Ch. 3 Notes

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Chapter 3: Propositional Logic The logical connectives Symbol Reading Name  P Not P Negation P & Q P and Q Conjunction (P, Q: conjuncts) P  Q P or Q (or both) Disjunction (P, Q: disjuncts) P  Q If P then Q Conditional (P: antecedent, Q: consequent) P  Q P if and only if Q Biconditional Well-formed formulas (wffs) in propositional logic (1) Any sentence letter is a wff. (2) If ϕ is a wff, so is ~ ϕ. (3) If ϕ and ψ are wffs, so are (ϕ & ψ), (ϕ  ψ), (ϕ  ψ), and (ϕ  ψ). (Note the parentheses.) (4) Unofficially, the outermost parentheses may be omitted.

The truth tables for the logical connectives Negation P  P T F F T Conjunction, disjunction, conditional, and biconditional P Q P & Q P  Q P  Q P  Q T T T T T T T F F T F F F T F T T F F F F F T T

Various concepts A wff can be (1) tautologous (a tautology, logically true); (2) inconsistent (contradictory, or logically false); or (3) contingent. An inconsistent (contingent) wff in propositional logic is called truth-functionally inconsistent (contingent). A wff  is tautologous  it is impossible for  to be false  there is no truth value assignment (i.e., no row in the truth table) in which  is false. A wff  is inconsistent  it is impossible for  to be true  there is no truth value assignment in which  is true. A wff  is contingent  is neither tautologous nor inconsistent  it is possible for  to be true and also for  to be false there is a truth value assignment in which  is true, and there is also a(nother) truth value assignment in which  is false. A pair of wffs  and  are equivalent (or  ::  )  it is impossible for one to be true and the other false  there is no truth value assignment in which one is true and the other is false. An argument from the set Г of premises to the conclusion C is valid  it is impossible for all members of Г to be true and the conclusion C to be false

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 there is no truth value assignment in which all members of Г are true and the conclusion C is false. A set Г of wffs is consistent  it is possible for all members of Г to be true at the same time  there is no truth value assignment in which all members of Г are true. Some famous equivalence relations :  P :: P P & P :: P P  P :: P  ( P & Q ) ::  P   Q  ( P  Q ) ::  P &  Q De Morgan’s Laws P  Q ::  P  Q  ( P  Q ) :: P &  Q P  Q :: ( P  Q ) & ( Q  P ) :: ( P & Q )  (  P &  Q ) P  Q ::  Q   P Contraposition P  ( Q  R ) :: ( P & Q )  R P & ( Q  R ) :: ( P & Q )  ( P & R ) Distributive Laws P  ( Q & R ) :: ( P  Q ) & ( P  R ) Distributive Laws Some famous valid argument forms : P  Q , P Ͱ Q Modus Ponens P  Q ,  Q Ͱ  P Modus Tollens P  Q , Q  R Ͱ P  R Hypothetical Syllogism P  Q ,  P Ͱ Q Disjunctive Syllogism P  Q , P  R , Q  R Ͱ R Constructive Dilemma