Consistency

Logical consistency is important. If my positions fail to be consistent, this is clear proof that at least one of them is mistaken.

Nonetheless, logical consistency is not the supreme value. It would not be the supreme value even for someone who cared about truth alone. The reason for this is that someone may see that his position is inconsistent, and know that it follows that his position is partially mistaken, without knowing any concrete way to modify his position which will improve it. His best estimate of any particular modification may be that his position would end up containing a greater amount of error. In this case, caring about the truth will entail that he preserve his position without modification, despite recognizing that his position contains some falsity.

This happens to all of us, as we can see from one particular case. I would be foolish and arrogant if I believed, “All of my beliefs are true.” In order for that to be the case, I would have to be essentially infallible. So it is far more reasonable for me to believe that I am mistaken about some things. But then if we take all of my beliefs together, including the belief that “some of my beliefs are mistaken,” we can derive a contradiction.

And it is clear, in the general case, that I have no way to improve my position to eliminate the inconsistency. Believing that all of my beliefs are true would not improve it. It would add some consistency, but only by means of adding a definite falsity. Likewise, changing some other belief would not help, since even if some of my beliefs are mistaken, I do not know which ones those are. Changing a belief at random is likely to worsen my position overall. And if I examine my beliefs and come to the conclusion that I should modify one belief or another, this will not and should not change my conviction that some of my beliefs are mistaken.

 

Making Arguments vs. Manipulating Symbols

There is still another problem with Spinoza’s manner of argumentation. Spinoza is trying to get geometrical certainty about metaphysics by a logical arrangement of his claims. But this cannot work even in principle. If you take the rules of logic and the forms of the syllogisms, and fit sentences into them using their mere verbal patterns, without thinking about what you are saying, what it means and in what sense it is true or untrue, then you are manipulating symbols, not actually making arguments, and it may well mean that your conclusion is false, whether or not each of your premises is true in some way.

Alexander Pruss, in a recent blog post, argues for the existence of God from certain facts about language:

This argument is valid:

  1. All semantic truths are knowable to members of the community of language users.
  2. There are semantic truths that are not knowable to human language users.
  3. Therefore, there is at least one non-human language user.

There is some reason to accept (1) in light of the conventionality of language. Premise (2) is going to be quite controversial. I justify it by means of a standard argument for epistemicism. Consider Queen Elizabeth II. There are 88 statements of the form:

  • Elizabeth was not old at age n but she was old at age n+1

where n ranges from 1 to 88. It’s a straightforward matter of classical logic to show that if all 88 statements are false, then:

  1. Elizabeth was old at age 1 or Elizabeth is not old at age 89.

But (4) is clearly false: Her Majesty is old now at age 89, and she surely wasn’t old at age one. So, at least one of the 88 statements is false. This means that there is a sharp transition from being not old to being old. But it is clear that no matter what we find out about our behavior, biology and other relevant things, we can’t know exactly where that transition lies. It seems very plausible that the relevant unknowable fact about the transition is a semantic fact. Hence, (2) is true.

The most plausible candidate for the non-human language user who is capable of knowing such semantic facts is God. God could institute the fundamental semantic facts of human language and thereby know them.

(“So, at least one of the 88 statements is false” should be “So, at least one of the 88 statements is true.”) I would consider this to be more a case of manipulating symbols than of making a serious argument.

An atheist is likely to say about this argument, “Wait a minute. Maybe it’s not obvious to me what is wrong with your argument. But there’s just no way you can prove the existence of God from simple facts about how words are used. So there must be something wrong with the argument.”

I agree with the hypothetical atheist that reasonable intuitions would say that you cannot prove the existence of God in such a way, and that this is a reason for doubting the argument even if you cannot formally point out what is wrong with it.

But in fact I think there are two basic problems with it. In the first place, Pruss seems to be failing to consider the actual meaning of his premises. He says, “There is some reason to accept (1) in light of the conventionality of language.” What does this mean? A semantic fact is a fact about the meaning of words or sentences. Pruss is arguing that all facts about the meanings of words or sentences should be knowable to members of the community of language users, and that we should accept this because language is conventional. In other words, human beings make up the meanings of words and sentences. So they can know these meanings; whatever they cannot know about the meaning is not a part of the meaning, since they have not invented it.

But later Pruss says:

This means that there is a sharp transition from being not old to being old. But it is clear that no matter what we find out about our behavior, biology and other relevant things, we can’t know exactly where that transition lies. It seems very plausible that the relevant unknowable fact about the transition is a semantic fact. Hence, (2) is true.

But if this is right, it undercuts the justification for believing the first premise. For the only reason we had to believe that all of the semantic facts are knowable to the community of language users, was a reason to believe that they were knowable to human beings. If they are not knowable to human beings, we no longer have a reason to believe that they are knowable to anyone, or at any rate not a reason that Pruss has given.

This illustrates my point about the necessity of considering the meaning of what you are saying. The argument for the first premise is in fact an argument that human beings can know all of the semantic facts; thus if they cannot, we no longer have a good reason to accept the first premise. We cannot simply say, “This argument is logically valid, we’ve given a reason for the first and a reason for the second, that gives us reason to accept the conclusion.” We need to think about what those reasons are and how they fit together.

The second problem with this argument is that the “standard argument for epistemicism” is just wrong. And likewise, the argument consists of precisely nothing but manipulating words, without thinking about the meaning behind them. It is a “a straightforward matter of classical logic” in the sense that we can fit these words into the logical forms, but this does not mean that this process is telling us anything about reality.

To see this, consider this new word that we can construct by convention, namely “zold.” A person who is between 80 and 90 years old is said to be zold; a person who is between 1 and 10 years old is said not to be zold.

Now consider the 88 statements of the form, “Elizabeth was not zold at age n but she was zold at age n+1.” It’s a straightforward matter of classical logic to show that if all 88 statements are false, then either Elizabeth was zold at age 1, or she was not zold at age 89. But this is clearly false according to the conventions already defined. So at least one of the statements must be true, and there is a sharp transition from being not zold to being zold. It is obvious that no matter what we find out about human beings, that will not tell us where the transition is; the transition must be a semantic fact, a fact about the meaning of the word “zold.”

Obviously, in reality there is no such semantic fact. The convention that we used to define the word simply does not suffice to generate a sharp transition. The problem with the argument for the sharp transition is that the rules of logic presuppose perfectly well defined terms, and this term is not perfectly well defined.

And it is not difficult to see that the word “old” does not differ in a meaningful way from the word “zold.” In reality the two come to have meaning in very similar ways, and in a such a way that there cannot be a sharply defined transition, nor can classical logic force there to be such a sharp transition.

It is not enough to fit your sentences into a logical form. If you want the truth, the hard work of thinking about reality cannot be avoided.

Liar Game

While this is the name of a certain story, it is also the name I am giving to the game I am about to propose. The rules are that I propose a certain number of statements, and the player has to categorize them as true or false. The player wins if all of them are correctly categorized, and fails if he does not categorize them all, or if he mistakenly categorizes a true statement as false, or a false statement as true. It is against the rules for him to place a statement in both categories.

The statements I propose are the following:

  1. 2+2=4.
  2. 2+2=5.
  3. The player will categorize this as false.

It can easily be seen that the player is guaranteed to lose the game. If the player does not categorize the third statement, then it is false, and he has failed to categorize them all. On the other hand, if he categorizes it as true, it is false, and if he categorizes it as false, it is true. In any case either he fails to categorize it, or he categorizes it incorrectly.

It is evident that this is related to the paradox of the Liar, but there is a significant difference. The original liar statement is paradoxical, in the sense that applying the ordinary rules of logic results in a contradiction regardless of whether one considers the statement to be true or false.

This is not the case here. There is nothing paradoxical about the statement, in this sense. Given an actual player and an actual instance of playing the game, the statement will plainly be true or false in an objective sense, and without any contradiction being implied. It is just that the player cannot possibly categorize it correctly, since its truth is correlated with the player categorizing it as false.

 

The Paradox of the Heap

The paradox of the heap argues in this way:

A large pile of sand is composed of grains of sand. But taking away a grain of sand from a pile of sand cannot make a pile of sand stop being a pile of sand. Therefore if you continually take away grains of sand from the pile until only one grain of sand remains, that grain must still be a pile of sand.

A similar argument can be made with any vague word that can differ by an apparently continuous number of degrees. Thus for example it is applied to whether a man has a beard (he should not be able to change from having a beard to not having a beard by the removal of a single hair), to colors (an imperceptible variation of color should not be able to change a thing from being red to not being red), and so on.

The conclusion, that a single grain of sand is a pile of sand, or that a shaven man has a beard, or that the color blue is red, is obviously false. In order to block the deduction, it seems necessary to say that it fails at a particular point. But this means that at some point, a pile of sand will indeed stop being a pile of sand when you take away a single grain. But this seems absurd.

Suppose you don’t know the meaning of “red,” and someone attempts to explain. They presumably do so by pointing to examples of red things. But this does not provide you with a rigid definition of redness that you could use to determine whether some arbitrary color is an example of red or not. Rather, the probability that you will call something red will vary continuously as the color of things becomes more remote from the examples from which you learned the name, being very high for the canonical examples and becoming very low as you approach other colors such as blue.

This explains why setting a boundary where an imperceptible change of color would change something from being red to being not red seems inappropriate. Red doesn’t have a rigid definition in the first place, and assigning such a boundary would mean assigning such a definition. But this would be modifying the meaning of the word. Consequently, if the meaning is accepted in an unmodified form, the deduction cannot logically be blocked, just as in the previous post, if the meaning of “true” is accepted in an unmodified form, one cannot block the deduction that all statements are both true and false.

Someone might conclude from this that I am accepting the conclusions of the paradoxical arguments, and therefore that I am saying that all statements are both true and false, and that a single grain of sand is a pile, and so on.

I am not. Concluding that this is my position is simply making the exact same mistake that is made in the original paradoxes. And that mistake is to assume a perfection in human language which does not exist. “True,” “pile,” and so on, are words that possess meaning in an imperfect way. Ultimately all human words are imperfect in this way, because all human language is vague. The fact that logic cannot block the paradoxical conclusions without modifying the meanings of our words happens not because those conclusions are true, but because the meanings are imperfect, while logic presupposes a perfection of meaning which is simply not there.

In a number of other places I have talked about how various motivations can lead us astray. But there are some areas where the very desire for truth can lead us away from truth, and the discussion of such logical paradoxes, and of the vagueness of human thought and language, is one of those areas. In particular, the desire for truth can lead us to wish to believe that truth is more attainable than it actually is. In this case it would happen by wishing to believe that human language is more perfect than it is, as for example that “red” really does have a meaning that would cause something in an a definitive way to stop being red at some point with an imperceptible change, or in the case of the Liar, to assert that the word “true” really does have something like a level subscript attached to its meaning, or that it has some other definition which can block the paradoxical deductions.

These things are not true. Nor are the paradoxical conclusions.

The Liar

The paradox of the Liar is a logical problem that results from a sentence that implies that the very sentence itself is false, or at least that it is not true. Consider the following statement:

(1) Statement (1) is not true.

Is statement (1) true or not? We might reason about it as follows.

(2) If statement (1) is true, then statement (1) is not true, since this is what it says.

(3) But this is absurd, since statement (1) would then be both true and not true.

(4) Therefore (1) is not true.

(5) But this is just what (1) says. So (1) is true.

And so on. There does not appear any way to avoid the conclusion that (1) is both true and not true, which is a contradiction. Nor is it helpful to say that it is neither true nor not true, since the same contradiction will follow: if something fails to be not true, it is surely true.

Any statement whatever will follow from a contradiction, so if one accepts this contradiction, one will be forced to accept that every statement is both true and false.

A. N. Prior discusses the idea of an analytically valid inference:

It is sometimes alleged that there are inferences whose validity arises solely from the meanings of certain expressions occurring in them. The precise technicalities employed are not important, but let us say that such inferences, if any such there be, are analytically valid.

One sort of inference which is sometimes said to be in this sense analytically valid is the passage from a conjunction to either of its conjuncts, e.g., the inference ‘Grass is green and the sky is blue, therefore grass is green’. The validity of this inference is said to arise solely from the meaning of the word ‘and’. For if we are asked what is the meaning of the word ‘and’, at least in the purely conjunctive sense (as opposed to, e.g., its colloquial use to mean ‘and then’), the answer is said to be completely given by saying that (i) from any pair of statements P and Q we can infer the statement formed by joining P to Q by ‘and’ (which statement we hereafter describe as ‘the statement P-and-Q’), that (ii) from any conjunctive statement P-and-Q we can infer P, and (iii) from P-and-Q we can always infer Q. Anyone who has learnt to perform these inferences knows the meaning of ‘and’, for there is simply nothing more to knowing the meaning of ‘and’ than being able to perform these inferences.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P and given Q we can infer R, and given R we can infer P and can also infer Q. But on the view we are considering such a doubt is quite misplaced, once we have introduced a word, say the word ‘and’, precisely in order to form a statement R with these properties from any pair of statements P and Q. The doubt reflects the old superstitious view that an expression must have some independently determined meaning before we can discover whether inferences involving it are valid or invalid. With analytically valid inferences this just isn’t so.

I hope the conception of an analytically valid inference is now at least as clear to my readers as it is to myself. If not, further illumination is obtainable from Professor Popper’s paper on’ Logic without Assumptions’ in Proceedings of the Aristotelian Society for 1946-7, and from Professor Kneale’s contribution to Contemporary British Philosophy, Volume III. I have also been much helped in my understanding of the notion by some lectures of Mr. Strawson’s and some notes of Mr. Hare’s.

He proceeds to draw some conclusions from this:

I want now to draw attention to a point not generally noticed, namely that in this sense of ‘analytically valid’ any statement whatever may be inferred, in an analytically valid way, from any other. ‘2 and 2 are 5’, for instance, from ‘2 and 2 are 4 ‘. It is done in two steps, thus:

2 and 2 are 4.

Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.

Therefore, 2 and 2 are 5.

There may well be readers who have not previously encountered this conjunction ‘tonk’, it being a comparatively recent addition to the language; but it is the simplest matter in the world to explain what it means. Its meaning is completely given by the rules that (i) from any statement P we can infer any statement formed by joining P to any statement Q by ‘tonk’ (which compound statement we hereafter describe as’ the statement P-tonk-Q ‘), and that (ii) from any ‘contonktive’ statement P-tonk-Q we can infer the contained statement Q.

A doubt might be raised as to whether it is really the case that, for any pair of statements P and Q, there is always a statement R such that given P we can infer R, and given R we can infer Q. But this doubt is of course quite misplaced, now that we have introduced the word ‘tonk’ precisely in order to form a statement R with these properties from any pair of statements P and Q.

As a matter of simple history, there have been logicians of some eminence who have seriously doubted whether sentences of the form ‘P and Q’ express single propositions (and so, e.g., have negations). Aristotle himself, in De Soph. Elench. 176 a 1 ff., denies that ‘Are Callias and Themistocles musical?’ is a single question; and J. S. Mill says of ‘Caesar is dead and Brutus is alive’ that ‘we might as well call a street a complex house, as these two propositions a complex proposition’ (System of Logic I, iv. 3). So it is not to be wondered at if the form ‘P tonk Q’ is greeted at first with similar scepticism. But more enlightened views will surely prevail at last, especially when men consider the extreme convenience of the new form, which promises to banish falsche Spitzfindigkeit from Logic for ever.

His point is quite clear. Given the way the word “tonk” is defined, one cannot avoid drawing all possible conclusions. But this means the word “tonk”, defined in this way, is quite unacceptable in the first place.

If we define the word “true” by saying that “P is true” is a statement such that it necessarily follows from P, and such that P necessarily follows from “P is true,” and we consider this an acceptable definition, then the rules of logic will force us to accept all possible conclusions.

Like the definition of the word “tonk”, therefore, this definition of the word “true” is unacceptable, and in the same sense, namely that if the definition is accepted, all possible conclusions follow.

This explains why all solutions to the Liar paradox seem to fail, in the sense that in the end either they admit a contradiction, or they insist that we change the meaning of our language, as for example by talking about levels of truth and so on. For despite the consequences, the word “true” does basically have the meaning stated. The only real difference in comparison with the word “tonk” is that the latter word would never be used in any real language, because the consequences are obvious. In the case of “true,” the consequences are subtle and only follow in special circumstances, namely the kind that are found in the case of the Liar paradox, and so the word could be incorporated into human language, and basically with this meaning, before the implications were noticed.

Note that this is quite different from saying that the word “true” has an inconsistent meaning. The problem is even deeper than that. We could define the word “zrackled” to mean “white and not white in the same respect,” and the meaning would be inconsistent. The only consequence would be that nothing is zrackled, and no contradiction would follow. But if we said that “true” has an inconsistent meaning, and consequently that nothing is true, it would follow from the meaning stated that “nothing is true” is true, and consequently that “nothing is true” is not true. The problem is that we are attempting to define the word, at least in part, by certain rules of usage, and those rules themselves force a contradiction, and ultimately force one to draw all possible conclusions, as with the word “tonk.”

The Unexpected Hanging

Wikipedia tells the tale of the unexpected hanging:

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.

The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

Doubtless there are various ways to explain what is going on here. But the moral of the story is simply that no matter how solid your reasoning seems to you, no matter how absolutely conclusive, reality does not have to care. You can be wrong anyway.