REASONING A
TRUE OR FALSE
One proposition may be expressed by many different sentences.
TRUE
Example:
God loves the world.
The world is loved by God.
Deus mundum amat.
An uppercase letter that represents a single, given proposition.
A lowercase letter that represents any proposition.
What is a propositional constant and a propositional variable.
A proposition that has only one component part.
What is a simple proposition.
Example:
The truck is green.
John is a male.
Given that:
B means The boys are bad.
M means The man is mad.
G means The girls are glad.
S means The students are sad.
Translate: If not M and not S then G.
If the man is not mad and the students are not sad, then the girls are glad.
A proposition is ________ - ____________ when its truth value depends upon the truth values of its component parts.
What is truth-functional.
Lesson 1 (page 9)
A proposition that is always TRUE due to its logical structure.
What is a tautology.
Example:
B ~B ~~B B ≡ ~~B
A valid argument form which can be used to justify steps in a proof.
What is rule of inference.
Words that combine or modify simple propositions in order to form compound propositions.
What are logical operators.
Examples:
"and", "but" ∙
"or" ∨
Symbolize each of the following propositions on your whiteboards.
1. "Forest rangers and grizzly bears are not both friendly creatures."
2. "Both forest rangers and grizzly bears are not friendly creatures."
1. ~(F ∙ G)
2. (~F ∙ ~G)
In (1.) the "not" comes before the "both".
In (2.) the "both" comes before the "not".
A listing of the possible truth values for a set of one or more propositions.
What is a truth table.
A proposition that has more than one component part or is modified in some other way.
What is a compound proposition.
Example:
Dave is a male and he is tall.
Three fundamental logical operators.
What is negation (~), conjunction (∙), and disjunction (∨).
A proposition that is false by logical structure.
What is self-contradiction.
Name the nine rules of inference from memory.
What is...
1. Modus Ponens (M.P.)
2. Modus Tollens (M.T.)
3. Hypothetical Syllogism (H.S.)
4. Disjunctive Syllogism (D.S.)
5. Constructive Dilemma (C.D.)
6. Absorption (Abs.)
7. Simplification (Simp.)
8. Addition (Add.)
9. Conjunction (Conj.)
On your whiteboard write out a complete truth table for the following compound proposition.
~p ∨ (~q ∙ r)
p q r ~p ~q (~q ∙ r) ~p ∨ (~q ∙ r)
F
F
T
F
T
T
T
T
TRUE OR FALSE?
If either disjunct is true, the disjunction as a whole is true.
What is TRUE.
The following propositions are examples of which logical operator?
1. Fido is a dog implies that Fido is a mammal.
2. When you finish your dinner I will give you dessert.
3. Cheating during a test is a sufficient condition for your suspension.
What is a conditional (⊃)
(Also called hypothetical or material implication)
Given:
J means Joseph went to Egypt.
I means Israel went to Egypt.
F means There was a famine.
S means The sons of Israel became slaves.
Symbolize the following:
"Israel went to Egypt, but either Joseph did not go to Egypt, or there was a famine."
I ∙ (~J ∨ F)
TRUE OR FALSE?
If the antecedent is FALSE and the consequent is FALSE the whole proposition is TRUE.
What is TRUE.
Example:
"If a poodle is a tiger, then a poodle is a feline.
F F
Though the component parts are both FALSE the argument is nonetheless TRUE.
A valid argument which presents a choice between two conditionals.
What is a dilemma.
TRUE or FALSE?
Variables in the rules of inference can represent very complicated compound propositions.
What is TRUE.
Example:
[(F ⊃ ~C) ∨ (Q ≡ X)] ⊃ (~E ∨ J)
[(F ⊃ ~C) ∨ (Q ≡ X)] / ∴ ~E ∨ J
IS...
Modus ponens
p ⊃ q
p / ∴ q
Lesson 14 (page 109)
Use shorter truth-table to determine validity.
"If I teach you what you know, then you learn nothing new. If I teach you what you do not know, then you cannot understand. I either teach you what you know or what you don't know. Therefore, you either learn nothing new or you cannot understand." (K, N, U)
What is VALID.
K ⊃ N ~K ⊃ U K ∨ ~K ∴ N ∨ U
The contradiction is under first conditional - should be F.
Remember, assume all premises to be T and the conclusion F, if there is a contradiction the argument is VALID.
Use the shorter truth-table method to determine the validity of the propositional argument.
p ⊃ q p ∨ r q ⊃ ~r ∴ ~(p ≡ r)
What is VALID.
See Lesson 9
In this problem, two guesses are required.
1. Assume p in the conclusion is T.
2. Assume p in the conclusion is F.
Both will lead to a contradiction.
Use a shorter truth table to determine if the propositions are EQUIVALENT.
p ⊃ ~q ~p ∙ ~q
What is NO.
Remember, we assume the 1st premise is T and the 2nd F, IF we can work through with no contradiction they are indeed NOT EQUIVALENT.
Determine the CONSISTENCY of the set of propositions.
"Either Lefty or Capone is responsible for the fire. If Capone is responsible for the fire, then he knew about the dangerous conditions but did nothing about them. Capone knew about the dangerous conditions only if he did something about them. Lefty is not responsible for the fire." (L, C, K, D)
What is INCONSISTENT.
L ∨ C C ⊃ (K ∙ ~D) K ⊃ D ~L
The contradiction is in the 3rd premise "K ⊃ D".
This is customary to END a proof with.
What is Q.E.D.
quod erat demonstrandum = What was to be demonstrated.
Determine which rule of inference is used in the argument.
"If you love God then you love your neighbor. Judas did not love his neighbor. It is obvious that Judas did not love God."
What is M.T.
Provide the justification for each step of the proof.
1. E ⊃ F
2. G ⊃ H
3. E ∨ G
4. ~F / ∴ H
5. (E ⊃ F) ∙ (G ⊃ H) _________________
6. F ∨ H ____________________
7. H. __________________
5. 1, 2 Conj.
6. 5, 3 C.D.
7. 6, 4 D.S.
Translate the following argument into symbolic form. Then, determine it's validity.
"I am a member of a club only if the club is willing to have me and I join it. I do not join any club if the club is willing to have me. So I am not a member of a club." (M, W, J)
What is VALID.
M ⊃ (W ∙ J) W ⊃ ~J ∴ ~M
Write a formal proof of validity for the given argument.
1. (R ⊃ S) ∙ (T ⊃ U)
2. V ∨ ~S
3. ~V / ∴ ~R
4. R ⊃ S ________________
5. ~S _______________
6. ~R _______________
4. 1 Simp.
5. 2, 3 D.S.
6. 4, 5 M.T.
Justify each step...
1. (P ∙ Q) ⊃ R
2. ~R / ∴ P ⊃ ~Q
3. ______________ 1, 2 M.T.
4. ______________ 3 DeM.
5. ______________ 4 (Material Implication) Impl.
3. ~(P ∙ Q)
4. ~P ∨ ~Q
5. P ⊃ ~Q
Using the Rules of inference and the Rules of replacement complete the following formal proof.
1) (P ∨ Q) ⊃ [R ∙ (S ∙ T)]
2) Q / ∴ R ∙ S
3) Q ∨ P _________________
4) P ∨ Q _________________
5) R ∙ (S ∙ T) __________________
6) (R ∙ S) ∙ T ___________________
7) R ∙ S ___________________
3) 2, Add.
4) 3, Com.
5) 1, 4 M.P.
6) 5 Assoc.
7) 6 Simplification
Justify each step in the proof using the rules of inference or replacement.
1) A ∙ ~B / ∴ B ⊃ C
2) ~B ∙ A ______________
3) ~B ________________
4) ~B ∨ C ________________
5) B ⊃ C __________________
2) 1 Com.
3) 2 Simp.
4) 3 Add.
5) 4 Impl.
Justify each step using Rules of inference.
1) X ⊃ Y
2) W ⊃ Z
3) X ∨ W / ∴ Y ∨ Z
4) ______________ 1, 2 Conj.
5) Y ∨ Z _________________
4) (X ⊃ Y) ∙ (W ⊃ Z)
5) 4, 3 C.D.
Justify each step using the Rules of Inference.
1) F ⊃ G
2) (F ⊃ H) ⊃ (I ∨ G)
3) (F ∙ G) ⊃ H
4) ~I / ∴ G
5) _______________ 1, Abs.
6) F ⊃ H _________________
7) ________________ 2, 6 M.P.
8) G __________________
5) F ⊃ (F ∙ G)
6) 5, 3 H.S.
7) I ∨ G
8) 7, 4 D.S.