Predicate logic, also known as first-order logic, extends propositional logic by introducing predicates, quantifiers, and variables to reason about properties and relations of objects in a more expressive manner. Inference theory in predicate logic deals with the rules and methods for deriving conclusions from premises using logical inference.

Predicate Formulas:

In predicate logic, formulas are constructed using predicates, variables, quantifiers, and logical connectives. Here are the main components:

• Predicates: Represent properties or relations that can be true or false. They are usually denoted by symbols followed by one or more variables.
  • Example:
    
    $$

    P(x) could represent “x is a prime number.”

• Variables: Represent objects or entities in the domain of discourse.
  • Example:$x$, $y$,$z$ are variables representing numbers, individuals, or objects.
• Quantifiers: Specify the scope of variables in a formula, indicating whether the formula applies universally or existentially to objects in the domain.
  • Universal Quantifier ($\forall$): Asserts that a statement holds for all elements in the domain.
  • Existential Quantifier (
    
    $\mathrm{}$

    ∃): Asserts that a statement holds for at least one element in the domain.

• Logical Connectives: Include logical connectives such as AND (
  
  $$

  ∧), OR (
  $V<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>$

  ¬), implication (
  $\to <span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">,\; etc.,\; for\; forming\; compound\; formulas.</span>$

Quantifiers:

• Universal Quantifier ($\forall$): Indicates that a statement is true for all values of a variable.
  • Example:
    
    $\mathrm{}$

    ∀xP(x) means “For all
    $$

    x, P(x) is true.”

• Existential Quantifier (
  
  $\mathrm{}$

  $\exists$): Indicates that a statement is true for at least one value of a variable.

  • Example:
    
    $\mathrm{}$

    ∃xP(x) means “There exists an
    $$

    x such that
    $$

    P(x) is true.”

Inference Theory of Predicate Logic:

Inference theory in predicate logic concerns the rules and methods for deriving valid conclusions from given premises using logical inference. Some key aspects include:

• Inference Rules: Just like in propositional logic, predicate logic has inference rules that allow for the derivation of new formulas from existing ones.
  • Example: Modus Ponens, Universal Instantiation, Existential Generalization, etc.
• Validity: A conclusion derived from premises using inference rules is considered valid if it logically follows from the premises. Validity ensures that if the premises are true, the conclusion must also be true.
• Soundness: An inference system is sound if it only derives conclusions that are logically entailed by the premises. In other words, a sound inference system guarantees that all derived conclusions are true whenever the premises are true.
• Completeness: An inference system is complete if it can derive every valid conclusion from the given premises. Completeness ensures that all valid conclusions can be reached using the inference rules of the system.
• Proofs: Inference theory involves constructing formal proofs, which are sequences of logical steps that justify the derivation of conclusions from premises using inference rules.

In summary, inference theory in predicate logic deals with the rules and methods for deriving valid conclusions from premises using logical inference, quantifiers, and predicates. It provides the foundation for reasoning about properties and relations of objects in a formal and rigorous manner.