Natural Deduction Proving Validity

The Basics of Deduction  Argument forms are instances of deduction (true premises guarantee the truth of the conclusion). p > q p q p > q p p > q ___ q

Argument Forms as “Rules”  A “rule” for deduction tells you what you may do, given the presence of certain kinds of statements or premises.  Since all argument forms do just this (tell us what we may infer, given certain kinds of premises), they function as rules.

A Simple Example (p. 344, #3) 1. R > D 2. E > R _______ ____ 2. E > R 1. R > D E > D___ _HS_ We use the first line to record the conclusion the “rule” lets us draw. To do so correctly, we have to know what rule to follow – or what argument form this argument illustrates. Use the second line to record this.

Another Example (p. 344, #6) ~ J v P ~ J S > J _________ ____ Since our argument forms each have only two premises, look for the two that you can use; ignore the other. ~ J v P ~ J S > J _________ ____ ~ J v P ~ J S > J ~ S______ _MT_

Your Strategy for Finding Conclusions  You are just looking for any two premises that follow the “rules” or argument forms; they don’t need to be near each other, or in the standard order.  Use the “left side/right side” technique for dealing with tildes (statement variables stand for anything on either side of an operator). Example: p. 344, #13.

Multiple-Step Deductions  You sometimes need to use more than one argument form (rule of implication) in a given argument to derive the conclusion.  The strategy is to find the conclusion to be derived in one of the given premises, and “work backward” from there.  More strategy suggestions are on p. 343 and 344

Example 1 – p. 346, #12 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T Our conclusion is ~ T 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T If we could “isolate” (B v ~ T) from premise 1, and find a way to get ~ B, then we could deduce ~ T. B v ~ T ~ B ~ T p v q (q = ~ T) ~ p q

Example 1, continued 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T (B v ~ T) is in a disjunctive statement. To “isolate” it, we need a disjunctive syllogism. To get one disjunct, we need to have the negation of the other (DS). Premise 3 gives us the negation of ~ M, in premise ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T

Example 1 – Partial Deduction 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T 5. B v ~ T 1, 3 DS Now we need to get ~ B, so we can deduce ~ T through another DS.

Example 1 – Full Deduction 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T 5.B v ~ T 1, 3 DS 6.~ B 2, 4 MT Look at premises 2 and 4. Through MP, we can uses these premises to deduce ~ B. We are ready for our final step – using our previously established conclusions to derive the final, or main, conclusion through DS. 1. ~ M v ( B v ~ T) 2. B > W 3. ~ ~ M 4. ~ W / ~ T 5.B v ~ T 1, 3 DS 6.~ B 2, 4 MT 7.~ T 5, 6 DS

Multiple Step Deductions - Conclusions  Use your knowledge of argument forms to apply the (first 4) rules of implication to show validity through deduction.  Be flexible in finding premise pairs; they may be anywhere in the argument.  Be stringent in your justifications; every line must show premise numbers and the rule through which you connect them to the derived statement.