Peirce’s 1903 Lowell Lectures • Comment 7

Cf: Laws Of Form Discussion • JA
Re: Peirce List Discussion • JA

I’ll go ahead and copy out the first part of the article on Logical Implication, as I find I am still pleased with all I was able to say in such a short space.

Logical Implication

The concept of logical implication encompasses a specific logical function, a specific logical relation, and the various symbols that are used to denote this function and this relation. In order to define the specific function, relation, and symbols in question it is first necessary to establish a few ideas about the connections among them.

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

$\begin{array}{l} p ~\text{implies}~ q. \\[6pt] \text{if}~ p ~\text{then}~ q. \end{array}$

Here ${}^{\backprime\backprime} p {}^{\prime\prime}$ and ${}^{\backprime\backprime} q {}^{\prime\prime}$ are propositional variables that stand for any propositions in a given language. In a statement of the form ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},$ the first term, $p,$ is called the antecedent and the second term, $q,$ is called the consequent, while the statement as a whole is called either the conditional or the consequence. Assuming that the conditional statement is true, then the truth of the antecedent is a sufficient condition for the truth of the consequent, while the truth of the consequent is a necessary condition for the truth of the antecedent.

Note. Many writers draw a technical distinction between the form ${}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}$ and the form ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.$ In this usage, writing ${}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}$ asserts the existence of a certain relation between the logical value of $p$ and the logical value of $q,$ whereas writing ${}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}$ merely forms a compound statement whose logical value is a function of the logical values of $p$ and $q.$ This will be discussed in detail below.

Reference

Logical Implication • InterSciWiki • Wikiversity

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