Logic, Sets, and Functions - Lecture Notes | PHL 313K, Study notes of Mathematical logic

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Derivable rules for Ñ, &

Indirect proof (second form)

n. Show Ç n+1. —Ç AIP

p. Í q. —Í

—(—P&Q), Q ˇ P sl.

Commutativity of Conjunction (&C)

n. (Ç & Í) m. (Í & Ç) &C, n

Associativity of Conjunction (&A)

n. ((Ç & Í) & ˚) m. (Ç & (Í & ˚)) &A, n

Rules for the ‡ and the ‰

Conditional Exploitation

n. (Ç á Í) m. Ç p. Í áE, n, m

Conditional Proof

n. Show (Ç á Í) n+1. Ç ACP

n+ p. Í

Derivable rules for ‡, ‰

Conditional Exploitation* (‡E*)

n. (Ç á Í) m. —Í p. —Ç áE*, n, m

Biconditional Exploitation* (‰E*)

n. (Ç â Í) m. —Ç (or —Í) p. —Í (or —Ç) âE*, n, m

Negation-Conditional (Ñ ‡)

n. —(Ç á Í) m. (Ç & —Í) —á, n

Negation-Biconditional (щ)

n. —(Ç â Í) m. (—Ç â Í) (or Ç â —Í) —â, n

—(PáQ) ˇ (P&—Q) sl.

Derivable rule for ˆ De Morgan’s Laws Negation-Conjunction

n. —(Ç & Í) m. —Ç √ —Í —&, n

Negation-Disjunction n. —(Ç √ Í) m. —Ç & —Í —√, n

(P√Q) ˇ (Q√P) sl.

Conditional-Disjunction n. (Ç á Í) m. (—Ç √ Í) Commutativity of Disjunction n. (Ç √ Í) m. (Í √ Ç) √C, n

Associativity of Disjunction n. ((Ç √ Í) √ ˚) m. (Ç √ (Í √ ˚)) √A, n

Disjunction Exploitation* (ˆE, MTP) n. (Ç √ Í) m. —Ç (or —Í) p. Í (or Ç) √E, n, m

To exploit: Try: —Ç Use áE, √E, ——, —&,—á, —â, or —√. (Ç & Í) Use &E to get Ç and Í.

(Ç √ Í) (a) Prove —Ç or —Í, and use √E*, or (b) Prove Ç á Á and Í á Á, and use √E,

(Ç á Í) Prove Ç or —Í and use áE or áE*.

(Ç â Í) Prove Ç, —Ç, Í, or —Í and use âE or âE*.

1. When in doubt, use indirect proof.
2. If you've started an indirect proof and don't know what to do next, pick a sentence letter and try to prove it by another indirect proof.