1 / 12

Natural Deduction

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”.

chyna
Download Presentation

Natural Deduction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Natural Deduction Proving Validity

  2. 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

  3. 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.

  4. A Simple Example (p. 344, #3) 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. 1. R > D 2. E > R _______ ____ 2. E > R 1. R > D E > D___ _HS_

  5. Another Example (p. 344, #6) 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 _________ ____ ~ J v P ~ J S > J ~ S______ _MT_

  6. 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.

  7. 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

  8. Example 1 – p. 346, #12 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 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

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

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

  11. Example 1 – Full Deduction Look at premises 2 and 4. Through MP, we can uses these premises to deduce ~ B. • 1. ~ M v ( B v ~ T) • 2. B > W • 3. ~ ~ M • 4. ~ W / ~ T • B v ~ T 1, 3 DS • ~ B2, 4 MT • 1. ~ M v ( B v ~ T) • 2. B > W • 3. ~ ~ M • 4. ~ W / ~ T • B v ~ T 1, 3 DS • ~ B 2, 4 MT • ~ T 5, 6 DS We are ready for our final step – using our previously established conclusions to derive the final, or main, conclusion through DS.

  12. 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.

More Related