Logic Help
Deadline for Week IV: Tuesday, January 23rd.
Be sure to email me your work by Tuesday midnight.
My email: kamillasmith27@gmail.com
Week IV
Day 14.
Exercise 1.
Derive conclusions using MP, MT, HS, DS, DM, DN (double negation), and Simp (Simplification). Do not forget to write what method you are using and from which lines you obtained given letters. Do one step at a time.
Remember, the arrow ( ->) indicates a conditional sentence (If…., then…..).
a)
1. – L -> - S
2. – L
___________
3.
b)
1. – S -> ( - A -> - Z)
2. – S
____________
3.
c)
1. – S -> ( - C -> - B)
2. – S
3. – C
____________
4.
5.
d)
1. – A -> ( - C -> - L)
2. – A
3. L
____________
4.
5.
e)
1. – ( B v Q )
2. – B - > A
_____________
5. (*hint - use here the method of DM, DeMorgan’s Law)
6.
7.
8.
f)
1. B v ( E -> P)
2. – ( B v Q )
3. E
_____________
4. (* hint – consider first converting line 2 by DM)
5. (* consider deriving a single letter from line 4 by simplification)
6.
7.
g)
1. B v ( E -> P)
2. – ( B v Q )
3. – P
4. – E - > A
_____________
5. (* hint – consider first converting line 2 by DM)
6.
7.
8.
9.
h)
1. – P - > Q
2. – P & - B
3. Q -> ( A v B )
______________
4. (* hint – first consider simplification of line 2)
5.
6.
i) * Construct and solve your own problem. Try to make it sufficiently complex by using the method of MP, MT,
DM, and Simplification.
Exercise 2.
a) Symbolize the following sentences. Use the letter of each word marked in red to symbolize the propositions. (To give you an example, the first sentence is already symbolized.)
1.
If the US pulls out of Afghanistan then either the British will have to commit troops for the long term or the Canadians will.
U - > ( B v C )
2.
If the British do make such a commitment, the US will not pull out.
3.
But if it is put to a Vote of Canadian public opinion, the Canadians will not commit troops to Afghanistan for the long term.
b) Symbolize the 3 sentences in the argument below.
1. Either she traveled at her OWN expense or it was paid for by Canadian TAXPAYERS.
2. If she had gone at her OWN expense, then she would have flown in a COMERCIAN jet.
3. But she flew in a GOVERNMENT yet.
c) Now, read the sentences again, including the added 4th sentence. Altogether, there are 4 premises and one conclusion (the conclusion is hinted with the number 5 in blue.)
1.Either she traveled at her OWN expense or it was paid for by Canadian TAXPAYERS.
2. If she had gone at her OWN expense, then she would have flown in a COMERCIAN jet.
3. But she flew in a GOVERNMENT yet.
4. Clearly, if she did that she did not fly in a COMMERCIAL jet, ( 5) so we can conclude that her travel was paid for by Canadian TAXPAYERS.
Translate now the entire story into symbols. You should have 4 premises and 1 conclusion.
1.
2.
3.
4.
________
5.
* (Again, be careful in symbolizing sentence 4. What is here the “that” in “If she did that”? )
Now that you turned the entire argument into symbols, notice that the conclusion was the letter T, which stands here for “Her travel was paid by Canadian taxpayers.” But how was this conclusion obtained? It is a logical conclusion?
Apply the methods of valid inference we learned so far and check whether the conclusion indeed follows the premises. This means you have to write the symbolic premises again and “derive” a conclusion (solve the problem) according to the rule of logic.
1.
2.
3.
4.
_____________
5.
6.
7.
___________________________________________________________________________________
Disjunction
* (Do not confuse with Disjunctive Syllogism)
Today, in our last day, we will add one more method to our rules of valid inference.
This rule is called Disjunction, abbreviated Disj.
This is not the same as DS, which stands for Disjunctive Syllogism.
This rule means that from a simple premise such as “I am a human being” we can derive a conclusion that “I am a human being or a bird.”
Why would I want to derive such a useless and perhaps a silly conclusion?
Well, because this addition will not change the truth of my statement and it will prove useful. So:
From: H
I can derive H v B
From: P
I can derive P v Q
From: A
I can derive A v B
And so on….
The premise H, in the first example, is true. It states that “I am a human being,” which is true.
Now, I am of course not a bird, but the sentence “I am a human being or a bird” remains still true even if it is not true that I am a bird. So technically I can create a conjunction from any single premise and that will not affect my true statement.
Moreover, my statement will remain true, even if I add the negation of my original statement.
For example, H v - H I am a human being or I am not a human being.
Well, this is true. I am or I am not a human being.
Likewise, in the sentences:
I am alive or I am not alive. A v – A
Even though is it not true that I am not alive, since I am alive and not dead, the sentence remains true. It is true that “I am alive or I am not alive.” After all, I am either alive or dead.
Similarly:
God exists or God does not exist. G v - G
But if I say “God exists and God does not exist” at the same time, then this will affect the truth of my first claim because I will create a contradiction.
Likewise:
I am a human being and I am not a human being.
H & - H
This is a conjunction. Here, clearly I cannot add the negation of H.
But if I create a disjunction from the H:
H v - H I will preserve the truth of the claim that I am a human being.
Here are several examples of valid conclusions derived by means of Disjunction (Disj).
a)
1. C
_______
2. C v B 1, Disj (I derived C v B from line 1 by Disjunction.)
b)
1. A
_______
2. A v P 1, Disj
c)
1. P v Q
________
2. ( P v Q ) v G 1, Disj
d)
1. A
_______
2. A v ( B & C) 1, Disj
e)
1. M & N
_______
2. ( M & N ) v A 1, Disj
Now your turn.
Create your own imaginative and original examples of disjunction derived from a single premise. Create 2 examples.
_________________________________________
Exercise 3
Finally, we will “prove” symbolic arguments without using full English sentences.
Suppose we have to “prove” the validity of the following abstract argument:
F B, (F B) H / H
This argument has 2 premises. The first premise is in blue, the second premise is in green. The conclusion is in red.
1. F B Premise 1
2. (FB) H Premise 2
The conclusion is simply H.
To prove that the argument is valid we will use the method of MP, Modus Ponens.
1. F B Premise 1
2. (FB) H Premise 2
________________________
3. H 1, 2 MP
One more example:
Prove the validity of this argument:
(B & G) (Q v R), - A (B & G), - A / Q v R
First, I have to number of all my premises. There are clearly 3 premises (separated by commas). The first one in orange, the second is in green, the third is in blue.
1. (B & G) (Q v R) Premise 1
2. - A (B & G), Premise 2
3. - A Premise 3
________
What do I do next? I will write in line 4 what I can obtain from the premises. I will start with something simple. How about line 3?
Given that it is the case that – A, what can I obtain from this line and line 2? I can obtain – (B & G) by Modus Ponens, like this:
1. (B & G) (Q v R) Premise 1
2. - A (B & G), Premise 2
3. - A Premise 3
________
4. ( B & G ) 2, 3 MP
But the conclusion in this argument is Q v R , so I am clearly not done. I have to do something else to end with the conclusion
Q v R.
So, I will now look at premise 1. Premise 1 states that if it is the case that – (B & G) , then it is the case that Q v R . In line 5, therefore, I can conclude by MP, that it is Q v R. Like this:
1. (B & G) (Q v R) Premise 1
2. - A (B & G), Premise 2
3. - A Premise 3
________
4. – ( B & G ) 2, 3 MP
5. Q v R 1, 4 MP
It turns out that my conclusion here Q v R is exactly the same as the conclusion given in the argument Q v R. So I just proved that this argument:
(B & G) (Q v R), - A (B & G), - A / Q v R
is indeed valid.
One more practice example:
Prove:
( M v W) & P, M Y , - ( Y & P) / W v X
(First number the premises. Which letter is the conclusion that you have to prove? It is the disjunction W v X)
1. ( M v W) & P Premise 1
2. M Y Premise 2
3. - ( Y & P ) Premise 3
__________________
4. – Y v – P 3, DM
5. P 1, Simpl
6. – Y 4, 5 DS
7. – M 2, 6 MT
8. M v W 1, Simpl of conjunction
9. W 7, 8 DS
10. W v X 9, Disj
Now your turn.
Prove that the following arguments are valid.
a)
A, A -> X , X -> C / C v D
(* start by listing and numbering the 3 premises)
(* Remember also that you can create a disjunction from any letter (premise). If you concluded C, that’s good. But you need the conclusion to be C v D. )
b)
– ( – H v A ) / H v M
( * Notice that here you only have one premise. What can you do with it? Consider applying the method of DM)
c) – C, (R & S ) C / - R v – S
(*Notice, you have here only 2 premises. List the premises. Then consider using the method of MT first.)
________________________________________________________________________________________
You have now completed all work for this week. Please submit your work by midnight, Thursday, June 28
to kamillasmith27@gmail.com
You are done with the Intro to Logic course.
Students at A+ level, please consider solving the problems below.
Prove that the following arguments are valid.
a) – ( - A & - Y) , - (Y v C), A -> D / D
b) O v X , O C , G , G - C / X
c) - A / - (A & C )
d) D v – E, A E, - D / - ( A & B )