Logic and Language: Translating Sentences into Logical Expressions

Translating the sentence 'Everybody can fool Fred' into a logical expression, we can use the universal quantifier to indicate the statement applies to everyone. The proper translation is: a) ∀x(CanFool(x,Fred)), which reads as 'For all x, x can fool Fred'. This statement is considered a universal affirmative statement. If we wanted to prove this wrong, a single counterexample where someone cannot fool Fred would suffice because it is a strong categorical claim that applies to all members of a particular set here represented by 'x'.

Conditional statements are frequently used to express logical relations, such as necessary and sufficient conditions. They are often in the form of if-then statements but can also be expressed universally. For example, the statement 'All dogs are mammals' is logically equivalent to 'If something is a dog, then it is a mammal'. As such, when rephrasing ordinary statements into logical expressions, identifying these conditions is essential.

It is worth noting that a statement and its negation are opposites, where the law of noncontradiction and the law of the excluded middle play significant roles. For instance, 'Fred cannot be fooled by anyone' would be the negation of our original statement and can be expressed as ¬∃x(CanFool(x,Fred)).