4.3 First-Order Language and Syntax

First-order language is the building blocks of formal logic. It uses symbols like constants, variables, and predicates to represent objects and relationships. Understanding these components is crucial for constructing well-formed formulas and expressing complex ideas.

Mastering the syntax of first-order language allows us to create precise logical statements. We learn to combine terms, use quantifiers, and apply connectives correctly. This foundation is essential for analyzing arguments and developing mathematical proofs in more advanced topics.

Components and Syntax of First-Order Language

Components of first-order language

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• Constants denote specific objects in domain (a, b, c)
• Variables act as placeholders for objects (x, y, z)
• Function symbols represent operations between objects $f(x, y)$
• Predicate symbols express properties or relations $P(x, y)$
• Logical connectives include $\neg$ (not), $\wedge$ (and), $\vee$ (or), $\rightarrow$ (if-then), $\leftrightarrow$ (if and only if)
• Quantifiers $\forall$ (for all) and $\exists$ (there exists) specify scope
• Parentheses group and clarify order of operations $(x + y) * z$

Construction of well-formed formulas

• Terms built from constants, variables, functions $f(g(x), y)$
• Atomic formulas combine predicates with terms $P(f(x), y)$ or equality $t_1 = t_2$
• Complex formulas formed by:
  1. Negating: $\neg \phi$
  2. Combining with connectives: $(\phi \wedge \psi)$, $(\phi \vee \psi)$, $(\phi \rightarrow \psi)$, $(\phi \leftrightarrow \psi)$
  3. Quantifying: $\forall x \phi$, $\exists x \phi$

Free vs bound variables

• Free variables not within quantifier scope, replaceable $P(x, y)$ where y is free
• Bound variables occur within quantifier scope $\forall x P(x, y)$ where x is bound
• Quantifier scope extends to subformula end or closing parenthesis
• Mixed examples: $\exists x P(x) \wedge Q(x)$ first x bound, second free

Precedence rules for connectives

• Precedence order: parentheses, quantifiers, $\neg$, $\wedge$, $\vee$, $\rightarrow$, $\leftrightarrow$
• Conjunction and disjunction are left-associative $P \wedge Q \wedge R = (P \wedge Q) \wedge R$
• Implication is right-associative $P \rightarrow Q \rightarrow R = P \rightarrow (Q \rightarrow R)$
• Override default with parentheses $(\forall x P(x)) \wedge Q(x)$ vs $\forall x (P(x) \wedge Q(x))$