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Notes on Logic: Difference between revisions
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Every proposition is essentially true-false. Thus a proposition has two poles (corresponding to case of its truth and case of its falsity). We call this the ''sense'' of a proposition. The ''meaning'' of a proposition is the fact which actually corresponds to it. The chief characteristic of my theory is: ''p has the same meaning as not-p'' (constituent = particular, component = particular or relation, etc.). [''Cf.'' 4.0621.] | Every proposition is essentially true-false. Thus a proposition has two poles (corresponding to case of its truth and case of its falsity). We call this the ''sense'' of a proposition. The ''meaning'' of a proposition is the fact which actually corresponds to it. The chief characteristic of my theory is: ''p has the same meaning as not-p'' (constituent = particular, component = particular or relation, etc.). [''Cf.'' 4.0621.] | ||
Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols. It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition. To understand a proposition p it is not enough to know that p implies "p is true", but we must also know that ~p implies "p is false". This shows the bi-polarity of the proposition. We understand a proposition when we understand its constituents and forms. [''Cf.'' 4.024.] If we know the meaning of "a" and "b" and if we know what " | Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols. It is clear that we understand propositions without knowing whether they are true or false. But we can only know the meaning of a proposition when we know if it is true or false. What we understand is the sense of the proposition. To understand a proposition p it is not enough to know that p implies "p is true", but we must also know that ~p implies "p is false". This shows the bi-polarity of the proposition. We understand a proposition when we understand its constituents and forms. [''Cf.'' 4.024.] If we know the meaning of "a" and "b" and if we know what "<span class="nowrap">x R y</span>" means for all x's and y's, then we also understand "<span class="nowrap">a R b</span>". I understand the proposition "<span class="nowrap">a R b</span>" when I know that either the fact that <span class="nowrap">a R b</span> or the fact that not <span class="nowrap">a R b</span> corresponds to it; but this is not to be confused with the false opinion that I understand "<span class="nowrap">a R b</span>" when I know that "<span class="nowrap">a R b</span> or not <span class="nowrap">a R b</span>" is the case. | ||
Strictly speaking, it is incorrect to say we understand the proposition p when we know that "p is true" ≡ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol ≡ were either both true or both false. We require not only an equivalence but a formal equivalence, which is bound up with the introduction of the form of p. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e. all the general indefinables involved. | Strictly speaking, it is incorrect to say we understand the proposition p when we know that "p is true" ≡ p; for this would naturally always be the case if accidentally the propositions to right and left of the symbol ≡ were either both true or both false. We require not only an equivalence but a formal equivalence, which is bound up with the introduction of the form of p. What is wanted is the formal equivalence with respect to the forms of the proposition, i.e. all the general indefinables involved. | ||
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Indefinables are of two sorts: names and forms. Propositions cannot consist of names alone, they cannot be classes of names. [''Cf '' 3.142.] A name cannot only occur in two different propositions, but can occur in the same way in both. Propositions, which are symbols having reference to facts, are themselves facts (that this inkpot is on this table may express that I sit in this chair). We must be able to understand propositions we have never heard before. But every proposition is a new symbol. Hence we must have ''general'' indefinable symbols; these are unavoidable if propositions are not all indefinable. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads us to an impenetrable thicket. | Indefinables are of two sorts: names and forms. Propositions cannot consist of names alone, they cannot be classes of names. [''Cf '' 3.142.] A name cannot only occur in two different propositions, but can occur in the same way in both. Propositions, which are symbols having reference to facts, are themselves facts (that this inkpot is on this table may express that I sit in this chair). We must be able to understand propositions we have never heard before. But every proposition is a new symbol. Hence we must have ''general'' indefinable symbols; these are unavoidable if propositions are not all indefinable. Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this doctrine leads us to an impenetrable thicket. | ||
A proposition must be understood when ''all'' its indefinables are understood. The indefinables in " | A proposition must be understood when ''all'' its indefinables are understood. The indefinables in "<span class="nowrap">a R b</span>" are introduced as follows: (1) "a" is indefinable, (2) "b" is indefinable, (3) whatever "x" and "y" may mean, "<span class="nowrap">x R y</span>" says something indefinable about their meaning. | ||
We are not concerned in logic with the relation of any specific name to its meaning and just as little with the relation of a given proposition to reality. We do want to know that our names have meanings and propositions sense, and we thus introduce an indefinable concept "A" by saying "'A' denotes something indefinable", or the form of propositions | We are not concerned in logic with the relation of any specific name to its meaning and just as little with the relation of a given proposition to reality. We do want to know that our names have meanings and propositions sense, and we thus introduce an indefinable concept "A" by saying "'A' denotes something indefinable", or the form of propositions <span class="nowrap">a R b</span> by saying: "For all meanings of 'x' and 'y', '<span class="nowrap">x R y</span>' expresses something indefinable about x and y." | ||
The form of a proposition may be symbolized in the following way: Let us consider symbols of the form " | The form of a proposition may be symbolized in the following way: Let us consider symbols of the form "<span class="nowrap">x R y</span>", to which correspond primarily pairs of objects of which one has the name "x", the other the name "y". The x's and y's stand in various relations to each other, and among other relations the relation R holds between some but not between others. I now determine the sense of "<span class="nowrap">x R y</span>" by laying down the rule: when the facts behave in regard to "<span class="nowrap">x R y</span>" so that the meaning of "x" stands in the relation R to the meaning of "y", then I say that these facts are "of like sense" (''gleichsinnig'') with the proposition "<span class="nowrap">x R y</span>"; otherwise, "of opposite sense" (''entgegengesetzt''). I correlate the facts to the symbol "<span class="nowrap">x R y</span>" by thus dividing them into those of like sense and those of opposite sense. To this correlation corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form "<span class="nowrap">x R y</span>" when I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I extract from all possible relations the relation R, as by a name, I extract its meaning from among all possible things. | ||
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 " | Symbols are not what they seem to be. In "<span class="nowrap">a R b</span>" "R" looks like a substantive but it is not one. What symbolizes in "<span class="nowrap">a R b</span>" is that "R" occurs between "a" and "b". Hence "R" is ''not'' the indefinable in "<span class="nowrap">a R b</span>". 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. | ||
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If p = not-not-p, etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that in analysing such propositions, we must not be guided by Russell's method of symbolizing. | If p = not-not-p, etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality of symbols with the same sense; and thence it follows that in analysing such propositions, we must not be guided by Russell's method of symbolizing. | ||
Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then "p or not-p" has no meaning because it does not divide the plane. But though a particular proposition, "p or not-p", has no meaning, a general proposition, "For all p's, p or not-p", has a meaning, because this does not contain the nonsensical function "p or not-p", but the function "p or not-q", just as "for all x's, | Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then "p or not-p" has no meaning because it does not divide the plane. But though a particular proposition, "p or not-p", has no meaning, a general proposition, "For all p's, p or not-p", has a meaning, because this does not contain the nonsensical function "p or not-p", but the function "p or not-q", just as "for all x's, <span class="nowrap">x R x</span>" contains the function "<span class="nowrap">x R y</span>". | ||
Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the "manner of deduction" is indifferent. [''Cf.'' 5.132.] | Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also be inferred from q, and the "manner of deduction" is indifferent. [''Cf.'' 5.132.] | ||
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If only those signs which contain proper names are complex, then propositions containing nothing but apparent variables would be simple. Then what about their denials? Propositions are always complex, even if they contain no names. | If only those signs which contain proper names are complex, then propositions containing nothing but apparent variables would be simple. Then what about their denials? Propositions are always complex, even if they contain no names. | ||
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 <span class="nowrap">"(x) . x = x"</span>, 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: <span class="nowrap">p . p ⊃ q . ⊃<sub>p,q</sub> . q.</span> | ||
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 | 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)ϕx: a-(x)-.a ϕxb.-(∃x)-b and | :For (x)ϕx: a-(x)-.a ϕxb.-(∃x)-b and | ||
for (∃x)ϕx: a-(∃x)-.a ϕxb.-(x)-b | :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 | A very natural objection to the way in which I have introduced, e.g., propositions of the form <span class="nowrap">x R y</span> is that by it propositions such as <span class="nowrap">(∃ x, y) x R y</span> and similar ones are not explained, which yet obviously have in common with <span class="nowrap">a R b</span> what cRd has in common with <span class="nowrap">a R b</span>. ''But'' when we introduced propositions of the form <span class="nowrap">x R y</span> 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,ϕ)ϕx" or "( | It is easy to suppose only such symbols are complex as contain names of objects, and that accordingly "(x,ϕ)ϕx" or "<span class="nowrap">(∃ x, y)x R y</span>" 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 "<span class="nowrap">~(∃ x, y).x R y</span>"? 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 <span class="nowrap">x R y</span> has been introduced, it must henceforth be understood in propositions of the form <span class="nowrap">a R b</span> just in the same way as in propositions such as (∃x,y)<span class="nowrap">x R y</span> 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 ϕ(a,a) by ϕ(a,b) . a = b. | It is impossible to dispense with propositions in which the same argument occurs in different positions. It is obviously useless to replace <span class="nowrap">ϕ(a, a) by ϕ(a, b) . a = b</span>. | ||
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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In regard to notation it is important to observe that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F scheme, what symbolizes must be the same. In "not-not-p", "not-p" does not occur; for "not-not-p" is the same as "p", and therefore, if "not-p" occurred in "not-not-p", it would occur in "p". | In regard to notation it is important to observe that not every feature of a symbol symbolizes. In two molecular functions which have the same T-F scheme, what symbolizes must be the same. In "not-not-p", "not-p" does not occur; for "not-not-p" is the same as "p", and therefore, if "not-p" occurred in "not-not-p", it would occur in "p". | ||
A complex symbol must never be introduced as a single indefinable. Thus, for instance, no proposition is indefinable. For if one of the parts of the complex symbol occurs also in another connection, it must there be reintroduced. And would it then mean the same? The ways in which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables ''alone.'' It is easy to introduce "all" and "some" in a way that will make the construction of (say) "(x,y). | A complex symbol must never be introduced as a single indefinable. Thus, for instance, no proposition is indefinable. For if one of the parts of the complex symbol occurs also in another connection, it must there be reintroduced. And would it then mean the same? The ways in which we introduce our indefinables must permit us to construct all propositions that have sense from these indefinables ''alone.'' It is easy to introduce "all" and "some" in a way that will make the construction of (say) "(x,y).<span class="nowrap">x R y</span>" possible from "all" and "<span class="nowrap">x R y</span>" ''as introduced before.'' | ||
One must not say "The complex sign ' | One must not say "The complex sign '<span class="nowrap">a R b</span>'" says that a stands in the relation R to b; but that "a" stands in a certain relation to "b" says ''that'' <span class="nowrap">a R b</span>. [''Cf.'' 3.1432.] | ||
Only facts can express sense, a class of names cannot. [''Cf.'' 3.142.] This is easily shown. In | Only facts can express sense, a class of names cannot. [''Cf.'' 3.142.] This is easily shown. In <span class="nowrap">a R b</span> it is not the complex that symbolizes but the fact that the symbol a stands in a certain relation to the symbol b. Thus facts are symbolized by facts, or more correctly: that a certain thing is the case in the symbol says that a certain thing is the case in the world. | ||