| n | Proposition | Symbol | Explanation |
| 1 | Peter is gay. | P1 | Premise |
| 2 | Peter is male. | Q | From 1 |
Valid? Perhaps. A truth table with show:
| P | Q | ¬Q | P∧¬Q |
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |
If the superconjunction of all the premises and the negation of the conclusion comes out as necessarily false (F on all rows), then the argument is valid.
Or equivalently: An argument is valid iff the superconjunction of all the premises and the conclusion is a necessary truth. (I.e. all rows come out as T.)2
In the above case the argument is apparently invalid. But still it is “valid” in a sense. Why is this? It is because there is a necessary connection between the two propositions: Necessarily if p, then q [□(P→Q)]. It is impossible that (Peter is gay and Peter is not male) [¬◊(P∧¬Q)].
Now consider another argument:
| n | Proposition | Symbol | Explanation |
| 1 | Peter is male. | P | Premise |
| 2 | Peter is gay. | Q | From 1 |
Suppose further that it just happens to be the case (contingent truth) that all males are gay in a possible world. Formally this means that P implies Q in that world. So, is the second argument valid or not? Is it valid only in a specific world?
If we go by the first proposed definition of validity, then both arguments are invalid. But still there seems to be a crucial difference between their “validity” is some sense. It seems to me that it may be the necessity of the connection between P and Q that makes the difference. In the first case there is a necessary connection [□(P→Q)] because it is true per definition that a gay person is a man.3 But it is not necessarily true that all males are gay, that is only a contingent truth. There is no necessary connection between P and Q in the second argument [¬□(P→Q)]. Thus, the first argument is “valid” and the second is not even though it happens “to work” in some possible world. I think validity should not be relative to what possible world one considers the argument in. No matter what world one considers the first argument in, it comes out as valid in this sense. The second argument does not.
I suspect there is some deep confusion in this way of thinking about validity and I have tried to locate it (if it is there) by being clear, but I don’t think I’ve succeeded. I’ve also failed to properly state the meaning of ‘validity’ used in the second sense. I will continue to think about this.
Validity as a function of argument form
I’m not convinced that it is not a good idea to define validity as a sole function of the argument form. If that is done, then both the arguments above are invalid because the form they share is invalid.
From a pragmatic point of view it would be smart if people always reasoned in valid forms as it would make it much easier to check for validity instead of having to think about hidden necessary connections between the propositions in the argument.
Recap
There are three proposed ways to define validity:
1. An argument is valid iff the superconjunction of (all the premises and the conclusion) is a necessary truth. (Standard definition.)
2. An argument is valid iff it comes out as valid in all possible worlds. (This is not very clear.)
3. An argument is valid iff it has a valid form.
1I’m aware that this argument could be expressed in predicate logic but I’ve tried to keep it simple by using only propositional logic.
2This one is often mentioned in textbooks though they use the word ‘tautology’ instead of ‘necessary truth’. Probably to avoid talk of modalities and stick to truth table talk. A tautology is defined in that context as a proposition that comes out with only Ts, and a contradiction is defined as a proposition that comes out with all Fs.
3‘Gay’ is defined here as male that is homosexual.