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" Conclusion can only be false if atleast one of the premises is also false. If both premises are true, then conclusion is also true. We will say that argument is valid if the premise cannot be all true without the conclusion being true as well. "

I am not able to understand meaning of these lines in textbook. Can anyone clarify ? Thank You

Mauro ALLEGRANZA
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Jessica Griffin
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3 Answers3

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  • You know contraposition. The contrapositive of " if A then B " is " if B is false, then A is false", or, more precisely, " if not-B, then not-A".

  • A proposition and its contrapositive sentence are equivalent, they mean exactly the same thing.

  • The definition of deductive validity says that a reasoning is valid just in case :

if all the premisses are true, then ( necessarily) the conclusion is true.

  • By contrapositon, it can also be phrased :

if the conclusion is false ( i.e. not true) , then not all the premises are true ( meaning that at least one premise is false).

So

(1) if I know that a reasoning is valid

(2) and that its conclusion is actually false

(2) then , I can claim with certainty that at least one of its premises is false ( one or more, possibly all).

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    if premises is true, then conclusion is true. Is this assumed this way ? – Jessica Griffin May 28 '20 at 16:25
  • Yes this is the definition of validity. Logicians asked themselves the question : what is a good deducton, from a purely logical standpoint. And they decided to adopt this answer : a deduction is valid when and only when " in case the premises were true , the conclusion should also be true". ( It would be impossible for the conclusion to be false in case the premises were true). –  May 28 '20 at 16:31
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    @JessicaGriffin That's what it means for an argument to be valid. Certainly there are examples where the premises are true but the conclusion isn't necessarily true -- we would call these invalid arguments. E.g. consider the statement "if it is cold outside, then it must be snowing". The premise is "it is cold outside" and the conclusion is "it must be snowing". This is an invalid argument: there are days when it is cold outside but not snowing; i.e. the premise can be true but the conclusion false. For an argument to be valid, the conclusion must follow from the truth of the premises. – twosigma May 28 '20 at 16:32
  • Say if premise is of type P or Q, then premise being true means that either P or Q must happen, is this right ? – Jessica Griffin May 28 '20 at 16:32
  • That's it : the conclusion has to follow logically from the premises. Your cold/snow example is not valid, as you correctly say. –  May 28 '20 at 16:34
  • Your example with " or" is not very good ( though valid) , for it is just an instance of P, therefore P. . One could give as example : (P OR Q) , not-Q, therefore P. –  May 28 '20 at 16:35
  • If one premise is P or Q and another is Q or R. then how can we draw conclusion ? – Jessica Griffin May 28 '20 at 16:37
  • With these premises, suppose Q is false, what would follow? –  May 28 '20 at 16:41
  • @RayLittleRock Both P AND R are true then. But why assume Q is false ? If we have P or Q , not Q then conclusion is P. There is no assumption in second deduction. Why in the first one ? PS - I hope i am not bothering you too much – Jessica Griffin May 28 '20 at 16:42
  • This is " conditional proof" . You assume something in order to draw a conditional conclusion . Here, the only " interesting" conclusion that can be drawn here is " if not-Q, then (P&R)" or ( equivalently) " Q OR ( P&R) " –  May 28 '20 at 16:44
  • "(P OR Q) , not-Q therefore P " is an inference rule called disjunctive syllogism. –  May 28 '20 at 16:46
  • @RayLittleRock Thanks you so much – Jessica Griffin May 28 '20 at 16:47
  • You are welcome. –  May 28 '20 at 16:48
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An argument is called valid if its premises imply its conclusions. In other words, if a valid argument's premises are true, its conclusion is true. (A valid argument with true premises is called sound.) We can restate this equivalently with the contrapositive: if a valid argument's conclusion is false, at least one of its premises is false.

J.G.
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I suspect that you might be especially confused with the tongue-twisting phrase “argument is valid if the premise cannot be all true without the conclusion being true as well”.

Perhaps things can be illustrated by an example. Suppose I tell you, “You’ll never see me in the park on a Sunday”. This just means “If it’s a Sunday, you won’t see me in the park.” Now, suppose it is a Monday and we run into each other at the park. Have I broken my promise/claim? No. Although the conclusion is false (i.e. you did catch me in the park), the premises weren’t fulfilled: it’s a Monday, not a Sunday. So I haven’t broken my promise/claim.

The only way you could invalidate my claim is if you find me at the park on a Sunday. In other words, you can only invalidate my claim if the premises are true (i.e. it’s a Sunday), but the conclusion is false (i.e. you did see me in the park). This is what is meant when we say that an argument/claim is valid “if the premise cannot be all true without the conclusion being true as well”. It means we cannot have the following: the premises all true, but the conclusion false. My claim that “You’ll never see me in the park on a Sunday” will be valid so long as whenever it’s a Sunday (i.e. the premise is true) you don’t see me at the park (i.e. the conclusion is also true). However, I could be at the park on any other day of the week: that doesn’t invalidate my claim.

twosigma
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