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Home , Modus ponens, Truth table, Validity (logic)

Mathematics for Computer Proving 6.042J/18.062J Instead of tables, The of can try to prove valid formulas symbolically using and deduction rules

Albert R Meyer February 14, 2014 propositional logic.1 Albert R Meyer February 14, 2014 propositional logic.2

Proving Validity Algebra for Equivalence

The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R)

Albert R Meyer February 14, 2014 propositional logic.3 Albert R Meyer February 14, 2014 propositional logic.4

1 Algebra for Equivalence Algebra for Equivalence for example, The of rules for ≡ in DeMorgan’s law the text are complete: ≡ NOT(P AND Q) ≡ if two formulas are , these rules can prove it. NOT(P) OR NOT(Q)

Albert R Meyer February 14, 2014 propositional logic.5 Albert R Meyer February 14, 2014 propositional logic.6

A System A Proof System Another approach is to Lukasiewicz’ proof system is a start with some valid particularly elegant example of this idea. formulas (axioms) and deduce more valid formulas using proof rules

Albert R Meyer February 14, 2014 propositional logic.7 Albert R Meyer February 14, 2014 propositional logic.8

2 A Proof System Lukasiewicz’ Proof System Lukasiewicz’ proof system is a Axioms: particularly elegant example of 1) (¬P → P) → P this idea. It covers formulas 2) P → (¬P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES (→) and NOT. The only rule:

Albert R Meyer February 14, 2014 propositional logic.9 Albert R Meyer February 14, 2014 propositional logic.10

Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the rule. the inference rule. To illustrate the proof system For example, to prove: we’ll do an example, which you P→ P may safely skip. Albert R Meyer February 14, 2014 propositional logic.12 Albert R Meyer February 14, 2014 propositional logic.13

3 A Lukasiewicz’ Proof A Lukasiewicz’ Proof 3rd : 3rd axiom:

(P → Q ) → (P → Q ) → (( Q → R) → (P → R)) (( Q → P) → (P → P))

replace R by P replace Q by ( P → P)

Albert R Meyer February 14, 2014 propositional logic.14 Albert R Meyer February 14, 2014 propositional logic.15

A Lukasiewicz’ Proof A Lukasiewicz’ Proof 3rd axiom: so apply modus ponens: Axiom 2) Axiom 2) (P → (P → P) ) → (P → (P → P) ) → (((P → P) → P) → (P → P)) (((P → P) → P) → (P → P))

Albert R Meyer February 14, 2014 propositional logic.16 Albert R Meyer February 14, 2014 propositional logic.17

4 A Lukasiewicz’ Proof A Lukasiewicz’ Proof so apply modus ponens: so apply modus ponens:

Axiom 1) (((P → P) → P) → (P → P)) (P → P) QED

Albert R Meyer February 14, 2014 propositional logic.18 Albert R Meyer February 14, 2014 propositional logic.19

LukaLukasiewsiieczw’ icProofz is SSystemound Lukasiewicz is Complete The 3 Axioms are all valid Conversely, every valid (verify by ). (NOT,→)-formula is provable: We know modus ponens is system is “complete” Not hard to verify but would take s ound . So every provable a full lecture; we omit it. formula is also valid.

Albert R Meyer February 14, 2014 propositional logic.20 Albert R Meyer February 14, 2014 propositional logic.21

5 validity checking still inefficient

Algebraic & deduction proofs in general are no better than truth tables. No efficient method for verifying validity is known.

Albert R Meyer February 14, 2014 propositional logic.22

6 MIT OpenCourseWare http://ocw.mit.edu

6.042J / 18.062J Mathematics for Computer Science Spring 2015

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