## "If P implies Q, and Q is false, then P is also false."

What is the basis for reason? And mathematics?

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ReliStuPhD
Posts: 627
Joined: Sat Jan 24, 2015 5:28 pm

### "If P implies Q, and Q is false, then P is also false."

Ran across this today in some podcast-listening, and I have to confess it's beyond me to get at the truth or falsity of this statement on strictly analytical grounds (I'm leaning toward "false"). Anyone care to weigh in on this?
Immanuel Can
Posts: 8766
Joined: Wed Sep 25, 2013 4:42 pm

### Re: "If P implies Q, and Q is false, then P is also false."

Not necessarily true.

Let P be "rain" and Q be "wet."

If it rains, it implies that the ground will be wet.
But the ground is not wet.
Therefore, it has not rained.

Wrong. I bought an awning or umbrella, or the rain fell as virga (rain which does not reach the ground).

Only if the relationship between P and Q is invariable and exclusive can the falsification of Q allow us to deduce the falsehood of P.

So:
Let P be "bachelor" and Q "single" (i.e. unmarried).

If you are a bachelor, you are single.
But you are not single.
Therefore, you are not a bachelor.
thedoc
Posts: 6473
Joined: Thu Aug 30, 2012 4:18 pm

### Re: "If P implies Q, and Q is false, then P is also false."

"Implies" is a rather tenuous link and leaves a lot of possibilities for P to be true or false. It might simply be a case of insufficient data.
Melchior
Posts: 839
Joined: Mon Apr 28, 2014 3:20 pm

### Re: "If P implies Q, and Q is false, then P is also false."

ReliStuPhD wrote:Ran across this today in some podcast-listening, and I have to confess it's beyond me to get at the truth or falsity of this statement on strictly analytical grounds (I'm leaning toward "false"). Anyone care to weigh in on this?
The syllogism is valid.

If P then Q
Not Q
Thus: Not P

Why? If there had been a P, there would have been a Q! Since there is no Q, there cannot have been a P!

If it rains, the streets will be wet
The streets are dry
Therefore it has not rained.
Wyman
Posts: 974
Joined: Sat Jan 04, 2014 2:21 pm

### Re: "If P implies Q, and Q is false, then P is also false."

In modern logic the material implication is only false when the premise is true and the consequent is false. This leads to some counter-intuitive results, such as that anything follows from a false premise -

'If 2 + 2 =5 then New York City is in France' is true. But it is more intuitive in the sense of 'If you're correct, I'll be a monkey's uncle.'

Also, there is sometimes no connection between premise and consequent when the premise is true and the consequent is also true, such as 'If 2 + 2 =4, then New York City is in America.'

In your example, since 'If P then Q is true' (your premise) then one of three truth tables is possible -1) P is false and Q is true, 2) P is true and Q is true or 3) P is false and Q is false. Your supposition is that Q is false, so the only way the premise could be true is option 3). In which case, P is false.

The examples of If it is raining, then it is wet being equivalent to If it is not wet, then it is not raining is an example of the converse of an implication always being true - which is a logical truth, but not the same as your OP.
ReliStuPhD
Posts: 627
Joined: Sat Jan 24, 2015 5:28 pm

### Re: "If P implies Q, and Q is false, then P is also false."

Immanuel Can wrote:Not necessarily true.

Let P be "rain" and Q be "wet."

If it rains, it implies that the ground will be wet.
But the ground is not wet.
Therefore, it has not rained.

Wrong. I bought an awning or umbrella, or the rain fell as virga (rain which does not reach the ground).

Only if the relationship between P and Q is invariable and exclusive can the falsification of Q allow us to deduce the falsehood of P.

So:
Let P be "bachelor" and Q "single" (i.e. unmarried).

If you are a bachelor, you are single.
But you are not single.
Therefore, you are not a bachelor.
Actually, your response (and the others) got my brain working, so I looked up "implies" when used in logical statements and got Wikipedia: http://en.wikipedia.org/wiki/List_of_logic_symbols
Now, I am certainly aware that Wikipedia is not the end-all-be-all, but it looks like it's using "implies" as a synonym for "if ... then." If I'm reading that right (and if Wikipedia is not in error), then "If P implies Q, and Q is false, then P is also false" is just another way of saying "If p then Q. If not Q, then not P," no?

(And thanks, Wyman, for laying out the implications. )
Wyman
Posts: 974
Joined: Sat Jan 04, 2014 2:21 pm

### Re: "If P implies Q, and Q is false, then P is also false."

ReliStuPhD wrote:
Immanuel Can wrote:Not necessarily true.

Let P be "rain" and Q be "wet."

If it rains, it implies that the ground will be wet.
But the ground is not wet.
Therefore, it has not rained.

Wrong. I bought an awning or umbrella, or the rain fell as virga (rain which does not reach the ground).

Only if the relationship between P and Q is invariable and exclusive can the falsification of Q allow us to deduce the falsehood of P.

So:
Let P be "bachelor" and Q "single" (i.e. unmarried).

If you are a bachelor, you are single.
But you are not single.
Therefore, you are not a bachelor.
Actually, your response (and the others) got my brain working, so I looked up "implies" when used in logical statements and got Wikipedia: http://en.wikipedia.org/wiki/List_of_logic_symbols
Now, I am certainly aware that Wikipedia is not the end-all-be-all, but it looks like it's using "implies" as a synonym for "if ... then." If I'm reading that right (and if Wikipedia is not in error), then "If P implies Q, and Q is false, then P is also false" is just another way of saying "If p then Q. If not Q, then not P," no?

(And thanks, Wyman, for laying out the implications. )
Yes, in the sense that '2 squared + an even integer = an even number' is another way of saying 'for any integer x, x squared + an even integer = an even number.' An instantiation of a general rule.

I think - ask Arising if I'm correct.
Owen
Posts: 25
Joined: Tue Apr 23, 2013 11:21 am

### Re: "If P implies Q, and Q is false, then P is also false."

ReliStuPhD wrote:Ran across this today in some podcast-listening, and I have to confess it's beyond me to get at the truth or falsity of this statement on strictly analytical grounds (I'm leaning toward "false"). Anyone care to weigh in on this?
((p -> q) & ~q) -> ~p is a tautology.
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..f..t..t..f..f...t...t
..t..f..f..f..t...t...f
..f..t..f..t..t...t...t
HexHammer
Posts: 3346
Joined: Sat May 14, 2011 8:19 pm
Location: Denmark

### Re: "If P implies Q, and Q is false, then P is also false."

This is excatly why programmers has difficulties to program human minds, because it can't take account for fraud or subjectiveness, and "rain men" has such difficulties to understand reality.
Systematic
Posts: 339
Joined: Wed Feb 04, 2015 5:29 am

### Re: "If P implies Q, and Q is false, then P is also false."

ReliStuPhD wrote:Ran across this today in some podcast-listening, and I have to confess it's beyond me to get at the truth or falsity of this statement on strictly analytical grounds (I'm leaning toward "false"). Anyone care to weigh in on this?
We know that Q false means P false if P ONLY IMPLIES Q.

If P implies Q or R and if Q is false, then we don't yet know whether P is false.
David Handeye
Posts: 459
Joined: Wed Mar 04, 2015 6:39 pm
Location: Italia

### Re: "If P implies Q, and Q is false, then P is also false."

Systematic wrote:We know that Q false means P false if P ONLY IMPLIES Q. If P implies Q or R and if Q is false, then we don't yet know whether P is false.
what the f... are you saying? Where did that R come from? Are you joking, say the truth...
Systematic
Posts: 339
Joined: Wed Feb 04, 2015 5:29 am

### Re: "If P implies Q, and Q is false, then P is also false."

David Handeye wrote:
Systematic wrote:We know that Q false means P false if P ONLY IMPLIES Q. If P implies Q or R and if Q is false, then we don't yet know whether P is false.
what the f... are you saying? Where did that R come from? Are you joking, say the truth...
"R" is the other thing implied by P. P => Q or R. In that case P may be true even if Q is false. Q=F; R=T => P and R are true.

EDIT: Pardon the elicit conversion. If R is true, then P might be true.
Rilx
Posts: 37
Joined: Mon Feb 09, 2015 8:54 pm
Location: Helsinki, Finland

### Re: "If P implies Q, and Q is false, then P is also false."

I think that applying formal logic to natural events is often incomplete in the sense that it leaves the axiomatic basis undefined, while applying it to another formal system (eg mathematics) has no problem because the latter uses same axioms.

In this case the problem is the Law of Excluded Middle. The rule should be written "if P then Q else not Q" which actually includes LEM. Otherwise it leads to interpretation that Q is only contingent: "if P the Q is a possible consequence" and Q has no implication to P.