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# opinions about a semantic-logical flaw

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I was going through my old discrete mathematics textbook and i saw something that has been bothering me since high school and wanted to know if i'm the only person who thinks this. The problem is simple, i disagree with the truth table for an implication. More over i think that the truth table for a biconditional should be the TT for a implication. Now this is how convention for the TT of an implication is. P Q P->Q F F T F T T T F F T T T My problem is basically with when P is false and Q is true; If the "then" part is true and the "if" part is false then you never really prove the implication to be true... it could be that it just so happens the "then" part is true. Even though you didn't really prove it (the implication on a whole) to be false either, in the case of a logical assertation its either all or nothing. Since the output can't be proved indefinately it should not default to true. At least that is how i see things logically. I do not know what to call the truthtable above maybe an "implication with the benefit of the doubt" but it seems to me an implication is a misnomer and that the truthtable of a biconditional suits the definition of an implication more suitibly. Well anyhow thats how i feel and i just wanted other programmers/mathematicians opinions on it. -Thanks in advance for and comments- (Timkin if I remember right you are a mathematican, I hope you comment on this... this really has bothered me for a while and a dicussion/debate with a mathematician would be nice.) Italicized stuff is the edit - for clarity. [edited by - Infinisearch on August 13, 2002 2:49:10 AM]

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The implication operator can be confusing in this sense. There are very good reasons why it works the way it does. First: you say that it would make more sense if the biconditional TT were used for the implication. Using wordy examples to demonstrate this: "If it is raining, there are clouds in the sky". By your suggesstion, then it also must always be true that: "If there are clouds in the sky, then it is raining." This does not follow logic as we think of it.

Now, as for the implication truth table, think of it this way: "If it is raining, there are clouds in the sky" is a true implication. However, I look out the window, and it is not raining now. P is false. This does not change the truth of the implication. So, in principle, any false statement can be substituted for P, making the implication true.

One can come up with some absurd implications (example from my Discrete Structs book). "If 2+2=5, then I am the king of England". By the definition of implication, this entire statement is true. Though, P implies Q here, I could not use this to prove that I am the king of England, because I could never prove that 2+2=5.

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quote:
Original post by MisterAnderson42
Using wordy examples to demonstrate this: "If it is raining, there are clouds in the sky". By your suggesstion, then it also must always be true that: "If there are clouds in the sky, then it is raining." This does not follow logic as we think of it.

Not at all MisterAnderson, I am not suggesting that if P->Q then Q->P. The best way i can explain myself is through example. By way of your example "If it is raining then there are clouds in the sky" i am saying the following: If "P" is the "if" part and "Q" is the "then" part of the above quoted statement and we have P->Q according to the classic TT if i look out the window and i see there are clouds in the sky (Q=T) but it is not raining (P=F) then my "quantified implication" as a whole it true. This is what seems wrong to me... an implication by nature is an assertion unlike a logical AND or OR. Therefore when using an impliction are you not checking the validity of the assertion (implication) itself? And if so the assertion does not necessarily contain circular logic like your example. It could be total foolishness but the question is whether there is an "apparent implicative correlation" between P and Q. Back to your example with my quantifiers you can not verify or deny an "imlicative correlation" between P and Q with those quantifiers so i am saying that it (the implication as a whole) should be false.

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as far as i can remember (and it''s been a few years now) but P -> Q can also be written as ¬P v Q ([not P] or Q)... if you follow that logic then you get the same truth table...

to be honest, i can''t really explain why regardless of the "P" part, if "Q" is true then the result is true... i know what you mean about "if it''s raining outside then it will be cloudy"...

the way i look at it is more a case of "it''s cloudy because it''s raining outside" which allows for other possible reasons as to why it''s cloudy... "it''s cloudy because the rain hasn''t fallen yet", for example... ...

hope that helps...

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Here''s another perspective on how to think about it.

By making a statement, you are asserting it is true. Little is gained in resolving systems with false statements, except as far as they aid in determining true statements.

So, if my system is:

P
~Q

then I have two true statements: P is true, and ~Q is true. We can argue "No, Q is false." Quite right, but what was actually stated is ~Q, and THAT is true.

So our systems consists of statements of truth. Next example:

P
Q
P->Q

Essentially, we are saying that all three are true, and now it falls to determine if there is any contradiction. In the above system, there is no contradiction, so we can say that the overall system holds true. Let''s look at another system.

P
~Q
P->Q

Now, we hit a contradiction. By making these statements, I''ve asserted that they are true, but obviously it doesn''t hold. The system as a whole is false. This is really what you''re looking at under the P->Q line in truth tables.

Finally, let''s look at one more system:

~P
Q
P->Q

Again, we make systems of assertions of truth. If we were attempting to derive the truth value of P->Q, we couldn''t. LIKEWISE, IF WE CAN''T DETERMINE THE TRUTH VALUE OF P->Q, IT SHOULDN''T BE IN THE SYSTEM BECAUSE IT IS NOT KNOWN TO BE A STATEMENT OF TRUTH. But because it''s there, we are assuming it''s true.

Now we see if the system holds true -- are there any contradictions? In this case, there are not, and so because we asserted these statements were true, the system holds true.

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