Must a valid rule of inference always lead to a true conclusion?
I am sorry I posted my question by mistake in Number Theory section. Please ignore it or delete it from there.
My question is: Must a valid rule of inference always lead to a true conclusion?
A rule of inference is a prescription to produce a (set of) valid statement(s) from a (set of) valid statement(s).
So by definition, the answer is "yes"
Yes, unless the system is inconsistent. Remember to use only valid terms, though -- no barbershop paradox!
If London is in England then Paris is in Spain.But London is in England thus Paris is in Spain
Here we have a valid argument with false conclusion
The rule of inference used here is M.Ponens,because if we put :
London is in England=p...............Paris is in Spain=q the above argument takes the form:
p----->q & p and using M.Ponens the conclusion is q which is false
Valid is NOT the same as true. A sequence of statements is "valid" if the truth of each implies the truth of the next. But if the first statement (the hypothesis) is false, a valid argument can lead to a false conclusion.
for example, "A=>B and B=> C, therefore A=> C" is a valid argument. If A is "A person has brown hair", B is "a Person has brown eyes", and C is "a person is 6 feet tall", the argument is still VALID but the conclusion "If a person has brown hair then a person is 6 feet tall" if false because the hypothesis is false.
The fact is that neither symbolic logic, nor mathematics in general is concerned with true statements. They are concerned with valid arguments.
This was also asked in "Sets, Probabilty, and Logic" so I am merging the two threads.
If Γ ⊢ P, then Γ ⊨ P.
Soundness tells that deductions lead only to "correct" conclusions.
If the deductive system is not sound, a proof might lead to a wrong conclusion.
I notice you do not use the word "valid", which was the crucial question! The link you post defines an argument to be "sound" if and only if both the argument is valid and the hypothesis are true.
The question was whether a valid argument must always lead to a true conclusion.
The answer to that question is "No". A valid argument, with a false hypothesis, can lead to a false conclusion.
It is, of course, true that a valid argument, with a true hypothesis must lead to a true conclusion- that's pretty much the definition of "valid" argument- but "validity" of an argument is independent of the truth or falsity of either its hypothesis or conclusion separately.
Surprise everyone. This was an extra credits question. And the answer is NO. I was disappointed too. But I realy didn't understand way is NO?
Ok, Thank you. Someone ggot it right.
So now you are saying you were lying to us by not posting this in the "homework and schoolwork" area?
What do you mean I was lying. This was not homework question at all. This was an supplementary exersices called for extra credits. I didn't not use this for school extra credits. I just wanted to know why the answer was given NO. I had my answer YES. I am sorry for the missunderstanding.
Well, don't worry about it too much. I fell for it, too.
I apologize. You said earlier it was an extra credit problem, not just that it was in a section labled "for extra credit".
I got one of my logic HW questions wrong can anyone help me prove the following?
-(-P v -Q) therefore (P & Q)
Yeah, that's pretty much one of De Morgan's laws.
The general formula for De Morgan law is :
~(AvB) ===> ~A& ~B SO if you put A= ~P and B=~Q you will get P&Q assuming of course that ~(~P)=P AND ~(~Q)=Q ,UNLESS you want a proof of the De Morgan law
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