Predicate Logic: Fill & Download for Free

Predicate logic is the general term for all logics that use predicates, e.g. [math]p(x)[/math]. Here, [math]p[/math] is a predicate; we say that [math]p[/math] is predicated of [math]x[/math]. For example, [math]quoran(josh)[/math] means "[math]quoran[/math] is predicated of [math]josh[/math]", or more loosely, "Josh is a quoran".Predicate logic is opposed to propositional logic, which simply uses symbols without the ability to do predication. For example: [math]p \land q[/math] means "p and q" or "p and q are both true", where p and q are propositions. Predicate logic is an extension of propositional logic: a proposition is a predicate with no arguments.Predicate logic also supports the ability to have variables, and quantifiers over variables. For example, [math]\forall x \exists y.p(x, y)[/math] means "For all x there exists a y such that the proposition p(x,y) is true".In first-order predicate logic, variables can appear only inside a predicate. That is, you can quantify over variables, but not predicates themselves. In second-order logic, you can also quantify over predicates, e.g. [math]\forall p \forall x.p(x) \lor \lnot p(x)[/math] is true: for every predicate p, either p(x) or not p(x) is true, regardless of what x is.In set-theoretic terms, a first-order logic quantifies over individuals, while a second-order logic quantifies over sets of individuals. Viewed this way, a predicate is identical to the set of individuals that it applies to. You can make higher-order logics by using metatypes, quantifying over sets of predicates. This kind of logic can be used to define type systems for predicate logics, though this kind of use is rather abstruse.

In propositional logic, we use letters to symbolize entire propositions. Propositions are statements of the form "x is y" where x is a subject and y is a predicate. For example, "Socrates is a man" is a proposition and might be represented in propositional logic as "S".In predicate logic, we symbolize subject and predicate separately. Logicians often use lowercase letters to symbolize subjects (or objects) and uppercase letter to symbolize predicates. For example, Socrates is a subject and might be represented in predicate logic as "s" while "man" is a predicate and might be represented as "M". If so, "Socrates is a man" would be represented "Ms".The important difference is that you can use predicate logic to say something about a set of objects. By introducing the universal quantifier ("∀"), the existential quantifier ("∃") and variables ("x", "y" or "z"), we can use predicate logic to represent thing like "Everything is green" as "∀Gx" or "Something is blue" as "∃Bx".I would say that's the most important difference.

Yes, you can discuss various kinds of logic without mentioning integers, but you'll have to use integers and other mathematics to analyze many interesting properties of those logics. Theorems about logic are called metalogic, and mathematical tools are needed to prove most of them.Symbolic logic such as propositional logic and predicate logic only involves variables, logical connectives, and quantifiers on those variables. It is not necessary to use numbers to describe and use such logics.However, if you want to prove completeness of first order predicate logic, as Gödel did, you'll need mathematics. You'll need it, too, to prove his incompleteness theorems.