9.1 Identity in Predicate Logic

Predicate logic uses identity to compare objects and express sameness. This concept is key for simplifying complex statements and deriving new information. It's like having a tool that lets you swap out equivalent terms in logical expressions.

The identity symbol (=) shows when two terms refer to the same thing. It has three important properties: reflexivity, symmetry, and transitivity. These properties help us reason about relationships between objects and make logical deductions.

Identity in Predicate Logic

Concept of identity in logic

• Fundamental concept in predicate logic enables comparison of objects or terms
• Expresses two terms refer to the same object or individual
• Crucial role in logical reasoning by:
  • Enabling substitution of equivalent terms in logical statements
  • Facilitating simplification of complex logical expressions ($P(a) \land Q(a)$ can be simplified to $P(a)$ if $P = Q$)
  • Allowing derivation of new information based on properties of identity (if $a = b$ and $P(a)$ is true, then $P(b)$ is also true)

Application of identity symbol

• Identity symbol ($=$) expresses two terms refer to the same object or individual
  • If "a" and "b" refer to the same object, write: $a = b$
• Identity statements true if and only if terms on both sides of equality symbol refer to the same object
  • $2 + 3 = 5$ is a true identity statement
  • $x = y$ is true if and only if "x" and "y" refer to the same object
• Identity symbol not to be confused with equivalence connective ($\equiv$) used to express logical equivalence between statements
  • $P \equiv Q$ means $P$ and $Q$ have the same truth value for all possible assignments of their variables

Properties of identity

• Identity has three important properties: reflexivity, symmetry, and transitivity
• Reflexivity: For any term "a", $a = a$ is always true
  • Every object is identical to itself ($2 = 2$, $x = x$)
• Symmetry: If $a = b$, then $b = a$
  • If two terms are identical, order in which they are written does not matter ($2 + 3 = 5$ implies $5 = 2 + 3$)
• Transitivity: If $a = b$ and $b = c$, then $a = c$
  • If two terms are identical to a third term, they are also identical to each other (if $x = y$ and $y = z$, then $x = z$)

Identity for logical simplification

• Identity used to simplify complex logical statements by replacing terms with their identical counterparts
  • If $a = b$ and $P(a)$ is a logical statement, can replace "a" with "b" to obtain $P(b)$
  • If $x = 2y$ and $Q(x)$ is a logical statement, can replace "x" with "2y" to obtain $Q(2y)$
• Identity used to derive new information from existing statements
  • If $a = b$ and $Q(a)$ is known to be true, can infer $Q(b)$ is also true
  • If $x = y$ and $P(x)$ is true, then $P(y)$ is also true
• When using identity to simplify or derive new information, essential to ensure substitution is valid and does not change meaning of original statement