Put simply, for an argument to be deductively valid it must not be possible for the premises to be true but the conclusion false. What this means is that if I have a valid argument I can know that if the premises are all true then the conclusion must be true too. So the argument that

• 
  All men are mortal.
  
• 
  Socrates is a man.
  

Therefore

• 
  Socrates is mortal.
  

is valid, since if it is the case that all men are mortal and that Socrates is a man then there is no way, without abandoning all our rules of logic, that Socrates can not be mortal.

Counter-intuitive Aspects of Deductive Validity

False Premises

Bear in mind, however, that for the purposes of deductive validity a valid argument does not have to have true premises, but must simply obey the rule that if they were true then there would be no way for the conclusion to be false. So

• 
  All men are florescent orange.
  
• 
  Socrates is a man.
  

Therefore

• 
  Socrates is florescent orange.
  

is perfectly deductively valid.

One way of looking at it is that in another possible world, where all men are orange, Socrates if he is a man, must be orange. The discussion of possible worlds termed 'modal logic' is fascinating, and enters many areas of logic, not just notions of validity.

Insufficient Premises

Another unobvious fact about this definition is that the argument that

• 
  The man fell from a fourth-floor window onto the concrete below.
  

Therefore

• 
  The man died.
  

is, in fact, invalid, even though it might seem to obviously follow. The reason that this argument is invalid is that whilst it may (generally) be true with things being how they are, it wouldn't break any laws of logic for the man to fall out of the window and survive. For example, if the man was wearing a parachute he might float gently to the ground, so he fell out of a window, but did not die - it isn't the case that if the premise is true, the conclusion must be. We should instead add a second premise, that all men who fall out of windows die, and in fact the argument

• 
  The man fell out of a fourth-floor window onto the concrete below.
  
• 
  All men who fall out of fourth-floor windows onto concrete below die.
  

Therefore

• 
  The man died
  

is valid. You might argue that we could still add another premise, that he floated down, but if this were the case, and this saved him, then the second premise 'All men who fall out of fourth-floor windows onto the concrete below die' would not be true. This shows two interesting things - you can't add premises to a valid argument and make it invalid; and any argument with contradictory premises is automatically valid, since its premises can never all be true together, and thus we can never have a situation where all the premises are true and the conclusion false.

Truth-Tables and Validity

A truth-table¹, unfortunately, are limited as to what they can tell us about validity. Whilst it is true that when the premises are true and the conclusion is false we know that the argument must be invalid, none of the other possibilities tell us anything about the validity. Any of the other possibilities can have either true or false arguments:

Validity vs. Soundness

In contrast to the notion of validity, where even an argument with false premises can be valid, a 'sound' argument is a valid argument with the added requirement that the premises are true, which fits better with intuitive notions of what makes a good argument. Note that just because we're cutting out arguments with false premises doesn't mean we just need to look at the conclusions - for an argument with true premises to be valid, and thus sound, it must have a conclusion that cannot be false - so one that has a coincidental true premise will still not be sound.

Does it sound like I've just said the same thing multiple times? Well, I'd claim that each time I said a slightly different thing, but repeating this is no criticism, since this is a vital concept to grasp if you are to get anywhere in logic, and, whilst counter-intuitive in many ways, logic relies on this definition of validity to get anywhere.