translations with explanations

# translations with explanations - More on Translations N...

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More on Translations N. Apostol Key: D: It's a dog C: It's a cat. M: It's a mammal. I. NEITHER this NOR that / NOT this AND NOT that ('this', 'that' means anything what-so-ever, no matter whether simple or complex, no matter what the topic) (a) It's neither a cat nor a dog. same as: (b) It's NOT a cat AND NOT a dog. Major operator/connective is a conjunction (between two negated atomic statements) ~ C & ~ D (c) It's NOT a cat OR a dog. ~ ( C v D ). Major connective is a negation ( of a disjunction between two atomic statements). All three statements above are logically equivalent: they propose/reveal the same content (information), they propose the same meaning, they communicate the same information. In other words, the same information/meaning/content can be read off apparently different logical forms when the atomic statements (C, D in this case) are given meaning/content (when they spell out something in English). Contructing the truth table for each of the above statements results in the two statements agreeing in truth-values in each line (identical columns under their respective major connective). Moreover, creating a bi-conditional between the two statements, the bi-conditional is a tautology. RULE of Bi-Conditional/Rule of replacement/Rule of logical equivalence : Two statements are logically equivalent IF AND ONLY IF their biconditional is tautologous. So, whenever this happens

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