Syntax

Components of predicate logic

• Atomic sentence, atomic formula, term, expression, wff

• Symbols

• Constants

• Variables, free and bound

• Predicates, predicate symbols

• Quantifiers

• Universe of discourse

There are 6 kinds of symbols in PL:

• constants: $a,b,c,\ldots$

• variables: $x,y,z,\ldots$

• predicates: $A,B,C,\ldots$

• connectives: ¬ ∧ ∨ →

• quantifiers: ∃ ∀

• parentheses

(first 3: possibly with subscripts). An expression is any string of symbols; symbols in any order form a PL expression.

Constant

A logical constant of a formal language is a symbol that has the same semantic value under every interpretation of that language.

Two important types of logical constants are logical connectives and quantifiers.

The equality predicate (usually as infix =) is also treated as a logical constant in many logic systems.

Predicates

Fundamental component in predicate logic is a predicate, symbolized by an uppercase letter called predicate symbol, which is an expression that, combined with a name (variables and constants), produces an atomic sentence.

A predicate is an expression like "is a man", which is not a sentence on its own and which doesn't have a truth value. In order to get a truth value we need to specify an object as an argument of this predicate.

Predicates translate 3 kinds of statements: singular, universal and particular.

Singular statements

A singular statement is an affirmative or negative statement that asserts something about a named object (person, place, time, etc.).

• Singular terms are constants and variables.

• Constants pick out specific individuals.

• Variables do not stand for any specific individual - they are needed for introduction of quantifiers.

• An individual variable differs from an individual constant in that it can stand for any item in the universe of discourse (UD).

• A proper name is a singular term that picks out an individual without describing it.

• A definite description picks out an individual by means of a unique description.

• A singular terms must refer to one specific thing in UD

The expression "a is P" is translated as $P(a)$, with $a$ denoting a constant. However, in the expression, $P(x)$, $x$ is a variable; because a variable ranges over all objects in UD, this means that all objects in UD have the property $P$.

Examples:

• "Anything is possible": $\forall x Px$

• "Unicorns are extinct": $\lnot \exists x Ux$ or $\forall x \lnot Ux$

Universal statement

A universal statement is either affirmative or negative statement that makes an assertion about every member of its subject class.

• Universal statements are translated as conditionals.

• Variable are used to form a universal quantifier.

• e.g. "All $S$ are $P$" is translated as $\forall x(Sx\to Px)$.

For example, "All bricks are thick" can be symbolized as $\forall x(Bx \to Tx)$, meaning "for all x: if x is a brick, then x is thick".

A symbol that indicates that an assertion goes for all members is called universal quantifier, and it is introduced along with a variable, e.g. $\forall x(Px \to Qx)$

Particular statement

A particular statement is a statement that makes an assertion about one or more unnamed members of the subject class.

• Particular statements are translated as conjunctions.

• e.g. "Some $S$ are $P$" is translated as $\exists x(Sx\land Px)$