Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/31

not asserted. In ordinary written language a sentence contained between full stops denotes an asserted proposition, and if it is false the book is in error. The sign "$\scriptstyle {\vdash }$" prefixed to a proposition serves this same purpose in our symbolism. For example, if "$\scriptstyle {\vdash (p\supset p)}$" occurs, it is to be taken as a complete assertion convicting the authors of error unless the proposition "$\scriptstyle {p\supset p}$" is true (as it is). Also a proposition stated in symbols without this sign "$\scriptstyle {\vdash }$" prefixed is not asserted, and is merely put forward for consideration, or as a subordinate part of an asserted proposition.

Inference. The process of inference is as follows: a proposition "$\scriptstyle {p}$" is asserted, and a proposition "$\scriptstyle {p}$ implies $\scriptstyle {q}$" is asserted, and then as a sequel the proposition "$\scriptstyle {q}$" is asserted. The trust in inference is the belief that if the two former assertions are not in error, the final assertion is not in error. Accordingly whenever, in symbols, where $\scriptstyle {p}$ and $\scriptstyle {q}$ have of course special determinations,

"$\scriptstyle {\vdash p}$" and "$\scriptstyle {\vdash (p\supset q)}$"

have occurred, then "$\scriptstyle {\vdash q}$" will occur if it is desired to put it on record. The process of the inference cannot be reduced to symbols. Its sole record is the occurrence of "$\scriptstyle {\vdash q}$." It is of course convenient, even at the risk of repetition, to write "$\scriptstyle {\vdash p}$" and "$\scriptstyle {\vdash (p\supset q)}$" in close juxtaposition before proceeding to "$\scriptstyle {\vdash q}$" as the result of an inference. When this is to be done, for the sake of drawing attention to the inference which is being made, we shall write instead

"$\scriptstyle {\vdash p\supset \vdash q}$,"

which is to be considered as a mere abbreviation of the threefold statement

"$\scriptstyle {\vdash p}$" and $\scriptstyle {\vdash (p\supset q)}$" and "$\scriptstyle {\vdash q}$."

Thus "$\scriptstyle {\vdash p\supset \vdash q}$" may be read "$\scriptstyle {p}$, therefore $\scriptstyle {q}$," being in fact the same abbreviation, essentially, as this is; for "$\scriptstyle {p}$, therefore $\scriptstyle {q}$" does not explicitly state, what is part of its meaning, that $\scriptstyle {p}$ implies $\scriptstyle {q}$. An inference is the dropping of a true premiss; it is the dissolution of an implication.

The use of dots. Dots on the line of the symbols have two uses, one to bracket off propositions, the other to indicate the logical product of two propositions. Dots immediately preceded or followed by "$\scriptstyle {\lor }$" or "$\scriptstyle {\supset }$" or "$\scriptstyle {\equiv }$" or "$\scriptstyle {\vdash }$," or by "$\scriptstyle {(x)}$," "$\scriptstyle {(x,y)}$," "$\scriptstyle {(x,y,z)}$"…or "$\scriptstyle {(\exists x)}$," "$\scriptstyle {(\exists x,y)}$," "$\scriptstyle {(\exists x,y,z)}$"…or "$\scriptstyle {\left[(\iota x)(\phi x)\right]}$" or "$\scriptstyle {\left[R^{\prime }y\right]}$" or analogous expressions, serve to bracket off a proposition; dots occurring otherwise serve to mark a logical product. The general principle is that a larger number of dots indicates an outside bracket, a smaller number indicates an inside bracket. The exact rule as to the scope of the bracket indicated by dots is arrived at by dividing the occurrences of dots into three groups which we will name I, II, and III. Group I consists of dots adjoining a sign of implication ($\scriptstyle {\supset }$) or of equivalence ($\scriptstyle {\equiv }$) or of disjunction ($\scriptstyle {\lor }$) or of equality by definition ($\scriptstyle {={\text{Df}}}$). Group II consists of dots following brackets indicative of an apparent variable, such as $\scriptstyle {(x)}$ or $\scriptstyle {(x,y)}$ or ($\scriptstyle {\exists x}$) or