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First-order predicate calculus
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First-order predicate calculus

First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as "there is at least one X such that..." or "for any X, it is the case that...", where X is an element of a set called the domain of discourse.

First-order logic is distinguished from higher-order 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, first-order logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. Its restriction to quantification over individuals makes it difficult to use for the purposes of topology, but 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 Second-order logic.

Like any logical theory, first-order 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 first-order 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 well-formed 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.

Table of contents
1 Well-formed formulas of first-order logic
2 Examples of well-formed formulas
3 Case study: first-order Peano axioms
4 References

Well-formed formulas of first-order logic

The well-formed formulas contain: Note that only , , and are needed for a complete set of logical connectives.

The object, predicate and function constants will typically depend on the particular domain we are talking about.

Examples of well-formed formulas

Instead of giving the lengthy definition of well-formed formulas, we will simply look at some examples from arithmetic together with their ordinary interpretation. Our domain here is the set of natural numbers:

(i.e.: forAll x, thereExists y suchThat y isGreaterThan x )
For every number x there exists a bigger number y. That's true.

(i.e.: thereExists y suchThat forAll x, y isGreaterThan x )

There exists a number y which is bigger than every number x. That's not true.

(i.e.: forAll x, ( thereExists y suchThat 6*y=x) implies (thereExists y suchThat 3*y=x) )

If a number x is divisible by 6, then it is also divisible by 3. True.

(i.e.: thereExists x suchThat (not ThereExists y suchThat y < x ) )

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 first-order 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. (See also Leon Henkin)

Case study: first-order Peano axioms

The Peano axioms for the natural numbers are sometimes formulated as second-order 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 first-order axioms. A first-order 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:

    • i.e.: forAll x, not (0 = x + 1)
    • i.e.: no number has 0 as its successor
    • i.e.: forAll x, forAll y, not(x=y) implies not(x + 1 = y + 1)
    • i.e.: if x ≠ y, then x+1 ≠ y+1
    • i.e.: forAll x, x + 0 = x
    • i.e.: forAll x, forAll y, (x + y) + 1 = x + (y + 1)
    • i.e.: for all x and y, (x + y) + 1 = x + (y + 1)
    • i.e.: forAll x, x * 0 = 0
    • i.e.: forAll x, forAll y, x * (y + 1) = x * y + x
    • i.e.: for all x and y, x * (y + 1) = x * y + x
  1. This is an axiom scheme consisting of infinitely many axioms. If P(x) is any formula involving the constants 0, 1, +, *, = and a single free variable x, then the following formula is an axiom:
    • i.e.: ( P(0) and (forAll x, ( P(x) implies P(x + 1) ) ) ) implies (forAll x, P(x) )
    • i.e.: if something is true for 0, and from its being true for x it follows that it is also true for x + 1, then it is true for all x (induction)

Axioms 1, 2 and 7 are the traditional Peano axioms while axioms 3-6 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.

References