- 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.
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