Home / first order logic proof calculator / First-Order Logic
First-Order Logic is the calculus one usually has in mind when using
the word ``logic''. It is expressive enough for all of mathematics,
except for those concepts that rely on a notion of construction or
computation. However, dealing with more advanced concepts is often
somewhat awkward and researchers often design specialized logics
for that reason.
Our account of first-order logic will be similar to the one of
propositional logic. We will present
? The syntax, or the formal language of first-order logic, that
is symbols, formulas, sub-formulas, formation trees, substitution,
? The semantics of first-order logic
? Proof systems for first-order logic, such as the axioms, rules,
and proof strategies of the first-order tableau method and refinement
? The meta-mathematics of first-order logic, which established
the relation between the semantics and a proof system
In many ways, the account of first-order logic is a straightforward
extension of propositional logic. One must, however, be aware that
there are subtle differences.
The syntax of first-order logic is essentially an extension of propositional
logic by quantification and . Propositional variables are replaced
by n-ary predicate symbols (P , Q, R) which may be instantiated with
either variables (x, y, z, ...) or parameters (a, b, ...). Here
is a summary of the most important concepts.
102 CHAPTER 10. FIRST-ORDER LOGIC
(1) Atomic formulas are expressions of the form P c1..cn where P is
an n-ary predicate symbol and the ci are variables or parameters.
Note that many accounts of first-order logic use terms built
from variables and function symbols instead of parameters. This
makes the formal details a bit more complex.
(2) Formulas are built from atomic formulas using logical connectives
Every atomic formula is a formula.
If A and B are formulas and x is a variable then (A), ?A, A B,
A B, A B, (x)A, and (x)A are formulas.
(3) Pure formulas are formulas without parameters.
(4) The degree d(A) of a formula A is the number of logical connectives
and quantifiers in A.
(5) The scope of a quantifier is the smallest formula that follows
In (x)P x Qx the scope of (x) is just P x, while Qx is outside
the scope of the quantifier. To include Qx in the scope of (x)
one has to add parentheses: (x)(P x Qx).
Note that the conventions about the scope of quantifiers differ
in the literature.
(6) Free and bound variables are defined similarly to Second-Order
A variable x occurs bound in A if it occurs in the scope of a
quantifier. Any other occurrence of x in A is free.
(7) Closed formulas (or sentences) are formulas without free variables.
This is the default from now on.
(8) Substitution: A|x
a (or A[a/x]) is the result of replacing every
free occurrence of the variable x in A by the parameter a.
The technical definition is similar to the one for P 2. However,
since the term being substituted for x does not contain variables,
capture cannot occur.
(9) Subformulas are defined similar to propositional logic.
The only modification is that for any parameter a the formula
a is an immediate subformula of (x)A and (x)A.
(10) The formation tree of a formula F is a representation of all
subformulas of A in tree format.
That is, the root of the tree is F .
The sucessor of a formula of the form ?A is A.
The successors of A B, A B, A B are A and B.
10.3. SEMANTICS 103
The successors of (x)A and (x)A are A|x for all parameters ai.
Note that quantifiers usually have infinitely many successors.
Atomic formulas have no successors.
The semantics of first-order logic, like the one of propositional
logic and P 2, is based on a concept of valuations. In propositional
logic, it was sufficient to assign values to all propositional variables
and then extend the evaluation from atoms to formulas in a canonical
fashion. In P 2, the semantics of quantified formulas was defined
in terms of the values of all immediate subformulas:
v[(p)A] = (v|p )[A] B (v|p
In first-order logic, we will proceed in the same way. However,
since we don't have propositional variables anymore, we have to explain
the meaning of atomic formulas first.
The standard approach is to interpret parameters by elements of
some universe U and n-ary predicates by subsets of Un. A closed
formula P a1..an then expresses the fact that the interpretations ki U
of the ai, taken together as n-tuple (k1, .., kn), form an element of
the interpretation of P .
Smullyan's approach is similar to the above idea but avoids set
theory altogether. Instead, he introduces U-formulas, where the
elements of the universe U are used as parameters and defines first-order
valuations as canonical extensions of boolean valuations on the set
EU of all closed U-formulas. The semantics of arbitrary formulas
is then defined by a mapping from the set of parameters into U.
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