Well-Formed Formulas of Predicate Logic
Predicate logic, also known as first-order logic, is a formal system used to represent relationships between objects and make logical inferences. A well-formed formula (WFF) in predicate logic is a syntactically correct expression conforming to the rules of the language. Let's explore the syntax, semantics, and examples of WFFs in predicate logic.
Syntax of Predicate Logic
The syntax of predicate logic consists of logical symbols, variables, quantifiers, predicates, and connectives.
- Logical Symbols: $\land$ (conjunction), $\lor$ (disjunction), $\lnot$ (negation), $\rightarrow$ (implication), $\leftrightarrow$ (biconditional).
- Variables: Represent objects or individuals. Typically denoted by lowercase letters ($x, y, z$).
- Quantifiers: $\forall$ (universal quantifier), $\exists$ (existential quantifier).
- Predicates: Express properties or relations among objects. Typically denoted by uppercase letters or symbols ($P(x), Q(x, y)$).
- Connectives: Logical operators used to combine predicates and propositions.
Examples of Well-Formed Formulas
1. Simple Predicate
$P(x)$
Where $P(x)$ represents a unary predicate, indicating a property of the variable $x$.
2. Conjunction
$P(x) \land Q(y)$
Represents the conjunction of predicates $P(x)$ and $Q(y)$.
3. Disjunction
$P(x) \lor Q(y)$
Represents the disjunction of predicates $P(x)$ and $Q(y)$.
4. Negation
$\lnot P(x)$
Represents the negation of predicate $P(x)$.
5. Implication
$P(x) \rightarrow Q(y)$
Represents the implication where $P(x)$ implies $Q(y)$.
6. Universal Quantification
$\forall x \, P(x)$
Indicates that predicate $P(x)$ is true for all values of $x$.
7. Existential Quantification
$\exists x \, P(x)$
Indicates that there exists at least one value of $x$ for which $P(x)$ is true.
Semantics of Predicate Logic
The semantics of predicate logic define how well-formed formulas are interpreted. It involves assigning meanings to variables, predicates, and quantifiers.
Conclusion
Predicate logic provides a powerful framework for expressing and reasoning about relationships between objects and properties. Well-formed formulas in predicate logic adhere to syntactic rules and can be used to represent a wide range of logical statements and arguments.