Hypothetical Derivations Kareem Khalifa Department of Philosophy Middlebury

Hypothetical Derivations Kareem Khalifa Department of Philosophy Middlebury College

Overview • • What is a hypothetical derivation? Everyday examples Formal proofs and hypothetical derivations Sample Exercises

What is a hypothetical derivation? • A proof made on the basis of a temporary assumption or hypothesis, which is not asserted to be true, but only supposed for the sake of argument.

Everyday examples: • Suppose that you major in philosophy. Then your analytical reasoning skills will be honed, which in turn will increase your chances of rocking the LSAT. As a result, if you major in philosophy, you will be more likely to rock the LSAT. • Suppose that Khalifa is a real SOB. Then he’d give us a pop quiz every day. Fortunately, he doesn’t do this. So Khalifa must not be an SOB.

Two Central Observations 1. Hypotheses are not asserted to be true, but only supposed for the sake of argument. 2. Nevertheless, these suppositions yield propositions which can be asserted and not merely supposed, specifically: – Conditional statements: If you major in philosophy, your chances of rocking the LSAT increase. – Negative statements: Khalifa is not an SOB.

How to represent these two observations formally • Whenever you suppose something, indent it and draw a vertical line next to it. – You have now entered “the world of hypothesis. ” • • Now proceed as if you had this hypothesis as an additional premise. In order to leave the world of hypothesis, you must follow one of two rules…

The two rules • I: Given a hypothetical derivation of ψ from Ф, end the derivation and infer (Ф ψ). – In this case, (Ф ψ) can be asserted “outside” of the world of hypothesis • ~I: Given the hypothetical derivation of any formula of the form (ψ & ~ψ) from Ф, end the derivation and infer ~Ф. – In this case, ~Ф can be asserted “outside” of the world of hypothesis

Back to the everyday examples… • Let P = you major in philosophy; Q = Your analytical skills improve; R = You’re more likely to rock the LSAT. • The argument: P Q, Q R ├ P R

The Proof 1. P Q 2. Q R P R 3. |P 4. |Q 5. |R 6. P R A A H for I 1, 3 E 2, 4 E 3 -5 I Suppose that you major in philosophy Then your analytical skills will improve Then you’re more likely to rock the LSAT

The Application of I 1. P Q 2. Q R P R 3. |P 4. |Q 5. |R 6. P R A A I: Given a hypothetical derivation of ψ from Ф, end the derivation and infer (Ф ψ). H for I 1, 3 E 2, 4 E 3 -5 I Let Ф = P Let ψ = R Lines 3 -5 are the hypothetical derivation.

The Las Vegas Rule… • What happens in the world of hypothesis, stays in the world of hypothesis. • In our previous proof, it would be invalid to assert anything inside the world of hypothesis outside of the world of hypothesis.

Example of what we CAN’T infer 1. P Q A 2. Q R A 3. |P H for I 4. |Q 1, 3 E 5. |R 2, 4 E 6. P & R WRONG! 3, 5 &I You’ve broken the Las Vegas Rule!

The wrongness in plain English… • If you major in philosophy, then your analytical skills will improve. • If your analytical skills improve, then you’re more likely to rock the LSAT. • Thus you major in philosophy and you’re more likely to rock the LSAT. • Clearly INVALID! • So don’t violate the Las Vegas Rule.

What we CAN infer 1. P Q 2. Q R 3. |P 4. |Q 5. |R 6. |P & R 7. P (P&R) A A H for I 1, 3 E 2, 4 E 3, 5 &I 3 -6 I

The other example… • 1. 2. 3. 4. 5. 6. Let P = Khalifa is a real SOB; Q= Khalifa gives us a pop quiz every day. ~I: Given the P Q A hypothetical derivation ~Q A of any formula of the form (ψ & ~ψ) from Ф, ~P |P H for ~I infer ~Ф. |Q 1, 3 E Let Ф = P, ψ = Q |Q & ~Q 2, 4 &I The hypothetical derivation goes from ~P 3 -5 ~I lines 3 -5.

Proof Strategies— VERY IMPORTANT! • Memorize the strategies on Nolt, p. 99. – They are the key to getting partial credit on any proof exercise, since they show that you have some clue about how to go about constructing a proof. • Start tackling every proof by asking the following two questions: 1. What is the main operator of my conclusion? 2. What is the proof strategy that corresponds to Very important! Very Important! Very that operator? Important! Very Important!

Sample Exercises, Nolt 4. 3. 2 1. (P v Q) R P R 2. |P 3. |Pv. Q 4. |R 5. P R A H for I 2 v. I 1, 3 E 2 -4 I

Sample Exercise, Nolt 4. 3. 7 1. 2. 3. 4. 5. 6. 7. (P&Q) R P (Q R) | P | |Q | |P&Q | |R | Q R P (Q R) A H for I 2, 3 &I 1, 4 E 3 -5 I 2 -6 I

4. 3. 10 1. ~~P v ~~Q Pv. Q 2. | ~~P 3. | P 4. | Pv. Q 5. ~~P (P v Q) 6. | ~~Q 7. | Q 8. | Pv. Q 9. ~~Q (P v Q) 10. P v Q A H for I 2 ~E 3 v. I 2 -4 I H for I 6 ~E 7 v. I 6 -8 I 1, 5, 9 v. E

4. 3. 16 1. ~(P&Q) P ~Q 2. | P 3. | | Q 4. | | P&Q 5. | | (P&Q) & ~(P&Q) 6. | ~Q 7. P ~Q A H for I H for ~I 2, 3 &I 1, 4 &I 3 -5 ~I 2 -6 I