Page 1 Proof Strats

Logic Strategies

  • The following strategies are guidelines to approach various logical derivations in proofs.

Proving an Atomic Sentence or Negation

  • Method: Indirect derivation (referred to as "ASS ID").

  • Steps:

    1. Assume the negation (or unnegation) of the statement (atomic sentence or negation) you are trying to prove.

    2. Box and cancel: When you reach two lines, n and m, where line n is the negation of line m, cite this with "n, m ID".

Proving a Conditional Statement

  • Method: Direct proof strategy starting with antecedent assumption (referred to as "ASS CD").

  • Steps:

    1. Assume the antecedent of the conditional you are trying to prove.

    2. Bring in premises: Incorporate the necessary premises to support your proof.

    3. Conclude with consequent: When you arrive at a line, n, that includes the consequent of the conditional being proved, box and cancel citing this as "n CD".

Proving a Biconditional Statement PQ

  • Method: Demonstrate both directions of the conditional.

  • Steps:

    1. First direction: Show P→Q on line n, using the command "Show cond".

    2. Second direction: After proving P→Q, initiate a new show line for Q→P on line m, again using "Show cond".

    3. Combine to biconditional: After establishing Q→P, apply conditionals-to-biconditional (citing lines n and m as CB) to obtain PQ on line k.

    4. Box and cancel: Use direct derivation to box and cancel, noting this as "k DD".

Proving a Conjunction P^Q

  • Method: Establish both conjuncts separately.

  • Steps:

    1. Show the conjunction: Right after the show line containing P^Q, initiate a new show line on line n for one of the conjuncts, using "Show conj" for P.

    2. Show the second conjunct: Start a new show line on line m for the other conjunct, Q, again using "Show conj".

    3. Combine conjuncts: Use the rule "m, n ADJ" to combine P and Q into P^Q on line k.

    4. Box and cancel: Apply direct derivation to box and cancel, documenting this as "k DD".

Proving a Disjunction PvQ

  • Method: Indirect derivation technique to analyze negation.

  • Steps:

    1. Assume negation: Apply ASS ID to obtain ~(PvQ).

    2. Apply De Morgan's Law: Transform the negation by using DM to derive ~P and ~Q.

    3. Simplify: Express the negations separately to support continuation with the proof process as per standard indirect derivation procedures.