Use MP, MT, DS, and HS to prove that the following arguments are valid. (1) 1. R 2. SDR/.. ~ S (2) 1. A S * 2. (A.S) DR/.. R (3) 1. (HK) ~ 2. Rv (H.K)/:. R (4) 1. (PvQ) (R. W) (9) 2. LD (PVO)/:. LD (RW) (5) 1. RDS 2. TOR 3. (6) 1. ~S/. ~T ~M 2. NDG 3. NvM/:. G (7) 1. DDE DDF 3. F. E 2 (8) 1. GvH 2. HVI 3. I/. G 1 ~GRAY BY ~ (10) 1. (AB) C DVA 2. 3. 4. ~A/. C (11) 1. 2. 3. 4. ~DD (ADB) AD (BDC) C (13) 1. 2. 3. 4. ~ ~ - DDA CvD/:. ~ B (12) 1. (D F) ~ 2. (LVM) VR 3. 4. ~T~(LVM) (D F) v~ T/:. R ~ (A v B) Ɔ (Bv C) (BDC) VA (BDC) (A v B) -Al.. Bv C ~ . 4) 1. (PQ) [Rv (T-S)] 2. (TvR) (P.Q) 3. 4. ~ (T.S) TVRI.. R

09:22 Question 3 3. [30] Write each of these propositions by using propositional variables and the logical connectives. a) For you to win the contest it is necessary and sufficient that you have the only winning ticket. b) It is necessary to walk eight miles to get to the top of Long's Peak.

Philosophical Logic

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: • P(z) : z was given the placebo • D(z) : z was given the medication • M(z): z had migraines Translate cach of the following statements into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until cach negation operation applices directly to a predicate and then translate the logical expression back into English. Sample question: Some patient was given the placebo and the medication. 3r (P(z) ^ D(z)) • Negation: ¬ar (P(z) ^ D(z)) Applying De Morgan's law: ¥z (¬P(z) V ¬D(z)) English: Every patient was either not given the placebo or not given the medication (or both).