Tk123

Thomas Kiley - Discrete Mathematics - Year 2 - University of Warwick

Preicate logic rework

For the specific term, see First-order logic.

In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified. Two common quantifiers are the existential ∃ ("there exists") and universal ∀ ("for all") quantifiers. The variables could be elements in the universe under discussion, or perhaps relations or functions over that universe. For instance, an existential quantifier over a function symbol would be interpreted as modifier "there is a function".

In informal usage, the term "predicate logic" occasionally refers to first-order logic. Some authors consider the predicate calculus to be an axiomatized form of predicate logic, and the predicate logic to be derived from an informal, more intuitive development.

Predicate logics also include logics mixing modal operators and quantifiers. See Modal logic, Saul Kripke, Barcan Marcus formulae, A. N. Prior and Nicholas Rescher.

Syntax

Predicate calculus expressions are created from the aforementioned predicates (∃, ∀) combined with variables. These variables depend on the universe the predication is set in.

• Constants name specific objects or properties in the domain of discourse. Thus George, tree, tall and blue are examples of well formed constant symbols. The constants $\top$ (true) and $\bot$ (false) are sometimes included.
• Variable symbols are used to designate general classes or objects or properties in the domain of discourse.
• Functions denote a mapping of one or more elements in a set(called the domain of the function) into a unique element of another set(the range of the function). Elements of the domain and range are objects in the world of discourse. Every function symbol has an associated arity, indicating the number of elements in the domain mapped onto each element of range.

One might read ∀ Mother M ∃

A function expression is a function symbol followed by its arguments. The arguments are elements from the domain of the function; the number of arguments is equal to the arity of the function. The arguments are enclosed in parentheses and separated by commas. e.g.:

• f(X,Y)
• father(david)
• price(apple)

are all well-formed function expressions.

Predicate logics may be viewed syntactically as Chomsky grammars. As such, predicate logics (as well as modal logics and mixed modal predicate logics) may be viewed as context-sensitive, or more typically as context-free, grammars. As each one of the four Chomsky-type grammars have equivalent automata, these logics can be viewed as automata just as well.

See also

• Propositional logic
• First-order logic

References

• A. G. Hamilton 1978, Logic for Mathematicians, Cambridge University Press, Cambridge UK ISBN 0-521-21838-1.
• Abram Aronovic Stolyar 1970, Introduction to Elementary Mathematical Logic, Dover Publications, Inc. NY. ISBN 0-486-64561
• George F Luger, Artificial Intelligence, Pearson Education, ISBN 978-81-317-2327-2

Category:Predicate logic Category:Systems of formal logic

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