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SYMBOLIC LOGIC :CONDITIONAL PROOF Conditional proof can only be used to deduce a conditional claim. That is, the deduction of any formula via conditional proof is always dependent upon the formula assumed. Or, in other words, what is deduced is always the consequent of the antecedent that was assumed. For example, if you assume p and deduce t you have proven p t, not t by itself. When using conditional proof, any formula derived within the conditional proof lines, that is, between the assumption and the conclusion, is dependent on the assumption made. You cannot take that formula and use it on any line that is not within the conditional proof lines. You can construct more than one conditional proof in a deduction, but, when you do so, they either need to be (1) nested or (2) separate entirely. We cannot have a case in which the conditional proof lines cross. When using CP to prove a conditional proof, assume as much as possible. For example, if you are trying to prove [(A A) B] B, assume (A A) B, the antecedent. MORE RULES FOR DEDUCTION Rule of Assumption We can introduce an assumption into a proof at anytime as long as we meet the following conditions: The assumption has to be indented from the left line The assumption has to be labeled as such in the commentary line of the deduction The assumption has to have an arrow pointed at its line number from left to right. All the lines of the deduction corresponding to that assumption need to be enclosed in a vertical line that stretches from the arrow pointing to the assumption to the line before the results of the deduction are stated. Rule of Conditional Proof (Gustason, p. 116) If p is the formula that starts a vertical line and if t is a step in a deduction with a continuous vertical line to the left of it, then we can write p t, provided that: The vertical line is discontinued immediately under the t and before the p t All steps within the vertical line are considered closed off such that we cannot use lines within the assumption line to justify lines outside of the assumption line. Conditional Proof Example 1. p [q V (r & s)] Premise 2. (~q s) t 3. p To show: p t Premise Assumption 4. q V (r & s) 5. ~q (r & s) 1,3 Modus Ponens 4 Implication 6. ~q Assumption 7. (r & s) 5, 6 Modus Ponens 8. s 7 Simplification 9. ~q s 6-8 Conditional Proof 10. 2, 9 Modus Ponens 11. pt t 3-10 Conditional Proof