When is If P then Q equivalent to Not the case that P or Q?

[tdottoker]

Been at this all day and cannot come up with an instance where in english sentences, If P then Q is equivalent to It is not the case that P or Q. Can anyone provides some hints.. I am really stuck. Thanks!

 

[Stephen Tashi]

If P then Q is equivalent to It is not the case that P or Q.

Do you mean "it is not the case that ( P or Q) " or do you mean " (it is not the case that P) or Q". In other words, is it "not (P or Q)" or is it "(not P) or Q" ?

If it is "(not P) or Q" then any English statements should do for P and Q.

If it is "not (P or Q" then I'm not sure what is expected of you. Technically, two true statements are equivalent and also two false statements are equivalent. So you could make both "If P then Q" and "not (P or Q)" true or make both false. If you make P a false statement then you make both those two statements true. If you make P true and Q false then you make both those statements false.

 

[tdottoker]

Hi Stephen,

Thanks for the reply. I was actually trying to figure this out on a truth table and realized what your saying. Now I am even more confused! The question simply states that "It is not the case that Q or P". Does this mean not q or p (as in only one must be false) or not (q or p) or (not p) or q? The question says it is an inclusive disjunction. Any advice? Thanks.

 

[Stephen Tashi]

The question says it is an inclusive disjunction

To my mind, that doesn't clear up the ambiguity. You should follow the format for homework questions and state the (entire) problem. If you are asked to use English sentences for P and Q that have obvious truth values, I think the problem is to find particular instances of P and Q that make the two statements either both true or both false.

 

[tdottoker]

Hi Stephen,

Thanks again for the help. Here is the entire question:

Consider the following sentences forms: “If P then Q” and “It is not the
case that Q or P.” Assume the first is a material conditional and the second
is contains an inclusive disjunction.
(a) Using a truth-table show that for arbitrary P and Q the above sentences
are logically inequivalent. Be sure to explain what it is about the truthtable that shows they are inequivalent.
(b) In some special cases sentences of the above forms are logically equivalent. Give an example of this: specify some sentence P and some
sentence Q that, when substituted into the above forms, yield logically
equivalent sentences. Briefly explain your answer. Remember that the
sentences you specify for P and Q should be grammatical sentences of
English.

I am assuming the teacher means Not (P or Q). So you are saying the equivalent will come from the actual wording of the sentences? I tried heuristically just throwing in several types of statements at it and then putting their output in tables. Nothing I can think of makes them equivalents.

 

[HallsofIvy]

The statement "if P then Q" is the same as "(not P) or Q" for all truth values of P and Q.

The OP is looking for English language examples of those that make it clear that those are the same.

Try "if rain is predicted for today, I will carry my umbrella".

Is that the same as "either rain is not predicted today or I will carry my umbrella"?

Look at all the possiblities. If rain is predicted today and I take my umbrella, both are true. If rain is NOT predicted today and I take my umbrella, both are true. Ir rain is predicted today and I do not take my umbrella, both are false. If rain is NOT predicted todah and I do not take my umbrella, both are true.

 

[tdottoker]

Hi Halls,

Thanks for the help. So you are saying "It is not the case that Q or P" is "(not P) or Q"?

In a different section, my teacher used "It is not the case that" as a substitution for

Not [ ]

 

[Stephen Tashi]

(a) Using a truth-table show that for arbitrary P and Q the above sentences
are logically inequivalent.

That tells us that the two statements must be "If P then Q" and "not (P or Q)".

If the second statement were "(not P) or Q" then there woud be no line the truth table where their truth values would differ.

(b) In some special cases sentences of the above forms are logically equivalent. Give an example of this:.

I don't like the way that your course materials are using the phrase "logically equivalent" in two (apparently) subtly different contexts. It says the "sentences" are not logically equivalent but "in some cases" the "forms" are logically equivalent.

Anyway, the task in (b) is to find a line of the truth table where the two statements differ and then give specific examples for the statements P and Q that satisfy the truth values you need in that line.

 

[tdottoker]

Hi Stephen,

Thanks again. So for part b) I still cannot think of anything in regard to your new insight.
For example;

P: 2 + 2 = 4
Q: I am a pig

"If 2 + 2 = 4, then I am a pig".

Does this work?

 

[Stephen Tashi]

P: 2 + 2 = 4
Q: I am a pig

"If 2 + 2 = 4, then I am a pig".

Does this work?

For (b)? Assuming Q is false it doesn't. Both the statements "If P then Q" and "not(P or Q)" are false in the case when P is true and Q is false.

Did you fill out the truth table yet? Look on a line where the two statements don't have the same truth value. For example, make both P and Q true.

 

[tdottoker]

Hi Stephen,

Sorry, so for b) it is only asking for the instance where both If P then Q and Not (Q or P) are false or a case when both are true (logically equivalent?)? Is it simply just asking me to substitue sentences for an instance where both P and Q are false?

1= true
0= false

P Q
1 0
0 1
1 1
0 0

If P then Q
0
1
1
1

Not (P or Q)
0
0
0
1

 

[Stephen Tashi]

for b) it is only asking for the instance where both If P then Q and Not (Q or P) are false or a case when both are true (logically equivalent?)?

Yes. My previous post was incorrect. Your example did work. I was still thinking about making the two statements inequivalent. Your example illustrates the 1st line the table where the two "forms" are equivalent.

 

[tdottoker]

Thank you. Well see what my teacher says!