1 Propositional Proofs 1

Problem 2

Deduction In deduction, the conclusion is true whenever the premises are true.  Premise: p Conclusion: (p ∨ q)  Premise: p Non-Conclusion: (p ∧ q)  Premises: p, q Conclusion: (p ∧ q) 3

Propositional Proofs Logical Entailment: A set of premises Δ logically entails a conclusion ϕ (written as Δ |= ϕ ) if and only if every interpretation that satisfies the premises also satisfies the conclusion {p} |= (p ∨ q) {p} |# (p ∧ q) {p, q} |= (p ∧ q) Mainly used when it is difficult to build the truth table Generate new lines whose truth value follows from, but is not identical to, the truth of the source lines. Can be applied ONLY to entire lines, not parts of lines. 4

Truth Tables and Proofs  The truth table method and the proof method succeed in exactly the same cases.  On large problems, the proof method often takes fewer steps than the truth table method. However, in the worst case, the proof method may take just as many or more steps to find an answer as the truth table method.  Usually, proofs are much smaller than the corresponding truth tables. So writing an argument to convince others does not take as much space. 5

Rules of Inference A rule of inference is a rule of reasoning consisting of one set of sentence patterns, called premises, and a second set of sentence patterns, called conclusions. 6

Rule Instances An instance of a rule of inference is a rule in which all meta- variables have been consistently replaced by expressions in such a way that all premises and conclusions are syntactically legal sentences. 7

Nine Basic Inference Rules Modus Ponens (MP)  From a conditional and a line identical to its antecedent, you may derive a line identical to its consequent Modus Tollens (MT)  From a conditional and the negation of its consequent, you may derive the negation of its antecedent p  q p  q p  q ~q  ~p 8

Nine Basic Inference Rules Disjunctive Syllogism (DS)  From a disjunction and the negation of one disjunct, you may derive the other disjunct Hypothetical Syllogism (HS)  From 2 conditionals, if the consequent of the first is identical to the antecedent of the second, you may derive a new conditional whose antecedent is identical to the antecedent of the first and whose consequent is identical to the consequent of the second. p  q q  r  p  r 9

Nine Basic Inference Rules Simplification  From a conjunction you may derive either conjunct.Conjunction  From any 2 lines you may derive a conjunction which has those lines as conjunctsAddition  From any line you may derive a disjunction with that line as a disjunct 10

Nine Basic Inference Rules 11 Constructive Dilemma  From a disjunction and 2 conditionals, if the antecedents match the disjuncts, you may derive a disjunction of the consequents Absorption  From a conditional you may derive a new conditional whose antecedent is that of the original and whose consequent is a conjunction of the original antecedent and the original consequent

Nine Basic Inference Rules Modus Ponens (MP) p  q P  q Modus Tollens (MT) p  q ~q  ~p Hypothetical Syllogism (HS) p  q q  r  p  r 12

Rules of Equivalence Equivalent expressions are true and false under exactly the same circumstances. So, one expression, or part of an expression, can be replaced with an equivalent expression (or part) without any change in meaning. Rules of equivalence are bi-directional 13

Rules of Equivalence Double Negation (DN) p :: ~ ~ p ~ ~ p :: p 14

Rules of Equivalence 15

Rules of Equivalence 16

Rules of Equivalence 17 Double Negation (DN) p :: ~ ~ p ~ ~ p :: p Contraposition (Cont) (p  q) :: (~q  ~p)

That’s It! We now have all 19 of the rules of inference and equivalence MP, MT, DS, HS, Conj, Simp, Add, CD, Abs  NOTE: there are no inference rules for dealing with biconditionals DN, DeM, Asso, Comm, Dist, Imp, Exp, Cont, Re, Equiv 18

Proof A proof is a finite set of formulae, beginning with the premises of an argument and ending with its conclusion, in which each formula following the premises is derived from the preceding formulae according to established rules of inference and equivalence. 19

Example of Inference 20

21 p  q ~q  ~p

To save space we also write this process as follows eliminating one of the ~W's: D W -> ~D ~W T V W T 22

23 Thank You!