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Notes on Logic: Difference between revisions
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<div style="width: | <div style="width: 50%; margin: 0 auto;"><p>[[#nol-0|Preliminary]]<br/> | ||
[[#nol-1|I. Bi-polarity of Propositions. Sense and Meaning. Truth and Falsehood]]<br/> | [[#nol-1|I. Bi-polarity of Propositions. Sense and Meaning. Truth and Falsehood]]<br/> | ||
[[#nol-2|II. Analysis of Atomic Propositions, General Indefinables, Predicates, etc.]]<br/> | [[#nol-2|II. Analysis of Atomic Propositions, General Indefinables, Predicates, etc.]]<br/> | ||
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[[#nol-4|IV. Analysis of General Propositions]]<br/> | [[#nol-4|IV. Analysis of General Propositions]]<br/> | ||
[[#nol-5|V. Principles of Symbolism: What Symbolises in a Symbol. Facts for facts]]<br/> | [[#nol-5|V. Principles of Symbolism: What Symbolises in a Symbol. Facts for facts]]<br/> | ||
[[#nol-6|VI. Types]]<br/> | [[#nol-6|VI. Types]]<br/></p> | ||
</div> | </div> | ||
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One is tempted to interpret "not-p" as "everything else, only not p". That from a single fact p an infinity of others, not-not-p, etc., follow is hardly credible. [''See'' 5.43.] Man possesses an innate capacity for constructing symbols with which ''some'' sense can be expressed without having the slightest idea what each word signifies. [''Cf.'' 4.002.] The best example of this is mathematics, for man has until recently used the symbols for numbers without knowing what they signify or that they signify nothing. | One is tempted to interpret "not-p" as "everything else, only not p". That from a single fact p an infinity of others, not-not-p, etc., follow is hardly credible. [''See'' 5.43.] Man possesses an innate capacity for constructing symbols with which ''some'' sense can be expressed without having the slightest idea what each word signifies. [''Cf.'' 4.002.] The best example of this is mathematics, for man has until recently used the symbols for numbers without knowing what they signify or that they signify nothing. | ||
The assertion-sign is logically quite without significance. It only shows, in Frege and in Whitehead and Russell, that these authors hold t e propositions so indicated to be true. "<math>\vdash</math>", therefore, belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true. [''Cf.'' 4.442.] Assertion is merely psychological. There are only unasserted pro positions. Judgment, command and question all stand on the same level; but all have in common the propositional form, and that alone interests us. What interests logic are only the unasserted propositions. When we say A judges that, etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form but not in the proper order. This shows that a proposition itself must occur in the statement to the effect that it is judged. For instance, however "not-p" may be explained, the question | The assertion-sign is logically quite without significance. It only shows, in Frege and in Whitehead and Russell, that these authors hold t e propositions so indicated to be true. "<math>\vdash</math>", therefore, belongs as little to the proposition as (say) the number of the proposition. A proposition cannot possibly assert of itself that it is true. [''Cf.'' 4.442.] Assertion is merely psychological. There are only unasserted pro positions. Judgment, command and question all stand on the same level; but all have in common the propositional form, and that alone interests us. What interests logic are only the unasserted propositions. When we say A judges that, etc., then we have to mention a whole proposition which A judges. It will not do either to mention only its constituents, or its constituents and form but not in the proper order. This shows that a proposition itself must occur in the statement to the effect that it is judged. For instance, however "not-p" may be explained, the question "What is negated?" must have a meaning. In "A judges (that) p", p cannot be replaced by a proper name. This is apparent if we substitute "A judges that p is true and not-p is false". The proposition "A judges (that) p" consists of the proper name A, the proposition p with its two poles, and A's being related to both these poles in a certain way. This is obviously not a relation in the ordinary sense. Every right theory of judgment must make it impossible for me to judge that "this table penholders the book" (Russell's theory does not satisfy this requirement). [''Cf.'' 5.5422.] The structure of the proposition must be recognized and then the rest is easy. But ordinary language conceals the structure of the proposition: in it relations look like predicates, and predicates like names, etc. | ||
One reason for supposing that not all propositions which have more than one argument are relational propositions is that, if they were, the relations of judgment and inference would have to hold between an arbitrary number of things. The idea that propositions are names for complexes has suggested that whatever is not a proper name is a sign for a relation. Russell, for instance, imagines every fact as a spatial complex, and since spatial complexes consist of things and relations only, therefore he holds all do. | One reason for supposing that not all propositions which have more than one argument are relational propositions is that, if they were, the relations of judgment and inference would have to hold between an arbitrary number of things. The idea that propositions are names for complexes has suggested that whatever is not a proper name is a sign for a relation. Russell, for instance, imagines every fact as a spatial complex, and since spatial complexes consist of things and relations only, therefore he holds all do. | ||
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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-whichfollows-from-it. But there is a similarity of the two, expressible thus: φa . ⊃ | 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-whichfollows-from-it. But there is a similarity of the two, expressible thus: φa . ⊃<sub>φ,α</sub>.a = a. | ||
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. | ||