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Firstorder predicate calculus or firstorder logic is a theory in symbolic logic that formalizes quantified statements such as "there exists an object such that..." or "for all objects, it is the case that...".
Firstorder logic is distinguished from higherorder logic[?] in that it does not allow statements such as "for every property, it is the case that..." or "there exists a set of objects such that...".
Nevertheless, firstorder logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. It is the classical logical theory underlying mathematics. It is a stronger theory than sentential logic[?], but a weaker theory than arithmetic, set theory, or secondorder logic[?].
Like any logical theory, firstorder calculus consists of
There are two types of axioms: the logical axioms which embody the general truths about proper reasoning involving quantified statements, and the axioms describing the subject matter at hand, for instance axioms describing sets in set theory or axioms describing numbers in arithmetic.
While the set of inference rules in firstorder calculus is finite, the set of axioms may very well be and often is infinite. However we require that there is a general algorithm which can decide for a given wellformed formula whether it is an axiom or not. Furthermore, there should be an algorithm which can decide whether a given application of an inference rule is correct or not.
The wellformed formulas contain
The object, predicate and function constants will typically depend on the particular domain we are talking about.
Instead of giving the lengthy definition of wellformed formulas, we will simply look at some examples from arithmetic together with their ordinary interpretation. Our domain here is the set of natural numbers:
There exists a number y which is bigger than every number x. That's not true.
If a number x is divisible by 6, then it is also divisible by 3. True.
There exists a number x such that there doesn't exist a smaller number y. True (take x=0).
Now one would have to write down 15 logical axioms and 2 inference rules to completely specify the firstorder calculus. How can one be sure that those axioms and rules are enough? This is the subject of Gödel's completeness theorem: if you start out with some subject matter axioms and you look at a certain statement, then it is possible to prove that statement using only the subject matter axioms, the 15 logical axioms and the two inference rules if and only if the statement is true in every domain in which the subject matter axioms are true.
The Peano axioms for the natural numbers are sometimes formulated as secondorder statements (the induction axiom talks about all "properties" or all "sets of numbers"), but this is not necessary if one is willing to allow infinitely many firstorder axioms. A firstorder version of the Peano axioms follows.
We are using the object constants 0 and 1, the function constants + and *, and the predicate constant =. Here are the axioms:
Axioms 1, 2 and 7 are the traditional Peano axioms while axioms 36 serve to define addition and multiplication. If one omits the function constant * and axioms 5 and 6 and allows in scheme 7 only formulas without *, then one gets the very interesting Presburger arithmetic.
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