Find appropriate predicates and their specification to translate the following into predicate logic: (a) All red things are in the box. (b) Only red things are in the box. (c) No animal is both a cat and a dog. (d) Every prize was won by a boy. (e) A boy won every prize.

We employ the predicates,

InBox(x):x is in the box

Red(x):x is red

Animal(x):x is an animal

Cat(x):x is a cat

Dog(x):x is a dog

Boy(x):x is a boy

Prize(x):x is a prize

Won(x, y):x won y

to standardise or formalize the above statement.

a. All red things are in the box.

∀x(Red(x)→InBox(x))

b. Only red things are in the box.

∀x(InBox(x)→Red(x))

c. No animal is both a cat and a dog.

¬∃x(Animal(x)∧(Cat(x)∧Dog(x)))

or∀x(Animal(x)→(¬Cat(x)∨ ¬Dog(x)))

d. Every prize was won by a boy.

∀x[Prize(x)→ ∃y(Boy(y)∧Won(y, x))]

e. A boy won every prize.

∃y[Boy(y)∧ ∀x(Prize(x)→Won(y, x))]

Predicate Logic as to do with predicates, it involves propositions that houses variables.

A predicate is an expression of a single or different variables expressed on certain domain. A predicate with variables can be taken as a proposition by placing a value to the variable or by estimating the variable using a quantifier.

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