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Lecture 9

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**preview**shows half of the first page. to view the full**2 pages of the document.**PHIL2110 – Elementary Symbolic Logic

2016/02/11

Lecture –Disjunction continued

Disjunction (Disj): From any statement, infer the disjunction with another statement (can

add whatever statement you want to help you solve the problem)

oIn symbols: from p (stated alone), infer p V q. From q (stated alone), infer p V q

oValidity: one cannot assert p to be true and at the same time deny that either p or q is

true; similarly, one cannot assert q to be true and at the same time deny that either p

or q is true

oP V q is a weaker statement than p so we wouldn’t normally infer it. But we need the

rule for our formal proofs – so to reason from p V q to p or q is invalid!

oExample:

1. E

2. (EvF)→¬G

3. H→G

4. E v F: 1 Disj

5. ¬G: 2,4 MP

6. ¬H: 3,5 MT

∴E&¬H: 1,6 Conj

Disjunctive syllogism involves v and ¬ so it is analogous to conjunctive syllogism which

involves & and ¬

oDS says: either one or the other, not one, therefore the other

oCS says: not both, one, therefore not the other

oDisjunction is the counterpart of conjunction. But remember you only need one of the

disjuncts to apply Disj!

oThere is no counterpart for simplification

2016/02/23

Lecture – Disjunction part 3

De Morgan’s Laws (DM):

o‘not either’ is equivalent to ‘neither’

o‘not both’ is equivalent to ‘either one or not the other’

oIn symbols: from ¬(pvq) infer ¬p & ¬q (and vice versa); from ¬(p&q) infer ¬p v¬q

(and vice versa)

oProof of formal validity will come later

oP V q is a weaker statement than p so we would not normally infer it, but we need it

for our formal proofs

Example:

o1. (A v B) → C Prem

o2. ¬C v D Prem

o3. ¬D Prem

o4. ¬C 2,3 DS

o5. ¬(A v B) 1,4 MT

o6. ¬A & ¬B 5 DM

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