Question 1201495: For each of the following lists of premises, derive the indicated conclusion and complete the justification. In problems 4 and 8 you can add any statement you choose.
(1) 1. S ∨ H 2. B • E 3. R ⊃ G 4. _____ ____, Simp
(2) 1. (N ⊃ T) • (F ⊃ Q) 2. (N ⊃ R) ∨ (F ⊃ M) 3. N ∨ F 4. _______________ ____, CD
(3) 1. D 2. W 3. ____ ____, Conj
(4) 1. H 2. ____ ____, Add
(5) 1. R • (N ∨ K) 2. (G • T) ∨ S 3. (Q • C) ⊃ (J • L) 4. _____________ ____, Simp
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
I'll just do the first 2
"Simp" = simplification means (p • q) ⊃ p and if you can skip
"commutation", then (p • q) ⊃ q, also.
Think of ⊃ as the same as the word "guarantees".
"Simp" says if you have the 1st and the 2nd, then that guarantees that you have
the 1st ---
PLUS ---
if you can skip commutation, (swapping p with q), then it also guarantees that
you have the 2nd as well. [Plain old common sense! "If you've got both, you've either one, separately".
(1)
1. S ∨ H
2. B • E
3. R ⊃ G
4. B 2, simp.
5. E 2, simp.
So if B is the 1st and E is the 2nd, then if you have
the 1st and the 2nd, that guarantees that you have the 1st.
It also guarantees that you have the 2nd. [plain old common sense!]
(2)
"CD" = constructive dilemma means
[(p ⊃ q) • (r ⊃ s) • (p ∨ r)] ⊃ (q ∨ s)
Think of ⊃ as the same as the word "guarantees".
When you know that
(the 1st guarantees the 2nd) and (the 3rd guarantees the 4th) and
you have (the 1st OR the 3rd), then that guarantees that you have either (the
2nd or the 4th).
So below from 1 you have (N guarantees T) and (F guarantees Q), and from 3 you
have (N or F), so that guarantees that you must have either T or Q, by
constructive dilemma, CD.
1. (N ⊃ T) • (F ⊃ Q)
2. (N ⊃ R) ∨ (F ⊃ M)
3. N ∨ F
4. T V Q , 1,2, CD
Logic is all common sense if you know and think about what the symbols mean.
Edwin
|
|
|
