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# sols3 - 3 SEPTEMBER 6TH MORE ABOUT LOGIC 5 3 September 6th...

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3. SEPTEMBER 6TH: MORE ABOUT LOGIC 5 3. September 6th: More about logic 3.1 . Translate into a logical expression the adage ‘All that glitters is not gold’, assuming that P ( x ) means x glitters’, and Q ( x ) means x is gold’. (Watch out! this is trickier than it looks. Make sure that your translation matches the meaning of the sentence, not just its apparent structure.) Answer: What ‘All that glitters is not gold’ means is that there are things that glitter as gold would, but they turn out to be of no value. The meaning involves an existential quantifier, so the translation is ( ) ( x ) : P ( x ) ¬ Q ( x ) . Note that taking the sentence too literally would lead to something like ( x ) , P ( x ) = ¬ Q ( x ), or a variation on the same theme. This says “ All the things that glitter turn out not to be made of gold”. A sensible statement, but not what the adage means. The construction used in this sentence is peculiar to English. The ‘not’ in ‘All that glitters is not gold’ does not negate ‘is’, as it would in an ordinary sentence. It negates the whole sentence: it means that ‘It is not true that all that glitters is gold’. You can take this as another way to translate it into logic: ( ) ¬ ( ( x ) : P ( x ) = Q ( x ) ) , another fine answer. If you want a follow-up exercise, check that ( ) and ( ) are

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