Consistency and truth in logic

Marc Meléndez Schofield

Logic is most often defined as the study of valid inference. Inference is simply the process of deriving a conclusion from a set of assumptions (premises). An argument is said to be valid if, by virtue of its form, true premises necessarily imply a true conclusion.

This is the starting point of many logic text books. It will not, however, be our starting point. There is, of course, no real disagreement between their point of view and ours, as we shall see; just a different approach.

Consistency

'And if you take one from three hundred and sixty-five, what remains?'
'Three hundred and sixty-four, of course.'
Humpty Dumpty looked doubtful. 'I'd rather see that done on paper,' he said.
Alice couldn't help smiling as she took out her memorandum-book, and worked the sum for him:

Humpty Dumpty took the book and looked at it carefully. 'That seems to be done right—' he began.
'You're holding it upside down!' Alice interrupted.
'To be sure I was!' Humpty Dumpty said gaily, as she turned it round for him. 'I thought it looked a little queer. As I was saying, that ''seems'' to be done right — though I haven't time to look it over thoroughly just now —'[...]
(Lewis Carroll, Through the Looking Glass)

Let's start with something obvious: we cannot believe something that we know for certain is not true. We can imagine that we are flies, for example, but we can't actually believe it (assuming that we are fairly sane). As a consequence, if two beliefs are incompatible, that is, if they cannot both be true at the same time, and we realise this, then we cannot hold both beliefs. Note that this is just a description of what we often do, an apparent fact of rationality, if you like, and that we are not asking the question of whether we ought to be logical or rational.

Consider the following exchange:

'I'm sure I saw you in Paris last week'

'That's impossible! I spent last week in London.'

We usually assume that people cannot be in two distant places at once, so we would probably suspect that whoever says 'I spent the whole week in London' and 'I spent the whole week in Paris' is lying if they mean the same week. Furthermore, we accept statements like 'I spent last week in London, therefore I was not in Paris last week'. One could doubt the truth of the previous statement, but could one genuinely doubt that the argument is correct? Could one say, as Humpty Dumpty might, 'you were in London, yes, that seems to prove that you were not in Paris, though I haven't time to look it over thoroughly just now.'? As a matter of fact, we don't.

When we think things through, we tend naturally to discard beliefs that are incompatible with others that we hold more dearly. This will be our starting point.

A set of beliefs is consistent if they can all be true simultaneously or, in other words, if there are no contradictions among them. In logic, we will consider consistency carefully.

Consistency should not be confused with common sense, although they are certainly related. If I believe that the aliens have filled my house with secret cameras with the help of the CIA, then perhaps I am being unreasonable, but I am not necessarily being inconsistent. As long as this belief does not contradict anything else that I believe in, I am doing fine from a logical point of view. However, consistency does impose some restrictions on what we may believe in. For example, if I think that all bears have brown fur, then I cannot also believe that the caged polar bear that I am looking at in the zoo is a bear. It would be quite natural to think 'I used to believe that all bears had brown fur, but this bear standing in front of me has white fur. Therefore, not all bears have brown fur'.

Inference

A new belief can be inferred from a set of beliefs when the negation of the former contradicts the latter. For example, "this bear has brown fur" can be inferred from "all bears have brown fur" because the belief that this particular bear does not have brown fur contradicts the belief about all bears having brown fur.

The above example also illustrates another point. The existence of the polar bear leads us to reject either the belief "all bears have brown fur" or the belief "this so called polar bear is a bear". Both options are equally valid from the point of view of consistency. There are some beliefs, though, that we would find very difficult to abandon.

Deductive and inductive inferences ^*

Something is consistent if it holds firmly together. If I believe that all bears are brown, consistency demands that I believe that this bear is brown. But, having seen a thousands black crows, may I infer that all crows are black?

We have seen that "this bear is white" contradicts "all bears are brown", so the latter belief must be false if the former is true. But how can one say that all bears are brown if one has not seen all bears that were, are and will be?

Logic studies inferences, but there are two different types of inference mentioned above. On the one hand, we find so-called inductive inferences (or induction), which proceed from particular cases to a general law, as when we infer that all crows are black from having seen so many black crows. Deductive inferences (or deduction), on the other hand, infer the negation of anything which is inconsistent with a given set of beliefs. It is usually understood that logic studies deductive inferences only.

But are these two types of inference incompatible? May it not be true that inductive inferences are just one kind of deductive inference? It is clear that there is no logical way to assure that the next crow we will come across will be black just because every other crow we have ever seen is black, so it is possible to believe that it might or might not be black. There is no way to deduce that all crows are black, then. It would be possible, of course, if I had actually seen each and every crow and verified that they were all black. In that case, induction and deduction lead to the same conclusion. In logic and mathematics it is sometimes possible to use this reference to all the relevant cases, even in situations where the number of cases is infinite! But without a reference to all the particular cases, induction is not valid from a logical point of view.

Although we cannot guarantee the validity of inductive inferences, scientific theories often state general laws based on a limited amount of particular experiences. This means that scientific theories are only consistent as long as no contrary evidence is discovered. This led Karl Popper to argue that science progresses not by inductive inferences from greater and greater quantities of empirical data, but by falsifying provisional theories.

It is important to note the following fact: given a system of beliefs, we are not allowed to infer anything that is consistent with it, but only the negation of that which is inconsistent with the system. From the sentence A) "No communist is a capitalist" we may infer that B) "Lenin (who was a communist) was not a capitalist", but not C)"1 + 1 = 2", even though A and C are perfectly consistent (it is possible to believe both without falling into contradiction).

The complicated question of truth

It is said that the Nobel Prize physicist Niels Bohr had a horseshoe nailed to his door. One day a physics student asked him indignantly if he thought that it would bring him good luck. 'Of course not,' answered Bohr, 'but I have been told it works even if you don't believe in it.'¹

We are often worried about the consistency of our beliefs. Why? Couldn't we just take apparent contradictions as a feature of our eccentric personalities and say, as Bohr did, that contraria non contradictoria, sed complementaria sunt (opposites are not contradictory, but complementary)? In arguments, however, we often try to make the other party realise that they are holding contradictory positions in order to prove that they must be wrong, because if two statements contradict each other they cannot both be true. That is the key: what we are really worried about is truth.

The word 'true' indicates the aim of logic as does 'beautiful' that of aesthetics or 'good' that of ethics. All sciences have truth as their goal; but logic is concerned with it in a quite different way from this. It has much the same relation to truth as physics has to weight or heat. To discover truths is the task of all sciences; it falls to logic to discern the laws of truth.²

If physics searches for truths about physical objects and biology for truths about living organisms, logic searches for truths about truth. This does not mean that logicians believe that they posses the ultimate truth and that they must dictate this truth to other disciplines, or that they can decide what will count as true and what will not. Logic does not discern the laws of truth in that sense.

The word 'law' is used in two senses. When we speak of laws of morals or the state we mean regulations which ought to be obeyed but with which actual happenings are not always in conformity. Laws of nature are the generalization of natural occurrences with which the occurrences are always in accordance. It is rather in this sense that I speak of laws of truth. This is, to be sure, not so much a matter of what happens so much as of what is.³

Some logicians may use categorical and seemingly intolerant statements, but logic takes a relatively humble point of view when it comes to examining truth, for it only seeks to determine which relations are consistent and which are not. Alfred Tarski wrote an influential article in which he proposed a definition of the term 'truth' that included it among semantic terms, but he included the following warning:

It is perhaps worthwhile saying that semantics as it is conceived in this paper (and in former papers of the author) is a sober and modest discipline which has no pretensions of being a universal patent-medicine for all the ills and diseases of mankind. You will not find in semantics any remedy for decayed teeth or illusions of grandeur or class conflicts. Nor is semantics a device for establishing that everyone except the Speaker and his friends are speaking nonsense.⁴

The concept of truth and the law of contradiction

"True" and "false" are adjectives that are used to describe statements or sentences. Aristotle defined these words as follows.

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true.⁵

Once it has been deciphered, the previous definition turns out to be absolutely obvious. We might as well have said that a sentence is true when it describes things as they are, and false if it does not. This seems to imply that the different parts of our sentences correspond to objects or relations in the real world and that our sentences are therefore true in some situations (when the objects we refer to are arranged in a way that corresponds to the arrangement of words in our sentence) and false in others. Wittgenstein thought this intuition through to its ultimate consequences in his famous Tractatus logico-philosophicus.

4.01 A proposition is a picture of reality. A proposition is a model of reality as we imagine it.
4.011 At first sight a proposition—one set out on the printed page, for example—does not seem to be a picture of the reality with which it is concerned. But neither do written notes seem at first sight to be a picture of a piece of music, nor our phonetic notation (the alphabet) to be a picture of our speech. And yet these sign-languages prove to be pictures, even in the ordinary sense, of what they represent.⁶

The preceding comments show that truth is not a property of things in themselves, but rather has to do with the correspondence between words and things (or thoughts and things).

For falsity and truth are not in things —it is not as if the good were true, and the bad were in itself false— but in thought.⁷

Aristotle admitted that a statement might be true in one sense and false in another, but he denied the possibility of it being simultaneously true and false in the same sense. This basic fact is known as the law of contradiction. Aristotle argued that it was impossible to prove this law because it was necessary to presuppose it in any demonstration.

Some indeed demand that even this [the law of contradiction] shall be demonstrated, but this they do through want of education, for not to know of what things one should demand demonstration, and of what one should not, argues want of education. For it is impossible that there should be demonstration of absolutely everything (there would be an infinite regress, so there would be no demonstration); but if there are things of which one should not demand demonstration, these persons could not say what principle they maintain to be more self-evident than the present one.
We can , however, demonstrate negatively that this view [that something is so and not so] is impossible, if our opponent will only say something; and if he says nothing, it is absurd to seek to give an account of our views to one who cannot give an account of anything, in so far as he cannot do so. For such a man, as such, is from the start no better than a vegetable. Now negative demonstration I distinguish from demonstration proper, because in a demonstration one might be thought to be begging the question, but if another person is responsible for the assumption we shall have negative proof, not demonstration.⁸

It is enough to say something to have already presupposed the law of contradiction, as it would be impossible for words to mean anything if each of our expressions both said and didn't say something. This comment must not be understood as proof, but rather as an explanation, because it too is based on the law of contradiction.

If it is impossible to prove this law of contradiction, how can it be defended? According to Aristotle, it can be "demonstrated negatively" by showing that those who do not accept this principle contradict themselves (that much is obvious, as they accept contradiction from the beginning) and that, consequently, their arguments stop being meaningful.

If, however, [meanings] were not limited but one were to say that the word has an infinite number of meanings, obviously reasoning would be impossible; for not to have one meaning is to have no meaning, and if words have no meaning, our reasoning with one another, and indeed with ourselves, has been annihilated; for it is impossible to think of anything if we do not think of one thing; but if this is possible, one name might be assigned to this thing.⁹

If Aristotle was right, it is necessary to be consistent (not to contradict oneself) for thought to be possible.

But just because a predicate is consistent with other beliefs, this does not automatically mean that it is true, because its truth ultimately depends on the objective situation that it describes (except in the case of tautologies).

It is not because we think truly that you are pale, that you are pale, but because you are pale we who say this have the truth.¹⁰

Even if things in themselves are neither true nor false (because truth/falsehood are in the relation between things and words or thoughts), it is ultimately these things that determine what is true and what is not.¹¹ Truth could, therefore, also be understood in a primary and fundamental sense as this "discovery" of things. Although we have said that logic studies truth, it is important to emphasize that it does not deal with truth in this second sense.

Borderline cases and bizarre situations

Up until now we have assumed that a sentence either corresponds or does not correspond with reality, but there are many cases in which it is difficult to decide on this "correspondence". The difficulty might be caused by our incapacity to examine the reality which we are speaking about, as when we say that "Shakespeare took longer to write the second act in Romeo and Juliet than the first act in Hamlet". This sentence is obviously true or false, but it might be impossible for us to determine which.

In other cases, the problem is caused by the "incapacity" of an expression to correspond to reality in a straightforward way.¹² Let us take, for example, the claim that "Clara is too thin". It is possible to imagine a controversy on the matter even if Clara were present for everyone to examine. Any dividing line between "too thin" and "not too thin" will be arbitrary in some way. If we imagined that Clara gained wait gradually, gram by gram, when would we be able to say that she had stopped being "too thin"? In borderline cases, the truth of a statement is a matter of opinion.

We sometimes encounter what we shall call bizarre situations, in which it is not even clear if our choice of words is appropriate. If I say that "there is a pen on top of the book", it is easy in everyday life to check the truth of my claim. But ordinary everyday life includes absolute notions of "up" and "down", and expressions like "under" and "on top of " are dependent on them. For astronauts in space these notions will loose their meaning, and they will not be able to determine whether "there is a pen on top of the book" or if "the constellation of Cignus is above the constellation of Draco".

The preceding reflections should give some idea of the limitations which we can encounter in philosophy when we deal with problems from a logical point of view.

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Endnotes

^* This section is an adaptation of a contribution written in Spanish for the Wikiversity by user Monimino and myself.
¹ James S. Trefil: The Nature of Science: an A-Z guide to the laws and principles Governing our Universe, Houghton Mifflin Company (2003), p. 52.
² Gottlob Frege: The Thought: a Logical Inquiry in Readings in the Philosophy of Language, MIT press (1997), p. 9. Frege was one of the founding fathers of the modern discipline of logic.
³ Ibid.
⁴ Alfred Tarski: The Semantic Conception of Truth
⁵ Aristotle: Metaphysics, Book IV, Chapter 7, 1011b. Translated by W.D. Ross.
⁶ Wittgenstein: Tractatus logico-philosophicus, 4.01-4.011. Wittgenstein eventually questioned this intuition and radically changed his conception of meaning.
⁷ Aristotle: Metaphysics, Book VI, Chapter 4, 1027b. Translated by W.D. Ross.
⁸ Ibid. Book IV, Chapter 4, 1006a.
⁹ Ibid. Book IV, Chapter 4, 1006b.
¹⁰ Ibid. Book IX, Chapter 10, 1051b.
¹¹ There is an excellent study of the problem in Rodríguez, R.: Del sujeto y la verdad, Chapter 10, Aristóteles y la verdad antepredicativa.
¹² This is clearly presented in Hodges, W.: Logic. An introduction to elementary logic., Penguin Books (1991), pp 31-36. The terms "borderline case" and "bizarre situations" are taken from this book.