Exploring the Fundamentals of Logic

  • 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

 

  • 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
  • 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.