# Form and content in logic

**FORM AND CONTENT**

**IN**

**LOGIC**

One of the main objects of the logicians' inquiries has traditionally been various provinces of what might be called logical truth

The most ancient and best known example is Aristotle's theory of the syllogism. Let us say a few words about it.

An Aristotelian syllogism is, e.g. 'if all Europeans are white men and some Europeans are Mohammedans, then some white men are Mohammedans.'

The syllogism is an if-then-sentence.^{1} The if-sentence consists in its turn of two sentences, viz. 'all Europeans are white men' and 'some Europeans are Mohammedans', joined by the word 'and'. They are called premisses. The then-sentence is called the conclusion.

*if*all Europeans are white men and some are Mohammedans,

*then*some white men are Mohammedans.

*x*are

*y*and some

*x*are

*z,*then some

*y*are

*z'*without saying what

*x*and

*y*and

*z*mean.

^{1}See Appendix III at the end of this paper.

L.S.-B

*x*are

*y*essentially depends upon what

*x*and

*y*mean. The premisses and the conclusion, unlike the syllogism, thus express truth or falsehood according to their content and not for reasons of form.

^{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.

*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'.

*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.

*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*.

*(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'.

*of the same kind as*'Europeans', 'white men', and 'Mohammedans'. This means that we must substitute for them names of some properties (classes).

*Principia Mathematica.*It still remains, however, a comparatively unexplored branch of logic.

*v*

*érités de raison*and

*v*

*érités de faits,*and with Kant's distinction between

*analytische Urteile*and

*synthetische Urteile.*

*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.

*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

*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.

^{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.

*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.

*Amer.*

*Journ. of Math.*

**43**, 1921.

*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.

*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.

*n*things we can use a truth-table to

*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.

*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.

^{1}But, as we have seen, this will hardly do as a sufficient criterion.

*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.

^{2}

*im Endlichen identisch.*

^{2}Since 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)

*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.

^{8}) 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.)

*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

*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 procedure

^{1}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.

*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.

^{1}There 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).

*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.

*not*find that it is their tautology.

*all possible*combinations. (Cf. above, p. 8.)

^{1}

*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

*Proceedings of the Aristotelian Society,*Supp. vol. 9, 1929. (1955)

*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.

*(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.

^{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.

*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.

*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.)

*(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.

*An Essay in Modal Logic*(Amsterdam 1951) and in the paper 'Deontic Logic' of the present collection.

*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.'

*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*

*(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

*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.