Chapter 2

Logic and truth

I am not attempting to make this section a kind of mini-course in formal logic; it is rather an attempt to show the relationship between logic and statements, and why logic operates as it does, to give an understanding of what logic is, rather than to try to improve anyone's logical skills. In modern parlance, what I will be doing is meta-logic rather than logic here. I will be going through many of the traditional operations of formal logic, but the focus will be on how these operations reflect what statements are all about; and based on the relationship I see, I will offer critiques of traditional approaches and will give new formulations for some of the terms and new approaches so that things can become clearer. In that sense, this chapter should be useful for those who want to improve logical skills.

But a course in logic is something like a course in grammar. We already know how to speak; and so the grammar we have in our heads is really as complex as the one in textbooks. It is just that those who have studied grammar see why we say certain things in certain ways, and have reduced these relationships to rules, so that when we get into difficult constructions, we can understand what to say. For instance, the expression "between you and me" is correct, and "between you and I" is not, because "between" takes the objective case. Similarly, we know how to reason, but we can't necessarily spot all fallacies, because we've learned reasoning through practice and don't necessarily see why the rules work, and when apparently following them actually violates them. I have an example just below. But the point is that this particular chapter is not so much concerned with what the rules are, but why they are what they are.

Let me begin by remarking that formal logic has been thought for centuries to be the way we connect concepts or judgments; but I think a little experiment will show that it deals with statements rather than the acts of the mind that statements stand for. I have often put the following on the blackboard:

Either it is raining or it is not raining

But it is not raining

Therefore ... ?

There has always been at least one student who would answer, "Therefore, it is raining." But no one who is thinking could make such a mistake, since how could it be raining because it is not raining? The fallacy comes from knowing intuitively the rules of this kind of syllogism (Either A or B, but not A, therefore B) and from not realizing that B in this case is negative ("It is not raining"); and so if you affirm it, you have to state it as it is. That is, the reasoning goes:

Either (A) it is raining or (B) it is not-raining

But it is not raining (not A)

Therefore, it is not-raining (B)

The point, of course, is that the confusion comes from the words and the way they are arranged, and not what the words stand for; which means that logic deals with the words primarily and the judgments only secondarily, as being what is expressed by the words.

This experiment not only shows that the earlier philosophers were wrong in thinking that logic was primarily the linking of concepts and judgments, but it also tends to refute those contemporaries who seem to be holding the opposite position: that logic deals with statements, but that there is really no distinction between statements and judgments. That is, from what I can decipher of what they have written, they are modern exponents of the nominalist fallacy that what is called "thought" is simply some supposedly spiritual something behind the words; but since they hold that what is spiritual is imaginary and unreal, the only thing that thought really is is the words. I dealt in Chapter 5 of Section 3 of the third part 3.3.5 with why this position can't be held consistently with the way we actually use words.

Language, actually, involves several different types of logic: (a) grammar, which is the logic of how words go together in the language to express the various acts of the mind; (b) style, which is the logic of how words and sentences go together to unite sound, appearance, and meaning; and (c) what is ordinarily called "logic": how the sentences go together so that the last one is understood to be related to what went before.

These logics are by no means the same. For instance, the sentence, "No dije nada" (Lit. "I didn't say nothing") is grammatical in Spanish, but illogical, because if you didn't say nothing, then it is false that you said nothing, which means that you said something. Standard English grammar is logical in this respect, though the grammar of non-standard English is different. The statement I overheard recently spoken by a Black woman, "Ain't nobody never told me about that" happens to be logical. What it means is, "Nobody ever told me about that," and the negative in the "ain't" cancels the negative in the "never." But that this isn't the point, grammatically, of the multiple negatives can be shown by the statement in that same dialect, "Ain't nobody never told me nothing about that," which means the same thing, but now logically would have to mean something like, "Somebody once told me something about that." In Black English, piling up negatives simply emphasizes the negativeness of the sentence; double negatives do not cancel each other.(1)

Style is at least in part an esthetic logic, dealing with how the words sound and/or look on the page, how long phrases and sentences should be to hold attention and lead from one sentence to the next, how to avoid having words call attention to themselves instead of what they represent, how the sound even of written words (as they are "heard" by the reader) is to be kept from getting in the way of the judgments conveyed, and so on. For instance, in a book like this, for readers who are sophisticated and intelligent, long sentences like the previous one are, I think, in order, as long as they are broken up by commas, semicolons, and dashes in such a way that the ideas can be at once recognizable and flow into each other showing the very large whole that they are parts of. Whether I am successful in this, I will leave to you. The point is that the style is not the same as what is usually called the logic of what is written. Kant's style is notoriously bad; but his books are logically arranged.

Then what is it that is called the "logic" in what is being said? Let me put it in the form of a definition:

Formal logic is the arrangement of statements in such a way that it is understood that the final statement cannot be denied without contradicting what has already been said.

That is, the logic of a group of sentences is the way they back you into a corner by means of the Principle of Contradiction, so that if you agree that what is being said is true, then you have to admit that the final statement is also true, or you have contradicted yourself in one way or another.

It sounds, therefore, as if logic deals with truth. But this is not the case, actually. It deals only with the way statements are arranged, not the truth of the statements, and with the particular trick connected with the fact that some arrangements of statements demand a particular statement under penalty of contradicting themselves. Of course, if the statements are true, and if the arrangement is of the logical type, then (he said, using a logical inference) the final statement--the conclusion'--not only is true but cannot be false.

And this connection logic has with the truth of the statements is, of course, why we use it. But it must be understood that the logic itself doesn't deal with this. There can be logical inferences (operations) that generate false conclusions and are perfectly valid, and illogical fallacies that generate true conclusions. For instance, "Every German shepherd is a dog, and every dog is an insect, and therefore every German shepherd is an insect" is valid, but its conclusion is false; while "Every German shepherd by nature has four legs and every dog by nature has four legs and therefore every German shepherd is a dog" has three true statements in it, but is invalid--as can be seen by replacing "German shepherd" with "Arabian stallion."

This rather tenuous connection with truth has caused a lot of confusion in logical theory, particularly in modern times, where inferences are checked by "truth-tables," as if the actual truth mattered in the logic of what is going on. I think that instead of T's and F's in these truth-tables, the letters should be A's (for "Affirm") and D's (for "Deny") to reflect more accurately what is going on. Let me define a number of terms here, to avoid clutter:

A proposition is a statement of fact "proposed for the sake of the argument" in a logical inference.

An affirmation is the acceptance of the proposition.

A denial is the rejection of the proposition.

That is, affirmation accepts the proposition as "true for the sake of the argument," and not necessarily factually true. Thus, in the inference, "If it is raining out then the cat is inside, and the cat is not inside, therefore it is not raining out," the first proposition might be affirmed, even though it is recognized as not always a statement of the way things actually are.

Note that when negative propositions are affirmed, they are accepted as they stand. That is, if in the inference above you think that the cat is not inside, then you affirm the second proposition (you accept that the cat is not inside). Of course, the point of the inference is that if you affirm both of the first two propositions, then you can't deny the third one without contradicting yourself. So some more terms are in order:

An argument is a logical inference.

An inference is an arrangement of propositions such that the conclusion cannot be denied without either denying the premises or declaring the logic invalid

An inference is valid if the conclusion cannot be denied without denying at least one of the premises.

An inference is invalid if the conclusion can be denied without denying any premise.

A premise is a proposition from which a conclusion is drawn.

A conclusion is a proposition whose affirmation or denial depends on an inference. The conclusion is said to "follow from" the premises.

Implication is the relation of premises to the conclusion. Premises are said to "imply" the conclusion.

As the inference from the cat's behavior to what the weather is shows, logic may or may not have anything to do with the way the world actually works, depending on the actual truth of the propositions. But this does not make logic an idle game, because the world works consistently with the Principle of Contradiction, as we saw in Chapter 7 of Section 1 of the first part 1.1.11; and so, as I said earlier, if the premises are in fact true and if the inference is valid, then the conclusion must in fact be true. But establishing the truth of the premises is outside logic; and this is why within logic we only deal with propositions and affirmation and denial, and not with statements and their truth and falsity.

There are those who say that logic is only a game, but for a different reason. Insofar, they reason (using logic, by the way), as logic draws a conclusion from premises that imply it, the implication is already known before the conclusion is drawn; and therefore, drawing the conclusion is otiose, and no new knowledge has been gained by it. Presumably they say this to convince people who think that logic does lead to new knowledge that they are wrong. But if so, then why would they offer that inference? If it is valid and the premises are true, then the people they are trying to convince already know that logic gets you nowhere, and they haven't told them anything new. On the other hand, if they are expecting to have their hearers say, "Oh! I didn't realize that!" then this new insight on the part of the hearers implies that the inference is invalid or one of their premises is wrong (because something new was learned, which is impossible--on this view--if the inference is valid and the premises are true).

So I think we can safely say that there's something faulty about their position. As generally presented, it rests on the erroneous assumption (that Hume is largely to blame for, though he didn't originate it) that you can't know the truth of general propositions ("All dogs have four legs") unless you have checked all the dogs there are to see that each of them does in fact have four legs. Obviously, if you've done that, and then you say that German shepherds are dogs, your "conclusion" that therefore German shepherds have four legs is indeed a waste of time, because you've already checked all the German shepherds in getting your original premise.

But we don't get general statements in this way. For instance, on being presented with a three-legged dog, a person doesn't say, "Look at that! So not all dogs have four legs," but says, "How did that dog's leg get cut off?" That is, "All dogs have four legs" is a different kind of statement from the statement, "All the living beings in this room are human," which is understood to be false if one discovers an ant on the floor. The first is what Arthur Pap (following Nelson Goodman and Roderick Chisholm) calls a "lawlike generalization," which supports "counterfactual inferences"--or in other words, which people still accept as true in spite of instances to the contrary. The latter is not "lawlike," and is falsified if any instance contrary to it is found.

Lawlike generalizations are not in fact made by checking every instance of what they talk about. How we can make them is the problem of induction, which I will discuss later in the section on science; but on the assumption that we do in fact make general statements from incomplete observation, then obviously conclusions drawn from these general statements are not necessarily already known to be true.

And it would seem obvious that, from seeing one relationship, it does not follow that you explicitly understand all the relationships that are tied to it somehow. And it is quite possible that by rearranging words in various propositions, a new relationship among the words (and among the objects they refer to) is discovered that wasn't understood as such before. So logic can lead to new knowledge.

Of course, a meaningless "proposition" can't be true or false. This is because, as I mentioned in Chapter 5 of Section 3 of the third part 3.3.5, the meaning of a sentence is the conscious act it stands for; and so a statement's meaning is the judgment it stands for. But if it is meaningless, it can't represent a judgment, and it is only through the judgment that a statement can be true or false (even though its truth or falsity does not depend, as I said, on whether the judgment is or is not mistaken). Any statement that contradicts itself is meaningless, because it can't represent a judgment, as we saw in Chapter 7 of Section 1 of the first part 1.1.7. Note that it is not meaningless if it contradicts some known fact; in that case, it is false.

For instance, "The statement I am now writing is false" can't be a statement, because if it is false it is true (because it says it is false, and that would then make it true), and if it is true it is false (because it says it is false). I mentioned under in discussing the principles of identity and the excluded middle in Section 1 of the first part that this was a complicated problem, but that this locution couldn't be a statement. Basically, it can't be one because the judgment it would represent would be the recognition of being mistaken because one is not mistaken; and this is impossible for a self-transparent act. But there is more to it than just this.

Those who want to bypass judgments altogether and go directly from statements to facts find it difficult to deal with the distinction between falseness and meaninglessness. Remember, the truth or falseness of a statement does not depend on the judgment it represents, but on whether it expresses a fact or not; but the meaningfulness of a statement (not surprisingly, given what meaning is) depends on whether it can express a judgment. But if you don't hold this distinction, then since some apparent statements are manifestly meaningless (what could be the meaning of "The cold door sneezed a purple eyeball"?) and not false, then you have to resort to saying that they are not "well formed."

For instance, Bertrand Russell first tried to solve the problem of "This statement is false" by giving the rule that a statement cannot meaningfully refer to itself. From this it follows that the statement "The statement I am now writing is in English" is meaningless--and so presumably could not be understood by anyone. But that is silly. I can even envision a context for it. I could make a list of different languages by writing things like, "Esta frase está en castellano," "Cette phrase est en franais," "This sentence is in English," and so on, and you could figure out what each of them meant by looking at the ones you knew.

But then, as others have pointed out, what do you do with this?: "The following statement is true. The preceding statement was false." The rule was then changed to say that a statement that talks about another is in a meta-language, and it can only refer meaningfully to the language below it (not to itself or to a meta-language referring to it). Obviously, in the conundrum before us, the first statement is in a meta-language referring to the second. But the second statement's referring to the first puts it in a meta-language with respect to the first, and so it is in a higher-order meta-language, and so the combination is meaningless. But again, suppose the second statement said, "The preceding statement was in English." Is the combination now unintelligible?

Granted, the combination dealing with truth and falsity is unintelligible. The question is why. I think that making rules about meta-languages and not being able to talk about a meta-language at or above the one one is using is an ad hoc solution that by decree makes a whole series of perfectly intelligible statements "meaningless non-statements."

The meaninglessness of the two statements dealing with their mutual truth comes from the fact that the combination (as can be seen from the one dealing with English) is understood in one judgment; but the judgment it would represent is again, "I am not thinking of what I am now thinking of," and such a judgment can't be made, since the judgment is self-transparent.

But then, is the statement, "This statement is true" meaningful? There doesn't seem to be any problem if it is part of a larger statement that has some content, like St. Paul's, "...and I stayed with [the Rock, Peter] two weeks, without seeing any other Representative except James, the Master's relative. This is no lie I am writing to you. Before God it is not." Clearly, he is saying that the part of the statement dealing with his staying with The Rock is not a lie.

But if you take, "This is no lie I am writing to you" absolutely, with no context, could it express a judgment? The question then is what judgment it would be expressing. It is the equivalent of, "This statement does in fact express my judgment"; but this shows that the judgment it expresses is itself the judgment of the fact that the statement expresses it. But that fact doesn't exist until after the judgment is made, and so it couldn't know the fact until after the expression. But the statement is not self-transparent or atemporal, and so comes after the judgment as something distinct, which means that the judgment couldn't be made as to its factuality (truth) before it was actually stated.

You don't have this problem with "This statement is in English," because that statement can be true or false irrespective of the judgment of the person making it. Suppose someone, for instance, said, "Esta frase no está en castellano," not realizing that "castellano" is the Spaniards' normal way of speaking of Spanish. What the speaker meant was "This sentence is not in Castilian, it is in Spanish," but he misunderstood the words. Hence, his statement is false. As the person is making the statement, he is judging what the statement says.

This is only slightly different from judging the statement's truth while you are making it; but the difference is day and night as far as the meaningfulness of the two are concerned. The fact understood in the case of the language is a fact about the statement itself, while the "fact" understood about the truth of the statement is its supposed relation to the judgment that it expresses. But that judgment couldn't, as I said, be made prior to the "statement." Note that this applies not only to single "statements" but to combinations that refer back to themselves, like "The next statement is true. The preceding statement was true." Here, the truth of the "next" statement is known only after it is made; but since it says that the preceding statement was true, it could not be made until after the preceding statement was known to be true--or in other words, after what was known after it.

"The next statement is true" (or false) can be meaningful when a person uttering it knows what he is going to say next. And the next statement can refer back to the one now being uttered, as long as it does not refer to its truth. That is the unique case in which what is known before would have to be known afterwards and not before.

Where, then, are we? I think we can clarify Russell's "rule" with something that is not arbitrary. A statement can refer to itself meaningfully except when it is referring to its truth or falsity. When it refers to its truth or falsity, then the fact it refers to as true or false is the fact that it is expressing a prior judgment about itself. But that judgment cannot be made except subsequently to the statement. Given that the meaningfulness of a statement is that it is the expression of a judgment, this contradiction precludes statements from being meaningful if their "meaning" is supposed to be their own truth or falsity.

Conclusion 1: A statement cannot be meaningful and refer, either directly or indirectly, to its own truth or falsity.

This is a conclusion, not a "rule," because I have shown how such a statement contradicts itself; and what contradicts itself cannot be either true or false.



1. In case this is construed to be an argument for teaching Black English to Blacks, I want to point out that it argues in the opposite direction. Blacks already know how to speak their dialect; but unless they are taught Standard English, they won't realize that it expresses itself differently; and therefore what they say in their dialect sometimes means in Standard English exactly the opposite of what they are saying--which, of course, is going to make it difficult to communicate with people who speak Standard English.

This has nothing to do with whether Standard English is "right" and Black English or other dialects are "wrong"; it is a question of whether you want to communicate with others or not. If you are a member of a subgroup of the larger society, you have no grounds for expecting the society as a whole to defer to you when you speak to them; and so you have to learn the standard way of expressing yourself in that society. The Québecois in Canada have the same problem, and have tried to "solve" it by demanding that the country be bilingual; but I write this shortly after the "Meech Lake" change in the Canadian constitution (recognizing Québec as a distinct society) was defeated; and it looks at the moment as if the country is going to split up over the issue. Certainly French is a legitimate language; but whether a minority in an English-speaking country can demand that they keep their French as they intermingle with English-speakers is what the issue really is. And they rightly see this as giving them the status of a separate society within the country.