This article is from the Atheism FAQ, by mathew meta@pobox.com with numerous contributions by others.

There's one very important thing to remember:

The fact that a deductive argument is valid doesn't necessarily

mean that its conclusion holds.

That may seem confusing, but it's because of the slightly

counter-intuitive nature of how implication works.

Obviously you can build a valid argument out of true propositions. But

you can also build a completely valid argument using only false

propositions. For example:

* All insects have wings (premise)

* Woodlice are insects (premise)

* Therefore woodlice have wings (conclusion)

The conclusion isn't true because the argument's premises are false.

If the argument's premises were true, however, the conclusion would be

true. So the argument is entirely valid.

More subtly, you can reach a true conclusion from false premises --

even ludicrously false ones:

* All fish live in the ocean (premise)

* Sea otters are fish (premise)

* Therefore sea otters live in the ocean (conclusion)

However, there's one thing you can't do: start with true premises, go

through a valid deductive argument, and arrive at a false conclusion.

(Remember the definition of a valid deductive argument.)

So, here's a "truth table" for implication. The symbol "=> " denotes

implication; "A" is the premise, "B" the conclusion. "T" and "F"

represent true and false respectively.

CAPTION: Truth Table for Implication

Premise Conclusion Inference

A B A => B

false false true

false true true

true false false

true true true

* If the premises are false and the inference valid, the conclusion

can be true or false. (Lines 1 and 2.)

* If the premises are true and the conclusion false, the inference

must be invalid. (Line 3.)

* If the premises are true and the inference valid, the conclusion

must be true. (Line 4.)

A sound argument is a valid argument whose premises are true. A sound

argument therefore arrives at a true conclusion. Be careful not to

confuse sound arguments with valid arguments.

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