Possible Possible Possible Conclusion False Impossible Possible Possible Possible Valid Argument Invalid Argument Page 5 of 5