p implies q logic

https://cdn.discordapp.com/attachments/322819040059850757/375036647466860545/unknown.png

the entry on the table on the bottom left which states that p is false while q is true i don't understand why the statement p implies q is true, is true. I don't get it because if there's no info on q how can it be true? You can't know that p implies q is true automatically from the simple fact that no info is given on the truth of q "so thus it can't be false".

I don't get it or is the automatic default for any "unknown" "tied" propositions always true?

It's just like saying if i go for a run, then i do it on monday this is in the form

P\Rightarrow Q

because going for a run is a sufficient cause for doing it on monday however simply because it is monday does not mean that you will always go for a run

now p is going for a run and q is of course on a monday

so if p is false then you do not go for a run and if q is true then it's a monday

then if you do not go for a run then it's a monday

the truth of that statement that if you do not go for a run then it's monday can't be true because p is false. now i can understand int he table inthe link that if both a false such that the statement reads if you do not go for a run then it is not monday is defo true it makes sense and pretty much is true because if you are not running then it is not a monday. Bc running is a sufficient cause for it being a monday to if you reverse it and make it false then it makes sense.

However determining whether the statement that if you don't run then it's monday can't possibly be true as for the day to be a monday the sufficient reason is that you go for a run. So then if you don't run then it ain't a monday. So how can that statement be true?

I'm not sure i can think of any other way to put it but i can't seem to convince myself otherwise unless there's a fact that it's assumed that all "tied" statements that could be either true or false are all automatically true. It just doesn't make sense to me can someone explain?

Reply 1

artful_lounger

Universities Forum Helper

21

It literally says it below that table.

"While P is false, P => Q gives no information about the truth of Q so P => Q cannot be false".

You can't infer anything about Q, so the proposition can't be false - therefore it must be true. This comes from the section above, which defines in this particular case P as a sufficient condition but not a necessary one. So P being true is sufficient for the proposition to be true - so when P is true then the proposition is true; but it's not necessary - so then if P is false, the truth of the proposition only depends on Q.

Essentially, because it's only sufficient and not necessary, while P is true, P => Q is true. While P is false, P => Q is true, no matter the "value" of Q.

(edited 6 years ago)

Reply 2

will'o'wisp2

OP

15

Original post by artful_lounger

It literally says it below that table.

"While P is false, P => Q gives no information about the truth of Q so P => Q cannot be false".

You can't infer anything about Q, so the proposition can't be false - therefore it must be true. This comes from the section above, which defines in this particular case P as a sufficient condition but not a necessary one. So P being true is sufficient for the proposition to be true - so when P is true then the proposition is true; but it's not necessary - so then the truth of the proposition only depends on Q.

so then in my example as long as it is monday then you run?

anyway so p is sifficient for q but not necessary like you say so then q is the deciding factor as to whether the statement is true or not? so if q is true then no matter if p is true or not then the statement p implies q is true regardless?

Reply 3

simon0

13

Original post by will'o'wisp2

https://cdn.discordapp.com/attachments/322819040059850757/375036647466860545/unknown.png

the entry on the table on the bottom left which states that p is false while q is true i don't understand why the statement p implies q is true, is true. I don't get it because if there's no info on q how can it be true? You can't know that p implies q is true automatically from the simple fact that no info is given on the truth of q "so thus it can't be false".

I don't get it or is the automatic default for any "unknown" "tied" propositions always true?

It's just like saying if i go for a run, then i do it on monday this is in the form

P\Rightarrow Q

because going for a run is a sufficient cause for doing it on monday however simply because it is monday does not mean that you will always go for a run

now p is going for a run and q is of course on a monday

so if p is false then you do not go for a run and if q is true then it's a monday

then if you do not go for a run then it's a monday

the truth of that statement that if you do not go for a run then it's monday can't be true because p is false. now i can understand int he table inthe link that if both a false such that the statement reads if you do not go for a run then it is not monday is defo true it makes sense and pretty much is true because if you are not running then it is not a monday. Bc running is a sufficient cause for it being a monday to if you reverse it and make it false then it makes sense.

However determining whether the statement that if you don't run then it's monday can't possibly be true as for the day to be a monday the sufficient reason is that you go for a run. So then if you don't run then it ain't a monday. So how can that statement be true?

I'm not sure i can think of any other way to put it but i can't seem to convince myself otherwise unless there's a fact that it's assumed that all "tied" statements that could be either true or false are all automatically true. It just doesn't make sense to me can someone explain?

The way I saw the material conditional symbol is that:

A \Rightarrow B,

I saw this as in effect one whole statement "If A happend, then B happens".
Meaning if A happens, then B has to follow.

e.g. "If it rains"

\Rightarrow

"The grass is wet".
Meaning if it rains, then the "grass being wet" has to happen.
If it does rain AND after the grass is wet, then this satisfies the logical statement.

--------------------------------------------------------------

If A does not happen, then we do not care (and

A \Rightarrow B

is vacously true.)

We only care about the sequence of events, that if A happens, then B happens as well.

If that is the case, then statement

A \Rightarrow B

is satisfied.
(Also means that A is a sufficient condition for B).

I hope this clears it up a little. If there is an error then tell me. :-)

(edited 6 years ago)

Reply 4

artful_lounger

Universities Forum Helper

21

Original post by will'o'wisp2

so then in my example as long as it is monday then you run?

anyway so p is sifficient for q but not necessary like you say so then q is the deciding factor as to whether the statement is true or not? so if q is true then no matter if p is true or not then the statement p implies q is true regardless?

For the second part, yes. You're essentially assuming the conclusion then.

It's easier to think about the other way around - there are four "states" for it to be in, for each combination of true and false. We know P=>Q, so the only way that would NOT be true (if P is sufficient but not necessary), is if Q is false. Anything else goes.

This is because for the case where P is true but Q is false - the proposition (P=>Q) is telling us that we have the sufficient condition for Q to be true - that P is true - but for some reason Q is false. This contradicts the proposition and so the proposition is false in that case.

Your conclusion of your worded statement is correct, from what I can tell. It's phrased really confusingly and it's kind of the reason I dislike wordy problems in logic, because it just obscures the actual simplicity of it.

p implies q logic

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