Propositional logic deals with simple statements that are either true or false. It uses logical connectives to build complex expressions.
Example: P ^ Q if both P and Q are true, then the whole expression is true.
Propositions Proof Theory focuses on the rules for logical deductions and includes completeness, soundness, and compactness.
Example: A proof might show that from p→q and p , one can deduce q
Predicate Language including quantifiers and predicates, allowing for expressions involving variable subjects.
Example: In ∀x(P(x)→Q(x)) if P(x) “x is a dog”, and Q(x) is “x has four legs”, then the expression asserts that all dogs have four legs
Predicate Proof Theory focuses on the rules for predicate logical deductions in and includes completeness, soundness, and compactness.
Example: A proof might show that that if ∀x(P(x)) is true, then P(a) must also be true for any specific element a
Model Theory can be used to interpret the truth of statements within logical systems, assessing the consistency and validity
Example: A model might define a universe where P(x) means x>0 and Q(x) means x is an integer, to validate the statement ∀x(P(x)→Q(x))
Herbrand Structures focuses on a specific approach to simplify the interpretations of quantifiers
Example: If the universe consists of natural numbers with zero and the successor function, Herbrand’s theorem helps in proving existential statements like ∃x(x=x+1) false.
