Logic / Unit 6 Logic 266 worked example, 60 to 90 seconds
Applying Quantifier Negation
The idea
Quantifier negation drives a negation sign inward past a quantifier, flipping the quantifier as it passes, after which the propositional replacement rules finish the job inside.
In practice
Start with the denial of a universal: tilde, for all x, open bracket, Fx horseshoe Gx, close bracket. Read it aloud first — not every F is a G. Step one: move the negation past the quantifier, which flips the universal to an existential and leaves the negation attached to what follows. There exists an x such that tilde, open bracket, Fx horseshoe Gx, close bracket. Step two: the negation now sits on a conditional, so implication rewrites that conditional as a disjunction — tilde, open bracket, tilde Fx wedge Gx, close bracket. Step three: De Morgan takes the negation inside, negating both parts and turning the wedge into a dot — tilde tilde Fx dot tilde Gx. Step four: double negation clears the front. There exists an x such that Fx dot tilde Gx. Read it back: some F is not a G. The whole stack worked as one machine.
Where people slip
The quantifier flips only when the negation actually crosses it. A negation already inside the brackets governs the predicate rather than the quantifier, so flipping there changes the claim rather than rewriting it, and the resulting line no longer follows from the one above it.
Your place on the trail
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