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I'm currently studying logic and I'm confused about what is meant by the truth or falsity of statements within an argument.

From Daniel J. Velleman's, How to Prove It, pg.9:

"Although we have no guarantee the conclusion is true, it can only be false if at least one of the premises is false ... We will say an argument is valid if the premises cannot all be true without the conclusion being true as well."

Having then read a wikipedia article about the validity of an argument, I am told the following:

"The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid:

All cups are green.
Socrates is a cup.
Therefore, Socrates is green."


I feel that there is some distinction being made here between the terms "truth" and "false" in the two instances that I'm not quite grasping.

In the wikipedia article I believe "false" refers to the general truth of each individual statement i.e. we know all cups are not green in reality so this is a "false" statement. However in the Velleman book I am not sure what he means by the "truth" or "falsity" of the premise and conclusion.

In sentential logic what is meant by the "truth" or "falsity" of a statement – the "empirical truth" of the statement, or does it refer to some other notion of "truth" and "falsity"?

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An argument is valid iff :

there is no possible situation in which the premisses would be true and the conclusion false.

An argument is valid because it shows a "pattern", like :

All $A$ are $B$

$s$ is $A$

Therefore, $s$ is $B$.

We can think to it as a "form" to be filled : filling the "form" in the correct way, we get an instance of that "argument-pattern", like :

"All $Men$ are $Mammals$

$socrates$ is a $Man$

Therefore, $socrates$ is a $Mammal$",

with the feature that : if all the premisses are true, then also the conclusion is.

We can think at this beautiful feature as a way to "transfer" truth from premises to conclusion : we cannot "transfer" it if it is not present in the premisses.

What happens if we "fill the form" correctly, but with false premisses ? The "argument-pattern" does not change, but the antecedent of the definition above : "if all the premisses are true, ..." is not satisfied, and thus it gives us no clue about the truth or falsity of the conclusion.

This is what happens with :

"All $Cups$ are $Green$

$socrates$ is a $Cup$

Therefore, $socrates$ is $Green$."

Both premises are false, and thus the falsity of the conclusion does not invalidate the "argument-pattern".

We can have also cases more tricky than that; consider :

"All $Cows$ are $Mammals$

$socrates$ is a $Cow$

Therefore, $socrates$ is a $Mammals$."

In this case the conclusion is true, but not all the premises are. The argument is still valid, but we have applied it in the "wrong way" and thus we are not licensed to assert the conclusion on the ground of the argument alone.

In other terms, if we want to assert the (true) fact that "$socrates$ is a $Mammal$", we can do it e.g. according to some empirical knowledge, but not on the ground of the above (valid) argument, because the corresponding "argument-pattern" licenses us to "flag" the conclusion as true only when we have inferred it from true premisses.


Regarding the question :

what does the "truth" or "falsity" of a statement refer to? The "general truth" of the statement, or some other type of "truth" and "falsity"?

there are no "multiple" meaning of true in mathematics and logic (until we do not try to elucidate the concept of truth at a more philosophical level).

The notion of truth (in the form of the predicate "is true") applied to mathematical statements amounts to saying that a statement is true if and only if things are as they are said to be in the statement; more formally :

for any mathematical statement $\phi$, "$\phi$ is a true mathematical statement" is equivalent to $\phi$ itself.

Applied to our discussion above, "$socrates$ is a $Mammal$" is true because $socrates$ is a $Mammal$, while "$socrates$ is $Green$" is false because $socrates$ is not $Green$.

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It just means ordinary-language truth. A valid argument can only have a false conclusion if it has false premises; Velleman and Wikipedia are making the same point.

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  • $\begingroup$ Velleman then goes on to say: "We will say an argument is valid if the premises cannot all be true without the conclusion being true as well". From what Velleman is saying it seems that valid arguments can't have false premises. The premises in the wikipedia article are false, but the argument is still valid; so i'm lead to believe each source refers to different kinds of ideas of truth and falsity. Or am I confused? $\endgroup$ – seeker Oct 15 '15 at 0:27
  • $\begingroup$ No, you're confused. A valid argument can certainly have false premises. It's just that, if the premises happen all to be true, then validity is the guarantee that the conclusion is true as well. $\endgroup$ – Kevin Arlin Oct 15 '15 at 0:46

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