The most basic formula in Predicate Calculus is of the Form P. It is a true-false statement represented by an uppercase letter. Alternatively, P can have n terms written Pp1,...,pn. An infinite list of atomic formula can be generated and given the rules for constructing wff's the set of all meaningful formulae in Predicate Calculus can be constructed. However, not all wff's are of interest. Examine the following.
Px, Qx, Lxy ∀x(Px), ∃x(Px), ∀x(Px ⇒ Qx), ∀x[∃y(Lxy)]
To begin to understand what these formulas mean, first define the set of objects the terms in each formula can refer to. I like mathematics, so, I use numbers. I call this set the Universe.
Use the following interpretation:
The Universe: The natural numbers
Px: x is prime.
Qx: x is rational.
Lxy: x is less than y.
Thus ∀x(Px) means that everything is prime. Clearly this is a false statement. There exists at least one composite number. ∃x(Px) means a prime number exists. This is a true statement. ∀x(Px ⇒ Qx) means that any object in the universe is a rational if it is prime. This too is a true statement. ∀x[∃y(Lxy)] is perhaps the most interesting. It says there is no largest number. This is true because one can always find a number y such that for any choice of x, x is less than y. In this context the universe is infinite but there is no reason why we can't restrict our universe. Or change the universe to say the integers or maybe even go as far as the real numbers.
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