# Pure logical argument weakens claims

By dkl9, written 2023-253, revised 2023-253 (0 revisions)

In formal logic, the law of addition says that, from true proposition P, you can correctly conclude PQ, for any propositions P and Q.

Some people don't like this. It invites annoying conclusions like "2 + 2 = 4, and/or grass is purple".

When trying to make this annoyance precise, one points out that the law of addition weakens your claims, i.e. PQ is weaker (less strict) than just P. This is true, but it's not unique to the law of addition. Every fundamental logical law produces conclusions at most as strong as its inputs, and often weaker.

• Modus ponens requires PQ (i.e. (P, Q) is one of (T, T), (F, T), (F, F)) and P, which intersect to be logically equivalent to the very strict PQ — then it gives you the weak conclusion Q.
• Conjunctive simplification takes a strict PQ to a weaker P or Q (either one individually).
• Syllogism takes two independent implications (PQ and QR), which are each true in 3/4 of logical worlds, and thus true together in 9/16 of logical worlds, and gives you a single implication PR, only as strong as one of the inputs.
• And on and on.

Any combination of logical laws concludes at most as precisely as its inputs, and usually less. Logical work just helps to interpret the set of worlds your observations accept as options. If pure logic could rule out possible worlds, you might rule out the one true world and not know it, thereby missing the point of valid argument.

Logic is still useful sith a weak claim focused on what we care about (like the mortality of Socrates) is often more useful than a more precise claim with confusingly irrelevant details (like the humanity of Socrates and the mortality of humans).

Applying modus ponens, conjunctive simplification, or syllogism doesn't get rid of the strong input claims P, PQ, or PQ, respectively — they don't diminish your total knowledge — but addition just as well keeps around the original P, and to ignore that is to judge logical laws by different standards.

I do still hate the law of addition, but for a better reason: the main use of a logical-OR assertion like PQ is to get to a conclusion about P or Q individually, but in these cases, you already have a statement about P on its own (P is true), and making that new conclusion won't help you with Q.