> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/logic-types/predicate-logic1.md).

# 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 $$\exists x(Sx\land Px)$$