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Notes on Logic: Difference between revisions
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There is no ''thing'' which is the ''form'' of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment. | There is no ''thing'' which is the ''form'' of a proposition, and no name which is the name of a form. Accordingly we can also not say that a relation which in certain cases holds between things holds sometimes between forms and things. This goes against Russell's theory of judgment. | ||
Symbols are not what they seem to be. In "aRb" "R" looks like a substantive but it is not one. What symbolizes in "aRb" is that "R" occurs between "a" and "b". Hence "R" is ''not'' the indefinable in "aRb". Similarly in " | Symbols are not what they seem to be. In "aRb" "R" looks like a substantive but it is not one. What symbolizes in "aRb" is that "R" occurs between "a" and "b". Hence "R" is ''not'' the indefinable in "aRb". Similarly in "ϕx" "ϕ" looks like a substantive but is not one; in "~p", "~" looks like "ϕ" but is not like it. This is the first thing that indicates there ''may'' not be logical constants. A reason against them is the generality of logic: logic cannot treat a special set of things. | ||
Russell's "complexes" were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like "simples". But this alone makes them unserviceable as logical types (forms), since there would then have been significance in asserting, of a simple, that it was complex. But a ''property'' cannot be a logical type. | Russell's "complexes" were to have the useful property of being compounded, and were to combine with this the agreeable property that they could be treated like "simples". But this alone makes them unserviceable as logical types (forms), since there would then have been significance in asserting, of a simple, that it was complex. But a ''property'' cannot be a logical type. | ||
A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact-which-follows-from-it. But there is a similarity of the two, expressible thus: <span class="nowrap"> | A false theory of relations makes it easily seem as if the relation of fact and constituent were the same as that of fact and fact-which-follows-from-it. But there is a similarity of the two, expressible thus: <span class="nowrap">ϕa . ⊃<sub>ϕ,α</sub>.a = a.</span> | ||
Every statement about complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. [''Cf.'' 2.0201.] How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic. To repeat: every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e. that proposition which is equivalent to saying the complex exists. | Every statement about complexes can be resolved into the logical sum of a statement about the constituents and a statement about the proposition which describes the complex completely. [''Cf.'' 2.0201.] How, in each case, the resolution is to be made, is an important question, but its answer is not unconditionally necessary for the construction of logic. To repeat: every proposition which seems to be about a complex can be analysed into a proposition about its constituents and about the proposition which describes the complex perfectly; i.e. that proposition which is equivalent to saying the complex exists. | ||
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There are no propositions containing real variables. Those symbols which are called propositions in which "variables occur" are in reality not propositions at all, but only schemes of propositions, which do not become propositions unless we replace the variables by constants. There is no proposition which is expressed by "x = x", for "x" has no signification. But there is a proposition "(x).x = x", and propositions such as "Socrates = Socrates", etc. In books on logic no variables ought to occur, but only general propositions which justify the use of variables. It follows that the so-called definitions in logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions. And similarly, the so-called primitive ideas (''Urzeichen'') of logic are not primitive ideas but schemes of them. The mistaken idea that there are ''things'' called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which, e.g., expresses that q follows from p and p ⊃ q is this: p. p ⊃ q. ⊃<sub>p,q</sub>.q. | There are no propositions containing real variables. Those symbols which are called propositions in which "variables occur" are in reality not propositions at all, but only schemes of propositions, which do not become propositions unless we replace the variables by constants. There is no proposition which is expressed by "x = x", for "x" has no signification. But there is a proposition "(x).x = x", and propositions such as "Socrates = Socrates", etc. In books on logic no variables ought to occur, but only general propositions which justify the use of variables. It follows that the so-called definitions in logic are not definitions, but only schemes of definitions, and instead of these we ought to put general propositions. And similarly, the so-called primitive ideas (''Urzeichen'') of logic are not primitive ideas but schemes of them. The mistaken idea that there are ''things'' called facts or complexes and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from arbitrary cases. It is a fact that the proposition which, e.g., expresses that q follows from p and p ⊃ q is this: p. p ⊃ q. ⊃<sub>p,q</sub>.q. | ||
Cross-definability in the realm of general propositions leads to quite similar questions to those in the realm of ab-functions. There is the same objection in the case of apparent variables to the usual indefinables as in the case of molecular functions. The application of the ab notation to apparent variable propositions becomes clear if we consider that, for instance, the proposition "for all x, | Cross-definability in the realm of general propositions leads to quite similar questions to those in the realm of ab-functions. There is the same objection in the case of apparent variables to the usual indefinables as in the case of molecular functions. The application of the ab notation to apparent variable propositions becomes clear if we consider that, for instance, the proposition "for all x, ϕx" is to be true when ϕx is true for all x's, and false when ϕx is false for some x's. We see that ''some'' and ''all'' occur simultaneously in the proper apparent variable notation. The notation is | ||
For (x) | For (x)ϕx: a-(x)-.a ϕxb.-(∃x)-b and | ||
for (∃x) | for (∃x)ϕx: a-(∃x)-.a ϕxb.-(x)-b | ||
Old definitions now become tautologous. | Old definitions now become tautologous. | ||
A very natural objection to the way in which I have introduced, e.g., propositions of the form xRy is that by it propositions such as (∃x,y)xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. ''But'' when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y) | A very natural objection to the way in which I have introduced, e.g., propositions of the form xRy is that by it propositions such as (∃x,y)xRy and similar ones are not explained, which yet obviously have in common with aRb what cRd has in common with aRb. ''But'' when we introduced propositions of the form xRy we mentioned no one particular proposition of this form; and we only need to introduce (x,y)ϕ(x,y) for all ϕ's in any way which makes the sense of these propositions dependent on the sense of all propositions of the form ϕ(a,b), and thereby the justification of our procedure is established. | ||
<p style="text-align: center;" id="nol-5">'''V. Principles of Symbolism: What Symbolises in a Symbol. Facts for Facts'''</p> | <p style="text-align: center;" id="nol-5">'''V. Principles of Symbolism: What Symbolises in a Symbol. Facts for Facts'''</p> | ||
It is easy to suppose only such symbols are complex as contain names of objects, and that accordingly "(x, | It is easy to suppose only such symbols are complex as contain names of objects, and that accordingly "(x,ϕ)ϕx" or "(∃x,y)xRy" must be simple. It is then natural to call the first of these the name of a form, the second the name of a relation. But in that case what ''is'' the meaning, e.g., of "~(∃x,y).xRy"? Can we put "not" before a name? Alternate indefinability shows the indefinables have not yet been reached. [''Cf.'' 5.42.] The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be introduced in all combinations in which it can occur. We cannot, therefore, introduce it first for one combination, then for another; e.g. if the form xRy has been introduced, it must henceforth be understood in propositions of the form aRb just in the same way as in propositions such as (∃x,y)xRy and others. We must not introduce it first for one class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases and there could be no ground for using the same manner of combining symbols in both cases. In short, for the introduction of indefinable symbols and combinations of symbols the same holds, ''mutatis mutandis'', that Frege has said for the introduction of symbols by definitions. [''Cf.'' 5.451.] | ||
It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace | It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace ϕ(a,a) by ϕ(a,b) . a = b. | ||
It can never express the common characteristic of two objects that we designate them by the same name but otherwise by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where, then, would be the common element in the designations? [''Cf.'' 3.322.] Nevertheless, one is always tempted, in a difficulty, to take refuge in different ways of designation. | It can never express the common characteristic of two objects that we designate them by the same name but otherwise by two different ways of designation, for, since names are arbitrary, we might also choose different names, and where, then, would be the common element in the designations? [''Cf.'' 3.322.] Nevertheless, one is always tempted, in a difficulty, to take refuge in different ways of designation. | ||
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Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, etc. An indefinable symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition. | Every proposition which says something indefinable about a thing is a subject-predicate proposition; every proposition which says something indefinable about two things expresses a dual relation between these things, and so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate proposition, etc. An indefinable symbol can only be a name, and therefore we can know, by the symbol of an atomic proposition, whether it is a subject-predicate proposition. | ||
A proposition cannot occur in itself. This is the fundamental truth of the theory of types. [''Cf.'' 3.332.] In a proposition convert all indefinables into variables, there then remains a class of propositions which does not include all propositions, but does include an entire type. If we change a constituent a of a proposition | A proposition cannot occur in itself. This is the fundamental truth of the theory of types. [''Cf.'' 3.332.] In a proposition convert all indefinables into variables, there then remains a class of propositions which does not include all propositions, but does include an entire type. If we change a constituent a of a proposition ϕ(a) into a variable, then there is a class <math>\hat{p}[( \exists x ) . \phi x = p]</math>. This class, in general, still depends upon what, by an ''arbitrary convention'', we mean by "ϕx". But if we change into variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is not now dependent upon any convention, but only upon the nature of the symbol "ϕx". It corresponds to a logical type. [''Cf.'' 3.315.] | ||
There are two ways in which signs are similar. The names "Socrates" and "Plato" are similar: they are both names. But whatever they have in common must not be introduced before "Socrates" and "Plato" are introduced. The same applies to a subject-predicate form, etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e. types are not indefinables. | There are two ways in which signs are similar. The names "Socrates" and "Plato" are similar: they are both names. But whatever they have in common must not be introduced before "Socrates" and "Plato" are introduced. The same applies to a subject-predicate form, etc. Therefore, thing, proposition, subject-predicate form, etc., are not indefinables, i.e. types are not indefinables. | ||
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Every proposition that says something indefinable about one thing is a subject-predicate proposition, etc. Therefore, we can recognize a subject-predicate proposition, if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone. | Every proposition that says something indefinable about one thing is a subject-predicate proposition, etc. Therefore, we can recognize a subject-predicate proposition, if we know it contains only one name and one form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol alone. | ||
What is essential in a correct apparent-variable notation is this: (1) it must mention a type of proposition, (2) it must show which components (forms and constituents) of a proposition of this type are constants. Take ( | What is essential in a correct apparent-variable notation is this: (1) it must mention a type of proposition, (2) it must show which components (forms and constituents) of a proposition of this type are constants. Take (ϕ).ϕ!x. Then, if we describe the ''kind'' of symbols for which ϕ stands, the which, by the above, is enough to determine the type, then automatically "(ϕ).ϕ!x" cannot be fitted by this description, because it ''contains'' "ϕ!x" and the description is to describe ''all'' that symbolizes in symbols of the ϕ!x kind. If the description is ''thus'' completed, vicious circles can just as little occur as can for instance (ϕ).(x)ϕ where (x)ϕ is a subject-predicate proposition. | ||
We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other. Types can never be distinguished from each other by saying (as is currently done) that one has these ''but'' the other has those properties, for this presupposes that there is a ''meaning'' in asserting all these properties of both types. [''Cf.'' 4.1241.] And, from this it follows that, at least, these properties may be types, but certainly not the objects of which they are asserted. | We can never distinguish one logical type from another by attributing a property to members of the one which we deny to members of the other. Types can never be distinguished from each other by saying (as is currently done) that one has these ''but'' the other has those properties, for this presupposes that there is a ''meaning'' in asserting all these properties of both types. [''Cf.'' 4.1241.] And, from this it follows that, at least, these properties may be types, but certainly not the objects of which they are asserted. | ||