Logical form (Lukasiewicz)
From Aristotle's Syllogistic, by Jan Lukasiewicz, Oxford University Press 1951.
What is formal logic?
'It is usual to say that logic is formal, in so far as it is concerned merely with the form of thought, that is with our manner of thinking irrespective of the particular objects about which we are thinking'. This is a quotation from the well-known text-book of formal logic by Keynes[1]. And here is another quotation, from the History of Philosophy by Father Copleston: 'The Aristotelian Logic is often termed formal logic. Inasmuch as the Logic of Aristotle is an analysis of the forms of thought – this is an apt characterisation'[2].
In both quotations I read the expression 'form of thought', which I do not understand. Thought is a psychical phenomenon, and psychical phenomena have no extension. What is meant by the form of an object that has no extension? The expression 'form of thought' is inexact and it seems to me that this inexactitude arose from a wrong conception of logic. If you believe indeed that logic is the science of the laws of thought, you will be disposed to think that formal logic is an investigation of the forms of thought.
It is not true, however, that logic is the science of the laws of thought. It is not the object of logic to investigate how we are thinking or how we ought to think. The first task belongs to psychology, the second to a practical art of a similar kind to mnemonics. Logic has no more to do with thinking than mathematics has. You must think, of course, when you have to carry out an inference or proof, as you must think, too, when you have to solve a mathematical problem. But the laws of logic do not concern your thoughts in a greater degree than do those of mathematics. What is called 'psychologism' in logic is a mark of the decay of logic in modern philosophy. For this decay Aristotle is by no means responsible. Throughout the whole Prior Analytics, where the theory of the syllogism is systematically exposed, there exists not one psychological term. Aristotle knows with an intuitive sureness what belongs to logic, and among the logical problems treated by him there is no problem connected with a psychical phenomenon such as thinking.
What is therefore, according to Aristotle, the object of logic, and why is his logic called formal? The answer to this question is not given by Aristotle himself but by his followers, the Peripatetics.
There was a dispute among the philosophical schools of Ancient Greece about the relation of logic to philosophy. The Stoics contended that logic was a part of philosophy, the Peripatetics said that it was only an instrument of philosophy, and the Platonists were of the opinion that logic was equally a part and an instrument of philosophy. The dispute itself is of no great interest or importance, because the solution of the disputed problem seems to be for the most part a matter of convention. But an argument of the Peripatetics, preserved by Ammonius in his commentary on the Prior Analytics, deserves our attention.
Ammonius agrees with the Platonists and says: If you take syllogisms with concrete terms, as Plato does in proving syllogistically that the soul is immortal, then you treat logic as a part of philosophy; but if you take syllogisms as pure rules stated in letters, e.g. 'A is predicated of all B, B of all C, therefore A is predicated of all C', as do the Peripatetics following Aristotle, then you treat logic as an instrument of philosophy[3].
It is important to learn from this passage that according to the Peripatetics, who followed Aristotle, only syllogistic laws stated in variables belong to logic, and not their applications to concrete terms. The concrete terms, i.e. the values of the variables, are called the matter, hyle, of the syllogism. If you remove all concrete terms from a syllogism, replacing them by letters, you have removed the matter of the syllogism and what remains is called its form. Let us see of what elements this form consists.
To the form of the syllogism belong, besides the number and the disposition of the variables, the so-called logical constants. Two of them, the conjunctions 'and' and 'if', are auxiliary expressions and form part, as we shall see later, of a logical system which is more fundamental than that of Aristotle. The remaining four constants, viz. 'to belong to all', 'to belong to none', 'to belong to some' and 'to not-belong to some' are characteristic of Aristotelian logic. These constants represent relations between universal terms. The medieval logicians denoted them by A, E, I, and O respectively. The whole Aristotelian theory of the syllogism is built up on these four expressions with the help of the conjunctions 'and' and 'if'. We may say therefore: The logic of Aristotle is a theory of the relations A, E, I, and O in the field of universal terms.
It is obvious that such a theory has nothing more in common with our thinking than, for instance, the theory of the relations of greater and less in the field of numbers. There are, indeed, some similarities between these two theories. Compare, for example, the syllogism Barbara:
If a belongs to all b
and b belongs to all c,
then a belongs to all c,
with the following arithmetical law:
If a is greater than b
and b is greater than c,
then a is greater than c,
There are, of course, differences between these two laws: the range of variables is not the same, and the relations are different. But both relations, although different and occurring between different terms, have one property in common: they are both transitive, i.e. they are particular cases of the formula:
If a has the relation R to b
and b has the relation R to c,
then a has the relation R to c,
It is a curious thing that this very fact was observed by the logicians of the later school of the Stoics. Arguments like 'the first is greater than the second, the second is greater than the third, therefore the first is greater than the third' were called by the Stoics, as Alexander declares 'non-methodically conclusive' and were not treated as syllogisms in the sense of their logic. Nevertheless, the Stoics regarded such arguments as similar (homoioi) to categorical syllogisms[4]. This observation of the Stoics, which Alexander tries to confute without producing convincing counter-arguments, corroborates the supposition that the logic of Aristotle was conceived as a theory of special relations, like a mathematical theory.
What is formalism?
Formal logic and formalistic logic are two different things. The Aristotelian logic is formal without being formalistic, whereas the logic of the Stoics is both formal and formalistic. Let us explain what in modern formal logic is meant by 'formalism'.
Modern formal logic strives to attain the greatest possible exactness. This aim can be reached only by means of a precise language built up of stable, visually perceptible signs. Such a language is indispensable for any science. Our own thoughts not formed in words are for ourselves almost inapprehensible and the thoughts of other people, when not bearing an external shape, could be accessible only to a clairvoyant. Every scientific truth in order to be perceived and verified, must be put into an external form intelligible to everybody. All these statements seem incontestably true. Modern formal logic gives therefore the utmost attention to precision of language. What is called formalism is the consequence of this tendency. In order to understand what it is, let us analyse the following example.
There exists in logic a rule of inference, called formerly modus ponens and now the rule of detachment. According to this rule, if an implication of the form 'If a then b' is asserted and the antecedent of this implication is asserted too, we are allowed to assert its consequent b. In order to be able to apply this rule we must know that the proposition a, asserted separately, expresses 'the same' thought as the antecedent a of the implication, since only in this case are we allowed to perform the inference. We can state this only in the case where these two a's have exactly the same external form. For we cannot directly grasp the thoughts expressed by these two a's, and a necessary, although not sufficient, condition for identifying two thoughts is the external equality of their expressions.
When, for instance, asserting the implication 'If all philosophers are men, then all philosophers are mortal' you would also assert as second premiss the sentence 'Every philosopher is a man', you could not get from these premisses the conclusion 'All philosophers are mortal', because you would have no guarantee that the sentence 'Every philosopher is a man' represents the same thought as the sentence 'All philosophers are men'. It would be necessary to confirm by means of a definition that 'Every A is B' means the same as 'All A's are B's'; on the ground of this definition replace the sentence 'Every philosopher is a man' by the sentence 'All philosophers are men', and only then will it be possible to get the conclusion. By this example you can easily comprehend the meaning of formalism. Formalism requires that exactly the same thought should always be expressed by means of exactly the same series of words ordered in exactly the same manner. When a proof is formed according to this principle, we are able to control its validity on the basis of its external form only, without referring to the meaning of the terms used in the proof. In order to get the conclusion b from the premiss 'If a, then b', and a, we need not know either what a or what b really means; it suffices to notice that the two a's contained in the premisses have the same external form.
Aristotle and his followers, the Peripatetics, were not formalists. As we have already seen, Aristotle is not scrupulously exact in formulating his theses. The most striking case of this inexactitude is the structural discrepancy between the abstract and concrete forms of the syllogisms. Take as an example the syllogism with opposite premisses quoted above, in our section 4[5]. Let B and C be 'science' and A 'medicine'. Aristotle states:
| In variables | In concrete terms |
|---|---|
| If B belongs to all A | If all medicine is science |
| And C belongs to no A, | and no medicine is science, |
| then C does not belong to some B [6] | Then some science is not science |
The difference of corresponding premisses, of which the two syllogisms consist, is evident. Take, for instance, the first premiss. To the formula 'B belongs to all A' would correspond the sentence 'Science belongs to all medicine', and to the sentence 'All medicine is science' would correspond the formula 'All A is B'. The sentence in concrete terms, given by Aristotle, cannot be regarded as a substitution of the abstract formula accepted by him. What is the cause of the difference?
Alexander gives three explanations of this problem[7]: the first may be omitted as unimportant, the last is a philosophical one and is, in my opinion, wrong; only the second deserves our attention. According to this explanation, in formulae with the verb 'to be predicated of something' and, we may add, with the verb 'to belong to something', the subject and the predicate are better distinguishable (gnorimoteroi) than, we may add again, in formulae with the verb 'to be'. In fact, in formulae with 'to be' the subject as well as the predicate is used in the nominative; in formulae preferred by Aristotle only the predicate is in the nominative, and the subject is either in the genitive or in the dative and therefore can be more easily distinguished from the predicate. Very instructive, too, is the final remark of Alexander, from which it follows to say 'Virtue is predicated of all justice' instead of the customary 'All justice is virtue' was felt in Ancient Greek to be as artificial as in modern languages.
There are still more cases of inexactitude in Aristotelian logic. Aristotle constantly uses different phrases for the same thoughts. I shall give only a few examples of this kind. He begins his syllogistic with the words 'A is predicated of all B', but shortly he changes these words into the phrase 'A belongs to all B', which seems to be regular. The words 'is predicated' and 'belongs' are frequently omitted, sometimes even the important sign of the quantity 'all' is dropped. Besides the form 'A belongs to some B' there are forms which may be translated 'A belongs to some of the B's'. The premises of the syllogism are combined by means of different conjunctions. Syllogistic necessity is expressed in different ways and is sometimes entirely omitted[8]. Although these inexactitudes have no bad consequence for the system, they contribute in no way to its clearness of simplicity.
This procedure of Aristotle is probably not accidental, but seems to derive from some preconceptions. Aristotle says occasionally that we ought to exchange equivalent terms, words for words and phrases for phrases[9]. Commenting on this passage, Alexander declares that the essence of the syllogism depends not on words but on their meanings[10]. This statement, which is manifestly directed against the Stoics, can be understood thus: the syllogism does not change its essence, i.e. it remains a syllogism, if some of its expressions are replaced by other equivalent expressions, e.g. if the expression 'to be predicated of all' is replaced by the equivalent expression 'to belong to all'. The Stoics were of a directly opposite opinion. They would say that the essence of the syllogism depends on words, but not on their meanings. If therefore the words are changed the syllogism ceases to exist. This is illustrated by Alexander with an example from the logic of the Stoics. The rule of inference called modus ponens:
If a, then b;
but a;
therefore b,
is the first 'indemonstrable' syllogism of the Stoics. Both the Stoics and the Peripatetics seem mistakenly to regard the phrases 'If a, then b' and 'a entails b' as having the same meaning. But if, in the syllogism above, you replace the premiss 'If a, then b' by 'a entails b', saying
a entails b;
but a;
therefore b,
you get according to the Stoics a valid rule of inference, but not a syllogism. The logic of the Stoics is formalistic[11].
Notes
- ↑ Op. cit., p. 2.
- ↑ Op. cit., p. 277.
- ↑ Ammonius 10.36
- ↑ Alexander of Aphrodisias (fl. 200 AD), Alexandri in Aristotelis Analyticorum Priorum Librum I Commentarium, ed. M. Wallies, Berolini, 1899, 21.30.
- ↑ See p. 9 n3
- ↑ The conclusion in variables is dropped in the Greek text.
- ↑ Alexander 54.21
- ↑ The phrase to kata pantos tou B (kategoreitai) is twice omitted) is used in the mood Barbara (see p.3, n.2), to A panti toi B (huparchei) is altogether omitted) is used in another formulation of the same mood (see p. 10, n.5). The phrase to to A tini toi B appears in the laws of conversion; elsewhere, e.g. in the mood Disamis, we have to A tini toi B (see p.9, n.1). The logically important word panti is altogether omitted in a formulation of the mood Barbara (see p.2, n.). The conjunction 'and' is for the most part denoted by kai (see p.3, n.2; p.19, n.5). Syllogistic necessity is as a rule expressed by anagke huparchein (see p.7, n. of p.9, n.1), in the mood Felapton it is denoted by huparchei ex anagkes (see p.9, n.4). In one case it is dropped (see p.10, n.5).
- ↑ An. Prior i. 39, 49b 3.
- ↑ Alexander 372. 29
- ↑ Alexander 372. 28
