Vocab
A diagram that shows a set of propositions being decomposed into their literals.
a Truth Tree
p ⊃ q
p
∴ q
Modus Ponens
Justify each step:
1) P ⊃ Q
2) R ⊃ ~Q / ∴ P ⊃ ~R
3) ~~Q ⊃ ~R
4) Q ⊃ ~R
5) P ⊃ ~R
2 Trans.
3 D.N.
1, 4 H.S.
How many techniques in lesson 24 are there?
Four!!
A path on a truth tree for which a contradiction has been found.
Closed Branch
p ⊃ q
q ⊃ r
∴ p ⊃ r
Hypothetical Syllogism
Justify each step
1) P ⊃ ~Q / ∴ ~Q v (Q & ~P)
2) ~~Q ⊃ ~P
3) Q ⊃ ~P
4) Q ⊃ (Q & ~P)
5) ~Q v (Q & ~P)
1 Trans
2 D.N.
3 Abs.
4 Impl.
What is the third technique for constructing truth trees?
Stop when the truth tree answers the question being asked.
A path on a truth tree which includes no contradictions.
Open branch
p ∨ q
~p
∴ q
Disjunctive Syllogism
Determine the validity of this argument
(p ⊃ q) & [(p & q) ⊃ r] p ⊃ (r ⊃ s) ∴ p ⊃ s
Valid
This technique will keep you from unnecessarily rewriting propositions.
Decompose non-branching members first
Simple propositions or the negation of simple propositions.
Literals
(p ⊃ q) = (~q ⊃ p)
Determine the validity of this argument
p ⊃ (q ⊃ r) q ⊃ (p ⊃ r) ∴ (p v q) ⊃ r
Valid
Using this technique will help you simplify your truth tree, because it allowed you to close branches sooner.
Decompose members which result in the closing of one or more branches.
This phrase means to determine the truth values of the simple propositions for which the propositions in the set would all be true.
Recover the truth values
~(p & q) = (~p v ~q)
~(p v q) = (~q & ~p)
De Morgan's Theorems
{ ~(J=K), K ⊃ (~L v M), L & ~(K & M) }
Give yourself plenty of room! Good luck 😏
just kidding
This technique should only be used when the other rules don't apply...
Decompose more complex propositions first.