Natural Deduction: Using simple valid argument forms –as demonstrated by truth-tables—as rules of inference. A rule of inference is a rule stating that whenever premises of certain forms occur, conclusions of a certain form follow necessarily.

If it’s sunny we’ll be able to go to the mountains for a hike, but if it rains we’ll have to stay home and watch tv all day. Weatherunderground says there’s only a 10% chance of rain tomorrow. What follows? We’ll be able to go to the mountains for a hike

1. V v F 2. ~V / F Either apples are vegetables or they’re fruits. But they’re not vegetables. So… They are fruits 1. V v F 2. ~V / F DS, 1,2

Either apples and bananas are both vegetables or else they are both fruit. But it’s false that Apples and Bananas are vegetables, so they must be fruit. 1. (A ∙ B) v (F ∙R) 2. ~(A ∙ B) / F ∙ R How many lines will this truth table require?

Focus on the main operator of the statements p . q p > q (A v B) . (C v D) __ > __ 1 > 2 (S v B) > ~W ((W . Y) v X) . (H > B) [(~K > P) . N] > (A . C) [F . (G v B)] . (J > I) Focus on the main operator of the statements to be clear on what kind of statement each is.

Modus Ponens MP p > q p ----- q (B v N) > M (B v N) ------------- M A > C A ------ C (L > N) > (B > V) L > N --------- (H . (J v N)) > (N . L) H . (J v N) -------- N . L B > V

Modus Tollens MT p > q ~ q ----- ~ p F > H ~H ----- ~F ~(J v L) (M . B) > (J v L) ------- ~( M . B) [G v (B . O)] > [(D . V) . R] ~[(D . V) . R] ------- ~[G v (B . O)]

Disjunctive Syllogism DS p v q ~p ----- q (K v L) v O ~(K v L) ------ O (A v (B . D)) ~A ----- B . D [(T v Y) > (L . D)] v {[(F . T) v O] > W} ~ [(T v Y) > (L . D) ------ [(F . T) v O] > W

Hypothetical Syllogism HS p > q q > r ------- p > r (N . B) > G G > ( L v ~E) ------------- (N . B) > (L v ~E) M > (B > G) (B > G) > (F . T) ----------- ~V > [(O . ~C) > P] (K v L) > ~V ----------------- M > (F . T) (K v L) > [(O . ~C) > P]

Simplification SM p . q ------ p (B v N) . (C > L) ----- B v N A . (B >C) ------ A

Conjunction CN p q --- p . q (M . N) v W D > K --- ((M . N) v W) . (D > K)

p > q p / q MP p > q ~q / ~ p MT p v q ~ p / q DS p > q q > r / p > r HS A > B ~A > (C v D) ~B ~C / D 5. ~A 1,3 MT 6. C v D 2, 5 MP 7. D 4,6 DS

MP MT DS HS p > q p > q p v q p > q p / q ~q / ~p ~ p / q q > r / p > r E > (K > L) F > (L > M) G v E ~G F / K >M 6. E 3, 4 DS 7. K > L 1, 6 MP 8. L > M 2, 5 MP 7,8 HS 9. K > M

MP MT DS HS p > q p > q p v q p > q p / q ~q / ~p ~ p / q q > r / p > r J > (K > L) L v J ~L / ~K J K > L ~K 2,3 DS 1, 4 MP 5, 3 MT

MP MT DS HS p > q p > q p v q p > q p / q ~q / ~p ~ p / q q > r / p > r 1. ~ (S  T) > (~P > Q) 2. (S  T) > P 3. ~P / Q 4. ~ (S  T) 5. ~P > Q 6. Q 2,3 MT 1, 4 MP 3,5 MP

p > q p > q p v q p > q MP MT DS HS p > q p > q p v q p > q p / q ~q / ~p ~ p / q q > r / p > r H > [~E > (C > ~D)] ~D > E E v H ~E / ~C H 6. ~E > (C > ~D) 7. C > ~D 8. C > E 9. ~C 3,4 DS xxx 5,1 MP 4, 6 MP 7, 2 HS 4, 8 MT

Addition AD p --- p v q Constructive Dilemma (p > q) . (r > s) p v r ----- q v s T v F T A conjunction of conditionals, plus the disjunction of their antecedents yields the disjunction of their consequents.

Rules of inference (8) MP p > q / p // q MT p > q / ~q // ~p HS p > q / q > r // p > r DS p v q / ~p // q SM p . q // p CN p / q // p. q AD p // p v q CD (p > q) . (r > s) / p v r // q v s

Rules of inference (mt, mp, ds, etc.) are “one way” rules. Rules of equivalence are “two way” rules, allowing substitution of a statement form for an equivalent statement form. Rules of equivalence are written using “::” to indicate two expressions are equivalent to one another.

Double Negation p :: ~ ~p Two tildes can be added or deleted from any statement with no effect on the truth-value. 1.A > ~ (B . C) 2. B . C ~ ~ ( H v K) ~ H 3. ~ ~ (B . C) 2 DN 4. ~A 1,3 MT 3. H v K 1 DN 4. K 2 ,3 DS

~ (p v q) :: ~ p . ~ q “neither” “not this and not that” ~ (p . q) :: ~ p v ~ q “not both” “either not this or else not that” DM DeMorgan’s Theorem

The order of statements around a dot or wedge is of no consequence CM Commutation (p . q) : : (q . p) (p v q) : : (q v p) The order of statements around a dot or wedge is of no consequence to the truth-value of the statement 1. (J > N) v (C v D) 2. ~C 3. ~D / J >N 4. ~ C . ~ D CONJ 2,3, 5. ~ (C v D) DM 4 6. J > N COMM, DS 1,5 6. (C v D) v (J >N) COMM, 1 7. J > N DS 5, 6

statements around dots and wedges is of no Association AS [ p . (q . r)] : : [ (p . q) . r] [ p v (q v r)] : : [ (p v q) v r] The grouping of simple statements around dots and wedges is of no consequence for truth-values 1. A . (B . C) / C 2. (A . B) . C AS 1 3. C . (A . B) CM 2 4. C SM 3