Form and content in logic

which has been inserted the word 'not', and similarly the sentence 'there is not thunder' consists of the sentence 'there is thunder', into which has been inserted the word 'not'.

 

Like our syllogism, the if-then-sentence now under consideration obviously expresses a true proposition.¹ This proposi­tion is true independently of whether it is true or not that there is thunder or lightning and also of whether it is true or not that there is thunder, if there is lightning, and of whether it is true or not that there is not lightning, if there is not thunder. As a matter of fact, there can be lightning without thunder. But it nevertheless remains true that if there is thunder, if there is lightning, then there is not lightning, if there is not thunder.

 

As in our syllogism, we can distinguish between variables and constants in the example which we are discussing.

The variables are those parts of the sentence for which we could substitute other fragments of language without affecting the truth of the proposition expressed by the if-then-sentence. It is plain that the variables here are the sentences 'there is thunder' and 'there is lightning'. We could also have said: 'If x, if y, then not y, if not x', without saying what x and y mean. The constants are the words 'if-then' and 'not'.

We might again say that the variables give to our sentence its content, and that the constants give to it its form, and also that the sentence expresses truth because of its form and indepen­dently of its content.

It was not till the middle of the nineteenth century that logicians became systematically interested in an important province of logical truth, of which the following sentence is an example: 'If somebody is teacher of somebody, then somebody is pupil of somebody.'

This sentence also obviously expresses a true proposition. It is plain that it has variable parts, i.e. contains words for which we could substitute other words without affecting the truth of what we say. It is further plain that the existence of variables has to do with the fact that we used teachers and pupils to

'This is not in contradiction with the account of conditionals, given in the seventh essay of the present volume. What is here called 'if-then-sentences' are material implication sentences. (1955)

3

 

illustrate something which would also be true of, say, parents and children.

On the other hand, the words 'teacher' and 'pupil' could not be regarded as two variables of our sentence. For if we say 'if somebody is x of somebody, then somebody is y of somebody', then we obviously say something the truth or falsehood of which essentially depends upon the meaning of x and y. What we say is true, if x and y mean teacher and pupil, or parent and child, or master and servant respectively. But it is not true, e.g., if x means teacher and y means servant. Somebody could be somebody's teacher without anybody being somebody's servant.

The problem of discovering what is variable and what is constant in our third example is thus somewhat more difficult than in the two first examples. This is due to an insufficiency of our language.

The relations of a teacher to his pupil, or a parent to his child, or a master to his servant have a common feature, for which, however, there is no word in ordinary language. The logician must invent a word for it. He speaks about relations and their converse relations. Whenever one person is another person's teacher, the latter is without exception the former person's pupil, and for this reason we call the relation of a pupil to his teacher the converse relation of the relation of a teacher to his pupil. Instead of 'pupil' a logician might say 'teacher converse', and similarly instead of 'child' he might say 'parent converse' and instead of 'servant' he might say 'master converse'. And if he then said 'if somebody is x of somebody, then somebody is x converse of somebody', he would say some­thing, the truth of which is independent of the meaning of x in the same sense in which the truth of 'if x, if y, then not y, if not x'is independent of the meaning of x and y.

Thus our third example contains one variable only, viz. the word 'teacher' (or 'pupil'). Its constants are the words 'if-then', 'somebody', 'is', and 'converse'. The constant 'converse', how­ever, does not occur explicitly in the sentence, because we say 'pupil' ('teacher') when we mean what the logician calls a 'teacher converse'.

We might again say that the variable gives to our sentence its content, and that the constants give to it its form, and that the

 4

 

sentence expresses truth because of its form and independently of its content.

It is of some interest to observe that there are logical constants for which ordinary language has no name and for which the logician must invent a technical term. This fact probably offers a partial explanation why some branches of logic were much retarded in their development before the rise of so-called symbolic logic.

What we have said about variables and constants above is still vague. The following remarks are intended to make the distinction somewhat more precise:

In our syllogism there occurred three variables. The syllogism, we said, expresses a true proposition independently of the meaning of the variables. This, however, is subject to an important restriction. We must not substitute for the variables words which would make the whole expression void of meaning. We could not, for example, substitute numerals for the variables. We must substitute for them words of the same kind as 'Europeans', 'white men', and 'Mohammedans'. This means that we must substitute for them names of some properties (classes).

Similarly, it is obvious that in our second example we can only substitute for the variables sentences, i.e. linguistic expressions of propositions. In our third example again we can only substitute for the variable a name of a relation.

These restrictions on the meaning of the variables offer a suitable basis for a classification of expressions.

First, there are expressions in which the variables mean pro­positions. Their study belongs to a discipline known as the Logic of Propositions. Its foundations were laid by the Stoics. Its systematic study in modern times dates from Frege. It is a province of logic which is today pretty thoroughly explored.

Secondly, there are expressions in which the variables mean properties (classes). Their study belongs to a discipline which can conveniently be called the Logic of Properties (Classes). It includes, but is by no means coextensive with, the theory of the syllogism.

Thirdly, there are expressions in which the variables mean relations. The Logic of Relations was hardly studied at all before de Morgan. Important contributions to it were made by

 

Peirce and Schröder and the authors of Principia Mathematica. It still remains, however, a comparatively unexplored branch of logic.

There are other provinces of logic (and logical truth) besides the three mentioned, but we shall not discuss them here.

The three provinces mentioned are hierarchically related. This is seen from a consideration of logical constants. All the constants of the Logic of Propositions occur among the con­stants of the Logic of Properties, and all the constants of the Logic of Properties among the constants of the Logic of Rela­tions. But some constants of the Logic of Properties, e.g. 'all' and 'some', do not belong to the Logic of Propositions, and some constants of the Logic of Relations, e.g. 'converse', do not belong to the Logic of Properties.

It is an old observation that truth is sometimes dependent upon 'form' and sometimes upon 'content'. This observation has obviously something to do, e.g., with Leibniz's distinction between vérités de raison and vérités de faits, and with Kant's distinction between analytische Urteile and synthetische Urteile.

The distinction between form and content, however, is far from clear. What we have said above to elucidate it does not give us any means of determining whether a given sentence expresses logical truth or not. On the contrary, in distinguishing between the form and content of expressions we pre-supposed an insight into the truth of the propositions which they express and how this truth may be affected by the substitution of parts in the expressions. Nor does what Leibniz or Kant said in order to elucidate their above-mentioned distinctions help us very much.

A substantial contribution to the clarification of the idea of logical truth was made by Wittgenstein in his Tractatus logico-philosophicus. Wittgenstein there tried to give an account of the idea of logical truth by using the idea of a tautology which in its turn is based on the idea of a truth-function.

Let us call truth and falsehood truth-values. We can then define a truth-function in the following way:

A given proposition is a truth-function of n propositions, if for every possible combination of truth-values in the n propo­sitions there is a rule determining the truth-value of the given

6

 

proposition. (There are many problems attached to this defini­tion which we shall not discuss here.)

As we know, the Logic of Propositions studies expressions the variable parts of which are sentences. It was known before Wittgenstein that these expressions could be regarded as expressing truth-functions of the propositions expressed by their variable parts. Wittgenstein's new observation could be described as follows:

Among the truth-functions of n propositions there is the proposition which has the truth-value 'true' for every possible combination of truth-values in the n propositions. We call this truth-function the tautology of the n propositions. It is plain that if an expression of propositional logic expresses the tauto­logy of the propositions expressed by its variables, then what it expresses is true independently of the content of the variables. For whatever propositions the variables mean, the expression itself means a true proposition.

What truth-function of the propositions expressed by its variables a given expression of propositional logic itself expresses can always be found out or decided 'mechanically' in so-called truth-tables. The idea of truth-tables was developed by Wittgen­stein and independently of him by Post.¹ The characterization of logical truths as tautologies gets part of its importance from this fact, that one can for any given expression of propositional logic decide whether it expresses a tautology or not.

The tautology of n propositions is a proposition which is true for every possible combination of truth-values in the n proposi­tions. The combinations are mutually exclusive, i.e. n given propositions are never true and false in more than one of these ways. The combinations are also collectively exhaustive, i.e. n given propositions are always true and false in one of these ways. The tautology of the n propositions is the proposition which is true independently of which of these combinations of truth-values actually is the case with the n propositions. Logical truth could therefore be said to consist in agreement with every one of a number of mutually exclusive and collectively exhaus­tive possibilities.

1Post, 'Introduction to a General Theory of Elementary Propositions', in Amer. Journ. of Math. 43, 1921.

 

It is the character of (mutual exclusiveness and) collective exhaustiveness of the possibilities which entitles us to speak of them as representing every possible combination of truth-values in the n propositions. And this character itself is a consequence of two principles which are basic in relation to the concept of a truth-function and the construction of truth-tables and thus also to the decision whether an expression of propositional logic expresses logical truth or not. These principles are (i) that every proposition has a truth-value (is true or false) and (ii) that no proposition has more than one truth-value (is not true and false). We might call the first of them the Principle of Excluded Middle and the second the Principle of Contradiction.

It is an illusion to think that these two principles could themselves be proved to be tautologies and hence logical truths in the same sense as, e.g. 'it is raining or it is not raining' expresses the tautology of the proposition that it is raining. I shall not discuss here what sort of truths the two principles in question are or might be.

The concept of a tautology works very well for the purpose of clarifying the idea of logical truth or 'independence of content' in propositional logic, i.e. in that province of logical truth where the variable parts of the expressions which we study are sentences. Can it be applied for a similar purpose in those provinces of logical truth where the variables mean properties or relations?

This question could be answered affirmatively, if it could be shown that the logical constants which occur in expressions about properties and relations can be somehow 'eliminated' and 'replaced' by logical constants of propositional logic.

Consider again our syllogism. Suppose there are only two men in the world. We call them 'Smith' and 'Jones'. Instead of saying 'all Europeans are white men' we can also say 'if Smith is a European, then Smith is a white man, and if Jones is a European, then Jones is a white man'. Instead of saying 'some Europeans are Mohammedans' we can say 'Smith is a European and Smith is a Mohammedan or Jones is a European and Jones is a Mohammedan'. And instead of saying 'some white men are Mohammedans' we can say 'Smith is a white man and Smith is

8

 

a Mohammedan or Jones is a white man and Jones is a Mohammedan'. If we substitute these equivalent sentences for the premisses and the conclusion in our syllogism, we get a new if-then-sentence. It contains the logical constants 'if-then', 'and', 'or', 'not' and the six sentences 'Smith is a European', 'Smith is a Mohammedan', 'Smith is a white man', 'Jones is a Euro­pean', 'Jones is a Mohammedan', and 'Jones is a white man'. The constants mentioned are constants of propositional logic. The new expression which we get, therefore, expresses a truth-function of the propositions expressed by the six sentences. If we examine in a truth-table which truth-function of them it expresses, we shall find that it expresses their tautology.

It is 'intuitively' obvious that if the number of men were, not two, but one or three or four or five or just any number n we should always get a tautology if we replaced, as above, the all-sentences and the some-sentences by new sentences, consisting of the words 'if-then', 'and', 'or', and 'not', and parts which are themselves sentences. Let us call a world with only one man, with only two men, with only three men, etc., a 'possible world'. We can then say that the syllogism turns out to be a tautology in all possible worlds, if we eliminate from it the words 'all' and 'some' as described above.

Similar remarks about the elimination of constants apply to our example from relational logic: 'If somebody is teacher of somebody, then somebody is pupil of somebody.' The new expression which we get after the elimination contains the logical constants 'if-then', 'and', and 'or', and parts which are themselves sentences. If we examine in a truth-table which truth-function of the propositions expressed by these sentences the expression itself expresses, we shall find that it expresses their tautology.

It might now be suggested that the concept of a tautology in all possible worlds gives the proper meaning to the idea of logical truth or 'independence of content' in the Logic of Properties and the Logic of Relations.

The concept of a tautology in all possible worlds has a certain intuitive plausibility. But it is far from clear. There are at least two difficulties worth observation.

For any given world of n things we can use a truth-table to

9

 

decide whether any given expression of the Logic of Properties or Relations expresses a tautology in this world or not. But we cannot use truth-tables to show that such an expression expresses a tautology in all possible worlds, since the number of possible worlds is unlimited. When we said above that our syllogism expresses a tautology in all possible worlds, we referred to 'intuition'. Sometimes, however, we can refer to considerations (other than truth-tables) which amount to a 'proof' that an expression expresses a tautology in all possible worlds. This could actually have been done in the case of the syllogism and also in the case of our example about teachers and pupils. But the very nature of these proofs is far from clear, and besides one has not been able to provide a general instrument of proof which would make it possible in the case of any expression to decide whether it expresses a tautology in all possible worlds or not.

I now come to the second difficulty.

Consider the expression 'every man is shorter than some man'. Whatever be the number of men in the world, this expresses a falsehood. For among any number n of men, there will always be a tallest man (or several tallest men). It follows from this that the expression 'not every man is shorter than some man' expresses a truth in all possible worlds of men. But is this really a logical truth? Substitute for the word 'man' the word 'prime (number)' and for the word 'shorter' the word 'smaller'. Then we get the expressions 'every prime is smaller than some prime' and 'not every prime is smaller than some prime' respectively. The latter expresses a truth in any world of n numbers. Yet it expresses a false proposition.

These considerations take us to a distinction between finite and infinite worlds. Something can be true of any finite world without being true of an infinite world. Among any finite number of numbers there is a greatest number, but among all numbers there is not. Many things, on the other hand, are true of any finite world and also of infinite worlds. The syllogisms, for example, are valid for numbers as well as for men.

By an infinite world we shall here exclusively understand the world of natural numbers.

What it means for an expression to express a tautology in a
10

 

 

possible finite world is clear. It means to express a tautology in the sense of propositional logic of a finite number of proposi­tions. Which these propositions are can be found out by applying to the expression in question a technique for eliminating from it logical constants like 'all' and 'some' and 'converse'. What it means for an expression to express a tautology in an infinite world, however, is so far from clear that it is probably quite senseless. For there is no technique for the elimination of logical constants which would give us a finite number of propositions, of which the expression in question expressed the tautology.

It does not seem unplausible to suggest that to express a tautology in all possible finite worlds is a necessary criterion of logical truth.¹ But, as we have seen, this will hardly do as a sufficient criterion.

A sufficient criterion might be obtained in the following way:

Take an expression which fulfils the necessary criterion, i.e. expresses a tautology in all possible finite worlds. Take the contradictory of this expression. Let it be the case that we can substitute for its variable parts names of properties and rela­tions from the world of numbers and in this way obtain a pro­position which is true for numbers. We then say that the expres­sion from which we started does not express logical truth. It would have expressed logical truth, if it had not been the case that we can substitute in the contradictory of it names of arithmetical properties and relations and in this way obtain a true proposition.

About the necessary criterion we observe that no general method is known for deciding whether a given expression satisfies it or not.²

About the suggested sufficient criterion we only observe here that it seems to make the idea of logical truth dependent upon the idea of truth about numbers. As is well known, there is no general technique for deciding whether an expression about

1Our concept of a tautology in all possible (finite) worlds answers to that which Hilbert calls im Endlichen identisch.

²Since this was written (1949), it has been proved that no such general method is possible. See The Journal of Symbolic Logic, 15, 1950, p. 229. The proof is due to a Russian logician, Trachtenbrot. (1955)

11

 

numbers expresses a true proposition or not. There are, moreover, convincing reasons for believing that such a technique is impossible.

I shall now try to say some words about a different way of dealing with the problem of form and content in the Logic of Properties and Relations. This way also uses the idea of a tautology, but not the questionable extension of it which we have called a tautology in all possible worlds.

I shall first speak about the Logic of Properties.

Consider again our syllogism: 'If all Europeans are white men and some Europeans are Mohammedans, then some white men are Mohammedans'. As already observed, the syllogism is a sentence constructed out of the three sentences 'all Europeans are white men', 'some Europeans are Moham­medans', and 'some white men are Mohammedans' by means of the words 'if-then' and 'and'. These words are logical constants of propositional logic. For this reason the syllogism expresses a truth-function of the propositions expressed by the three sentences. If, however, we construct a truth-table for the syllogism, relying only upon the technique for the construction of such tables in propositional logic, we shall not find that it expresses the tautology of the propositions expressed by the three sentences. This is clear, for if the two first sentences expressed true propositions and the third a false proposition, then the syllogism itself would express a false proposition.

To say that the syllogism expresses logical truth is thus tantamount to saying that the propositions expressed by the premisses and the conclusion cannot be independently true and false in any combination of truth-values, but that the combination under which the two premisses express true pro­positions and the conclusion a false proposition is an impossibility. With the difficulties caused by this impossibility, how­ever, the technique of truth-tables in propositional logic is incapable of coping.

We raise the following question: Could the decision-technique of propositional logic be improved or amplified so as to make the tautological character of the proposition expressed by the syllogism emerge from a truth-table? It is not difficult to see that this question can be answered affirmatively.

12

 

We begin by defining a certain set of eight conditions in the following way:

The first condition is satisfied by any man who is European, white, and Mohammedan, the second by any man who is European and white but not Mohammedan, the third by any man who is European and Mohammedan but not white, the fourth by any man who is European but neither white nor Mohammedan, the fifth by any man who is not European but white and Mohammedan, the sixth by any man who is neither European nor white but Mohammedan, the seventh by any man who is neither European nor Mohammedan but white, and the eighth by any man who is neither European nor white nor Mohammedan.

These conditions are mutually exclusive, i.e. no man can satisfy two of them. They are also collectively exhaustive, i.e. any man satisfies one of them. They can be independently satisfied or not in 256 (= 2⁸) different ways, beginning from the case when they are all satisfied and ending with the case when none of them is satisfied. (The last case would mean that there are no men at all; if we wish, we can omit it from consideration as being an 'awkward' case.)

Anything that can be said about Europeans, white men, and Mohammedans, using the logical constants of the Logic of Propositions and of Properties can also be expressed by saying something about the way in which the eight conditions are satisfied or not. To say that all Europeans are white men is to say that the third and fourth conditions are not satisfied. To say the some Europeans are Mohammedans is to say that the first or third conditions are satisfied. Since to say that all Europeans are white men is to exclude the third condition from being satisfied, it follows that to say that all Europeans are white men and some Europeans are Mohammedans is to say that the first condition is satisfied but not the third and fourth. To say that some white men are Mohammedans, finally, is to say that the first or the fifth conditions are satis­fied. What the syllogism itself says is, therefore, that if the first condition is satisfied but not the third and fourth, then the first or the fifth conditions are also satisfied. This is a logical truth. It is further a truth-function of the proposition that the first

13

 

condition is satisfied, the proposition that the third condition is satisfied, the proposition that the fourth condition is satisfied, and the proposition that the fifth condition is satisfied. If we construct a truth-table, we shall find that it is the tautology of these four propositions.

By this means it has been shown that the syllogism expresses the tautology of four propositions. These four propositions are each to the effect that there is a man who satisfies a certain condition. We shall, therefore, call them the existence-constituents of the proposition expressed by the syllogism.

The above considerations can be generalized. Take any expression in the Logic of Properties. It contains n variable parts. These variables are names of properties. By a procedure —illustrated above for 'Europeans', 'white men', and 'Mohammedans'—we can set up a number (=2n) of conditions, which are mutually exclusive, collectively exhaustive, and independently satisfiable. By another procedure¹ we show that the given expression in the Logic of Properties expresses a truth-function of some propositions each to the effect that there is a thing which satisfies a certain one of the conditions. These propositions we call the existence-constituents of the proposition expressed by the original expression. If the original expression expresses the tautology of its existence-constituents it expresses logical truth.

The truth-value of any proposition to the effect that there is a thing which satisfies a certain one of the conditions depends upon the content of the n names of properties which occur in the expression. The (truth-value of the) tautology of some of these propositions, however, is independent of the truth-values of the propositions themselves. Hence if the expression expresses the tautology of these propositions, it expresses truth independently of the content of the property-names.

It seems to me that the concept of a tautology of existence-constituents gives a fair account of the idea of logical truth or 'independence of content' in the Logic of Properties. It is connected with a universal decision-procedure and it avoids the

¹There is a detailed account of these procedures and of the idea of logical truth in the Logic of Properties in the next essay. Essentially similar ideas about decidability are found in Quine's paper 'On the Logic of Quantification' (The Journal of Symbolic Logic, 10, 1945).

14

 

difficulties and obscurities which are attached to the concept of a tautology in all possible worlds and which arise from the attempt to eliminate the logical constants peculiar to the Logic of Properties.

I shall now say some words about the Logic of Relations.

Consider again our example 'if somebody is teacher of somebody, then somebody is pupil of somebody'. It is a sentence constructed out of the two sentences 'somebody is teacher of somebody' and 'somebody is pupil of somebody' by means of the logical constant 'if-then'. The logical constant is one of propositional logic. For this reason our example expresses a truth-function of the propositions expressed by the two sen­tences. If, however, we construct a truth-table, relying upon the technique for the construction of such tables in propositional logic, we shall certainly not find that our example expresses the tautology of the propositions expressed by the two sentences. We therefore ask whether the amplified decision-technique of the Logic of Properties can cope with the case.

One could define a set of mutually exclusive and collectively exhaustive conditions in the following way:

The first condition is satisfied by any man who is somebody's teacher and also somebody's pupil, the second by any man who is somebody's teacher but not anybody's pupil, the third by any man who is somebody's pupil but not anybody's teacher, and the fourth by any man who is neither anybody's teacher nor anybody's pupil.

To say that somebody is teacher of somebody is to say that the first or the second conditions are satisfied. To say that somebody is pupil of somebody is to say that the second or third conditions are satisfied. Hence to say that if somebody is teacher of somebody, then somebody is pupil of somebody is to say that if the first or second conditions are satisfied, then also the second or third. This is a truth-function of the pro­position that the first condition is satisfied, the proposition that the second condition is satisfied, and the proposition that the third condition is satisfied. Since each of these propositions is to the effect that there is a thing (a man) satisfying a certain condition, we shall call them the existence-constituents of the proposition that if somebody is teacher of somebody, then some-
15

 

body is pupil of somebody. This last proposition is thus, like the proposition expressed by our syllogism, a truth-function of its existence-constituents. If, however, we construct a truth-table, we shall not find that it is their tautology.

It is easy to see why the decision-technique of the Logic of Properties cannot cope with the concept of logical truth in the case now under discussion. The failure is due to the fact that the four conditions about teachers and pupils, though they resemble our eight conditions above about Europeans, white men, and Mohammedans in being mutually exclusive and collectively exhaustive, differ from them in not being independently satisfiable or left unsatisfied. This again is due to the fact that our example, as we have already observed, contains a logical constant—peculiar to the Logic of Relations—which is not explicit in ordinary language, viz. the constant which we have called 'converse'. Or to put it in a different way: our example does not contain two variable relation-names, but only one. We can regard 'teacher' as the variable name, in which case we have for 'pupil' to read 'teacher converse'. Or we can regard 'pupil' as the variable name, in which case we have for 'teacher' to substitute 'pupil converse'.

There is a feature which is characteristic of any relation and its converse relation. We will call it their existential symmetry. This means in our example that if some of the conditions for the truth of the proposition that somebody is teacher of some­body is satisfied, then some of the conditions for the truth of the proposition that somebody is pupil of somebody is also satis­fied, and vice versa. From this it follows that of the sixteen different ways in which our four conditions above might be satisfied or left unsatisfied according to the Principles of Excluded Middle or Contradiction, four ways are excluded as impossible thanks to the existential symmetry of the rela­tions called 'teacher' and 'pupil'.

According to the above, there are thus 16 — 4 or twelve combinations of truth-values in the existence-constituents, representing the 16 — 4 or twelve different ways in which the respective conditions can be satisfied or left unsatisfied. These twelve combinations of truth-values are mutually exclusive, i.e. the existence-constituents are never true and false in more

16

 

than one of these twelve ways. The combinations are also collectively exhaustive, i.e. the existence-constituents are always true and false in some of these twelve ways. It is this character of (mutual exclusiveness and) collective exhaustiveness which entitles us to speak of the twelve combinations as all possible combinations. (Cf. above, p. 8.)

If we construct a truth-table, we shall find that our example expresses a proposition which is true of every one of the possible combinations of truth-values in its existence-constituents. It thus expresses their tautology.

(The truth-table used in this case differs from an 'ordinary' truth-table in that certain combinations of truth-values are missing in it.)¹

The above considerations can also be generalized. Take any expression in the Logic of Relations. It contains n variable parts. These variables are names of relations. By a procedure— not, however, adequately illustrated by our example above for 'teacher' and 'pupil'—we can set up a number of conditions which are mutually exclusive and collectively exhaustive, though not (in general) independently satisfiable. By another procedure we show that the given expression in the Logic of Relations expresses a truth-function of some propositions each to the effect that there is a thing which satisfies a certain one of the conditions. These propositions we call the existence-constituents of the proposition expressed by the original expres­sion. By a third procedure, the nature of which is determined by the logical constants peculiar to relational logic,² we exclude certain combinations of truth-values in the existence-con­stituents as being impossible. The remaining combinations of truth-values represent a set of mutually exclusive and collec­tively exhaustive possibilities. If the original expression expresses the tautology of its existence-constituents (i.e., a proposition which is true for every one of the possible combinations of

1Truth-tables of this mutilated type were, as far as I know, for the first time mentioned by Wittgenstein in 'Some Remarks on Logical Form' in Proceedings of the Aristotelian Society, Supp. vol. 9, 1929. (1955)

2There are two (primitive) logical constants of relational logic. One is the constant which we have called 'converse'. The other is usually known as 'relative product' or 'chain'. The relation called 'uncle', e.g., is the chain-relation of the relations called 'brother' and 'father', and the relation called 'grandfather' is the chain-relation of the relation called 'father' with itself.

L.S.-C                          

17

 

truth-values in the existence-constituents), it expresses logical truth.

The truth-value of any proposition to the effect that there is a thing which satisfies a certain one of the conditions depends upon the content of the n names of relations which occur in the expression. The tautology of some of these propositions, however, is independent of the truth-values of the propositions themselves. Hence if the expression expresses the tautology of these propositions, it expresses truth independently of the con­tent of the relation-names.

It would be premature to attempt to answer the question whether the concept of a tautology of existence-constituents can give a satisfactory account of the idea of logical truth or 'independence of content' in the Logic of Relations, before settling other problems which occur in this context. One such problem is whether the concept of a tautology of existence-constituents as a criterion of logical truth is equivalent to the necessary and sufficient criterion which we have previously suggested (p. 11) or to a similar criterion. I shall not discuss this problem here.

appendix I

The above outline of decision-procedures for the Logic of Propositions, Properties, and Relations mentions truth-tables. There is another technique which is equivalent to the construction of truth-tables. It consists in the transformation of a given expression into a certain 'normal form'. From this normal form can immediately be seen, whether the given expression expresses logical truth or not.

It is a well-known fact that any expression of propositional logic has what might be called a perfect disjunctive normal form (ausgezeichnete disjunktive Normalform). This normal form enumerates those of a number of mutually exclusive and collectively exhaustive possibilities with which the given ex­pression expresses agreement. If it enumerates all possibilities, the expression expresses logical truth.

It is not difficult to show that any expression in the Logic of Properties has an analogous disjunctive normal form. The technique by means of which we derive it can be described as a

 

twofold application of the technique for deriving the perfect disjunctive normal form of expressions of propositional logic.¹ That the application is 'twofold' is a consequence of the fact that any expression in the Logic of Properties expresses agreement with some of a number of mutually exclusive and collectively exhaustive ways of satisfying a number of conditions, which are themselves mutually exclusive and collectively exhaustive.

It can finally be shown that also any expression of relational logic has an analogous normal form. The technique by means of which we derive it is substantially the same as the technique for deriving normal forms in the Logic of Properties. The normal forms of expressions of relational logic, however, differ from the normal forms in the Logic of Properties in that they sometimes contain redundant parts, answering to impossible combinations of truth-values in the existence-constituents.

appendix II (1955)

The analogues in the Logic of Properties and of Relations to the disjunctive and conjunctive normal forms known from the Logic of Propositions may be called distributive normal forms. For the existence of these analogues essentially depends upon the distributivity of the existential quantifier with regard to disjunctions, and of the universal quantifier with regard to conjunctions. From the technical point of view the distributive normal forms may be contrasted with the so-called prenex normal forms.

The study of distributive normal forms in the Logic of Properties can be traced back to a paper by H. Behmann, 'Beiträge zur Algebra der Logik, insbesondere zur Entscheidungsproblem' in Mathematische Annalen, 86, 1922. In Hilbert-Bernays's Grundlagen der Mathematik (Vol. i, pp. 146-8) the technique of these normal form derivations is described under the name of 'Zerlegung in Primärformeln'. Essentially the same technique is also described by Quine in his paper 'On the Logic of Quantification'. (Cf. above p. 14.)

That the theory of distributive normal forms can be extended

1A description of this technique of transformation into normal forms is found in the next essay.

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to the Logic of Relations has been shown in detail by J. Hintikka in 'Distributive Normal Forms in the Calculus of Predicates (Acta Philosophica Fennica,6, 1953). Hintikka's thesis is the most comprehensive study of these normal forms that exists. Hintikka has also shown that the notion of a tautology of existence-constituents as a criterion of logical truth in the predicate calculus is in effect equivalent to the traditional criterion using the notion of satisfiability. His findings may be said to constitute an affirmative answer to the question, raised in the essay (p. 18), whether the idea of the tautology can give a satisfactory account of formal truth and 'independence of content' in the Logic of Relations.

Applications of the theory of distributive normal forms to Modal Logic are found in my book An Essay in Modal Logic (Amsterdam 1951) and in the paper 'Deontic Logic' of the present collection.

appendix III (1955)

Traditionally, syllogisms are formulated as inferences of this type: 'All Europeans are white men. Some Europeans are Mohammedans. Therefore some white men are Mohammedans.' The formulation of syllogisms as if-then-sentences is unusual. It is found, for example, in Łukasiewicz's work on Aristotle's Syllogistic (1951). Łukasiewicz strongly insists that this formu­lation is true to Aristotle's view of the syllogism. 'All Aristo­telian syllogisms', he says (p. 20), 'are implications of the type "If α and β, then γ", where α and β are the two premisses and γ is the conclusion.' Łukasiewicz also points out (p. 73 and passim) that 'no Aristotelian syllogism is formulated as a rule of inference with the word "therefore", as is done in the tradi­tional logic'. 'The difference', he says (p. 21), 'between the Aristotelian and the traditional syllogism is fundamental.'

Much as these opinions of one of the finest authorities on the history of logic support the correctness of presenting Aris­totelian syllogisms as done in this paper, I must own that I feel doubts about their historical truth. For all I can see, no conclusive support can be found in the original text, as we have it, either for the view that the formulation with 'therefore' or for the view that the formulation with 'if-then' and 'and' is the

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(historically) correct one. The use which Aristotle himself makes of 'if-then'-formulations in the context does not, in my opinion, establish Lukasiewicz's point. When, for example, Aristotle says (Analytica Posteriora 96a 12-14) that 'if A is pre­dicated universally of B and B of C, then A too must be pre­dicated always and in every instance of C', he is not formulating a syllogism, but speaking about one. And the same seems to me to hold for the only example (98^b 5-10) quoted by Lukasiewicz

(p. 2).

I should myself attach significance to the fact that Aristotle, as far as I have been able to find, nowhere speaks of the syllo­gism as a truth (true proposition), nor of the proof ofa syllogism. (Cf. Lukasiewicz, op. cit., p. 44). To say with Lukasiewicz (p. 44 and p. 73) that Aristotle's own syllogistics is an 'axiomatized deductive system' is certainly a very bold 'moderniza­tion' of Aristotle. It seems to me much more to the point to say that the notion of logical truth is unknown to Aristotle. This is not necessarily to blame Aristotle of ignorance. It is an interesting question, to what extent logic can be developed independently of the idea of logical truth. The importance of this problem, however, was not alive to me, when I wrote the essay on form and content in logic.

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