Mathematical Logic as based on the Theory of Types


Published in: American Journal of Mathematics, vol.30(1908), pp. 222-262
Transcribed into hypertext by Burtzev B.I., Jul., 07, 2003
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All and Any.

AV.Given a statement containing a variable x, say ("x = x" we may affirm that this holds in all instances, or we may affirm any one of the instances without deciding as to which instance we are affirming. The distinction is roughly the same as that between the general and particular enunciation in Euclid. The general enunciation tells us something about (say) all triangles, while the particular enunciation takes one triangle, and asserts the same thing of this one triangle. But the triangle taken is any triangle, not some one special triangle; and thus although, throughout the proof, only one triangle is dealt with, yet the proof retains its generality. If we say : "Let ABC be a triangle, then the sides AB, AC are together greater than the side BC" we are saying something about one triangle, not about all triangles; but the one triangle concerned is absolutely ambiguous, and our statement consequently is also absolutely ambiguous. We do not affirm any one definite proposition, but an undetermined one of all the propositions resulting from supposing ABC to be this or that triangle. This notion of ambiguous assertion is very important, and it is vital not to confound an ambiguous assertion with the definite assertion that the same thing holds in all cases.
AW.The distinction between (1) asserting any value of a prepositional function, and (2) asserting that the function is always true, is present throughout mathematics, as it is in Euclid's distinction of general and particular enunciations. In any chain of mathematical reasoning, the objects whose properties are being investigated are the arguments to any value of some prepositional function. Take as an illustration the following definition :
AX."We call f(x) continuous for x = a if, for every positive number σ, different from 0, there exists a positive number ε, different from 0, such that, for all values of δ which are numerically less than ε, the difference f(a + δ) Ч f(a) is numerically less than σ.
AY.Here the function f is any function for which the above statement has a meaning; the statement is about f, and varies as f varies. But the statement is not about a σ or ε or δ, because all possible values of these are concerned, not one undetermined value. (In regard to ε, the statement "there exists a positive number ε such that etc" is the denial that the denial of "etc" is true of all positive numbers.) For this reason, when any value of a prepositional function is asserted, the argument (e.g., f in the above) is called a real variable; whereas, when a function is said to be always true, or to be not always true, the argument is called an apparent variable.8) Thus in the above definition, f is a real variable, and σ, ε, δ are apparent variables.
AZ.When we assert any value of a propositional function, we shall say simply that we assert the propositional function. Thus if we enunciate the law of identity in the form "x = x;" we are asserting the function "x = x;" i. e., we are asserting any value of this function. Similarly we may be said to deny a propositional function when we deny any instance of it. We can only truly assert a propositional function if, whatever value we choose, that value is true; similarly we can only truly deny it if, whatever value we choose, that value is false. Hence in the general case, in which some values are true and some false, we can neither assert nor deny a propositional function.9)
BA.If φx is a propositional function, we will denote by У(x).φxФ the proposition Уφx is always true.Ф Similarly (x, y).φ(x, y) will mean Уφ(x, y) is always true,Ф and so on. Then the distinction between the assertion of all values and the assertion of any is the distinction between (1) asserting (x).φx and (2) asserting φx where x is undetermined. The latter differs from the former in that it can not be treated as one determinate proposition.
BB.The distinction between asserting φx and asserting (x).φ(x) was, I believe, first emphasized by Frege.10) His reason for introducing the distinction explicitly was the same which had caused it to be present in the practice of mathematicians; namely, that deduction can only be effected with real variables, not with apparent variables. In the case of EuclidТs proofs, this is evident: we need (say) some one triangle ABC to reason about, though it does not matter what triangle it is. The triangle ABC is a real variable; and although it is any triangle, it remains the same triangle throughout the argument. But in the general enunciation, the triangle is an apparent variable. If we adhere to the apparent variable, we can not perform any deductions, and this is why in all proofs, real variables have to be used. Suppose, to take the simplest case, that we know Уφx is always trueФ i. e., У(x).φxФ and we know Уφx always implies ψxФ i. e. У(x).{φx implies ψx}.Ф How shall we infer Уψx is always true,Ф i.e., У(x).ψx?Ф We know it is always true that if φx is true, and if φx implies ψx, then ψx is true. But we have no premises to the effect that φx is true and φx implies ψx what we have is : φx is always true, and φx always implies ψx. In order to make our inference, we must go from Уφx is always trueФ to φx, and from Уφx always implies ψxФ to Уφx implies ψx,Ф where the x, while remaining any possible argument, is to be the same in both. Then, from УφxФ and Уφx implies ψxФ we infer Уψx;Ф thus Уψx,Ф is true for any possible argument, and therefore is always true. Thus in order to infer У(x)ψxФ from У(x)φxФ and У(x).{φx implies ψx}Ф we have to pass from the apparent to the real variable, and then back again to the apparent variable. This process is required in all mathematical reasoning which proceeds from the assertion of all values of one or more propositional functions to the assertion of all values of some other propositional function, as, e. g , from Уall isosceles triangles have equal angles at the baseФ to Уall triangles having equal angles at the base are isosceles.Ф In particular, this process is required in proving Barbara and the other moods of the syllogism. In a word, all deduction operates with real variables (or with constants).
BC.It might be supposed that we could dispense with apparent variables altogether, contenting ourselves with any as a substitute for all. This, however, is not the case. Take, for example, the definition of a continuous function quoted above: in this definition σ, ε and δ must be apparent variables. Apparent variables are constantly required for definitions. Take, e. g., the following: УAn integer is called a prime when it has no integral factors except 1 and itself.Ф This definition unavoidably involves an apparent variable in the form : УIf n is an integer other than 1 or the given integer, n is not a factor of the given integer, for all possible values of n
BD.The distinction between all and any is, therefore, necessary to deductive reasoning, and occurs throughout mathematics; though, so far as I know, its importance remained unnoticed until Frege pointed it out
BE.For our purposes it has a different utility, which is very great. In the case of such variables as propositions or properties, Уany valueФ is legitimate, though Уall valuesФ is not. Thus we may say: Уp is true or false, where p is any proposition,Ф though we can not say Уall propositions are true or false.Ф The reason is that, in the former, we merely affirm an undetermined one of the propositions of the form Уp is true or false,Ф whereas in the latter we affirm (if anything) a new proposition, different from all the propositions of the form Уp is true or fals.Ф Thus we may admit Уany valueФ of a variable in cases where Уall valuesФ would lead to reflexive fallacies; for the admission of Уany valueФ does not in the same way create new values. Hence the fundamental laws of logic can be stated concerning any proposition, though we can not significantly say that they hold of all propositions. These laws have, so to speak, a particular enunciation but no general enunciation. There is no one proposition which is the law of contradiction (say); there are only the various instances of the law. Of any proposition p, we can say: Уp and not-p can not both be true;Ф but there is no such proposition as: УEvery proposition p is such that p and not-p can not both be true.Ф
BF.A similar explanation applies to properties. We can speak of any property of x, but not of all properties, because new properties would be thereby generated. Thus we can say: УIf n is a finite integer, and if 0 has the property φ and m + 1 has the property φ provided m has it, it follows that n has the property φ.Ф Here we need not specify φ; φ stands for Уany property.Ф But we can not say : УA finite integer is defined as one which has every property φ possessed by 0 and by the successors of possessors.Ф For here it is essential to consider every property,11) not any property ; and in using such a definition we assume that it embodies a property distinctive of finite integers, which is just the kind of assumption from which, as we saw, the reflexive contradictions spring.
BG.In the above instance, it is necessary to avoid the suggestions of ordinary language, which is not suitable for expressing the distinction required. The point may be illustrated further as follows: If induction is to be used for defining finite integers, induction must state a definite property of finite integers, not an ambiguous property. But if φ is a real variable, the statement Уn has the property φ provided this property is possessed by 0 and by the successors of possessorsФ assigns to n a property which varies as φ varies, and such a property can not be used to define the class of finite integers. We wish to say: У Сn is a finite integerТ means: СWhatever property φ may be, n has the property φ provided φ is possessed by 0 and by the successors of possessors.Т Ф But here φ has become an apparent variable. To keep it a real variable, we should have to say: УWhatever property φ may be, Сn is a finite integerТ means: Сn has the property φ provided φ is possessed by 0 and by the successors of possessors,Т Ф But here the meaning of Сn is a finite integerТ varies as φ varies, and thus such a definition is impossible. This case illustrates an important point, namely the following: УThe scope12) of a real variable can never be less than the whole propositional function in the assertion of which the said variable occurs.Ф That is, if our propositional function is (say) Уφx implies p,Ф the assertion of this function will mean Уany value of Сφx implies pТ is trueФ not У Сany value of φx is trueТ implies p.Ф In the latter, we have really Уall values of φx are true,Ф and the x is an apparent variable.



8) These two terms are due to Peano, who uses them approximately in the above sense. Cf., e.g., Formulaire Mathematique, Vol. IV, p. 5 (Turin, 1903).  back to text8

9) Mr. MacColl speaks of "propositions" as divided into the three classes of certain, variable, and impossible. We may accept this division as applying to propositional functions. A function which can be asserted is certain, one which can be denied is impossible, and all others are (in Mr. MacColl's sense) variable.  back to text9

10) See his Grundgesetze der Arithmetik, Vol. I (Jena, 1893), І17, p. 31.  back to text10

11) This is indistinguishable from "all properties."  back to text11

12) The scope of a real variable is the whole function of which Уany valueФ is in question. Thus in Уψx implies pФ the scope of x is not ψx, but Уψx implies p  back to text12

I.    The Contradictions.
II.   All and Any
III.  The Meaning and Range of Generalized Propositions.
IV.   The Hierarchy of Types.
V.    The Axiom of Reducibility.
VI.   Primitive Ideas and Propositions of Symbolic Logic.
VII.  Elementary Theory of Classes and Relations.
VIII. Descriptive Functions.
IX.   Cardinal Numbers.
X.    Ordinal Numbers.
I.    ѕарадоксы.
II.   “¬се, каждый”   и   “любой, произвольный”
III. —мысл и область применимости обобще...

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