Logic

Let ‘B’= to be; let ‘~B’=not to be. P1: B v ~B P2: ~B C: ~B P2 is the negation of the left disjunct in P1, not the affirmation of the right disjunct in P1. P1: To be or not to be. P2: Not to be. C: Not to be. It seems to me that, argumentatively, there’s a difference between affirming ‘not to be’, the right disjunct, and negating ‘to be’, the left disjunct. It just happens that, in this case, what’s affirmed and what’s negated are logically equivalent. Is there a convention for conveying that argumentative difference? Also, can you recommend any articles or books where I can learn more about issues like this? Thank you very much :)

Comments

Interesting question! I think

Interesting question! I think you're right that there's something peculiar about this disjunctive syllogism:

(1) B v ~ B
(2) ~ B
(3) ~ B

You say that (2) must be the negation of (1)'s left disjunct rather than the assertion of (1)'s right disjunct, even though both of those are syntactically the same. You may find allies in those who distinguish between (i) denying or rejecting a proposition and (ii) asserting the proposition's negation. See Section 2.5 of this SEP entry.

But here's a different diagnosis. Although (1)-(3) is a valid argument, and even a valid instance of disjunctive syllogism, the argument is informally defective because premise (1) is superfluous: (1) isn't needed for the argument's validity. Furthermore, anyone justified in asserting (2) is thereby justified in asserting (3) without need of (1). This argument is similar:

(4) ~ B v B
(5) ~ ~ B
(6) B

The claim that (5) is the negation of (4)'s left disjunct is at least as plausible as the claim that (2) is the negation of (1)'s left disjunct. But maybe the better diagnosis is that (4)-(6) is informally defective because (4) is unnecessary for the argument's validity and unnecessary for rationally proceeding from (5) to (6).

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