Logic / Unit 6 Logic 264 worked example, 60 to 90 seconds
Translation Practice
The idea
Translating into predicate notation means choosing the quantifier the sentence needs, applying the pairing rule that goes with it, then reading the symbols back into English as a check.
In practice
Six sentences. All dogs bark: universal quantifier, and universals pair with the horseshoe — for all x, Dx horseshoe Bx. No dogs fly: universal again, with the consequent negated — for all x, Dx horseshoe tilde Fx. Some dogs swim: existential, and existentials pair with the dot — there exists an x such that Dx dot Sx. Some dogs do not swim: existential with the second half negated — there exists an x such that Dx dot tilde Sx. Only members vote reverses the terms before anything else happens: for all x, Vx horseshoe Mx. No unregistered users may post: for all x, Ux horseshoe tilde Px, where U is being unregistered. Now read each one back. For all x, Dx dot Bx would say everything in the universe is a barking dog, which is not what the first sentence claimed — and the read-back is what catches that.
Where people slip
The pairing rule is not a stylistic preference. A universal joined by the dot asserts that everything whatever satisfies both predicates, and an existential joined by the horseshoe is satisfied by anything that fails the antecedent. Both mistranslations produce well-formed statements that say something the original sentence never said.
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