Making explicit things that are already implied by “A Short Lesson on Arguments and Logic”; totally optional. Click through to see…
A valid deductive argument is one where the conclusion logically follows from the premises, such that it is impossible for the premises to be true and the conclusion to be false at the same time.
Given the definition above, all of the truth combinations for the premises and conclusion are possible except for the one that’s been ruled out–the premises are true and the conclusion is false. In other words, valid arguments can have false premises leading to a true conclusion, false premises leading to a false conclusion, and of course, true premises leading to a true conclusion. For the invalid argument, all four combinations are possible. This means that generally, from the truth/falsity of the premises and conclusion, you won’t be able to tell if the argument is valid or invalid–the only exception being true premises and false conclusion–definitely invalid.
But what about a sound argument? Now, a sound deductive argument is one the premises of which are all true, and the argument is valid. Since the argument is valid, it can’t have true premises and a false conclusion. Since the premises are true, it has to have a true conclusion. But this also means that if the conclusion is false, the argument has to be unsound, which is just the “contraposition” (see below) of “if the argument is sound, the conclusion has to be true”. (The reverse doesn’t follow–you can’t derive “if the conclusion is true, the argument sound, or, if the argument is unsound, the conclusion is false.)
The above are all about how the truth/falsity of the premises and conclusion of an argument correlated with its validity/soundness. But you might ask–didn’t we teach you that if you want to object to an argument, you shouldn’t just assert that the conclusion is false–you should attack its validity or the truth of its premises? That’s correct, but a separate point. When you offer an objection to an argument, you aren’t just figuring out how the truth/falsity of the premises/conclusion affect each other–presumably, you are offering reasons why you reject the conclusion, or why others should reject the conclusion. In other words, distinguish between these two thoughts:
(1) When faced with an argument and believing that the conclusion is false, you (legitimately) infer that the argument is unsound. You are kinda obliged to do this if you maintain that the conclusion is false.
(2) When offering an objection to an argument, you do so by attacking either the premises or the validity, rather than the conclusion directly. This is ideally how a rational objection is offered, rather than just the assertion of a disagreement.
Both thoughts are correct and they don’t conflict with each other.
If you think through how the four forms of arguments to do with conditionals (“if…then…”) introduced in “A Short Lesson on Arguments and Logic”–the valid ones modus ponens and modus tollens, and the invalid ones affirming the consequent and denying the antecedent–you should be able to deduce a rule about the conversion of one conditional statement into another that is logically equivalent. This rule is called “contraposition“:
- If P then Q
- If not Q then not P (Contraposition; valid)
In other words, every conditional statement can be ‘flipped’–but you need to add a “not” in front of both the antecedent (the part that comes after the “if…”) and the consequent (the part that comes after the “then…”).
So for instance “If not R then S” becomes “If not S, then not not R”, which is just “If not S then R” (assuming standard logic, two “not”s cancel out each other–so, “not not R” = “R”).
How does this happen? Well, Contraposition is itself the derivative of modus tollens, and another very useful rule “Conditionalization” (below). Consider a standard modus tollens:
- If P then Q
- not Q
- not P (Modus Tollens; valid)
Normally, when you assert an argument, you assert that all the premises are true, and that they logically lead to the conclusion. So if you assert a modus tollens, you assert “If P then Q”, and “not Q”, and that “not P” follows from the first two. But what if you aren’t sure about, say, the “not Q” (the second premise)–you aren’t really ready to commit by asserting that it is true? One thing you can do is to offer a conditional version of your earlier argument. Imagine yourself saying: if that missing premise “not Q” is also true, then, “not P”–that’s the gist of “Conditionalization”. But when you do that, you have basically offered a Contraposition of what was the first premise in your original argument.
(Incidentally, with the above, you can also figure out what happens when you do the contraposition wrongly. For instance, you flip the original conditional statement without adding the “not” in front of the antecedent and consequent–it would be as if you are working on an invalid, affirming the consequent version of the above reasoning.)
Ok, so how does Conditionalization–in general–work? Let’s say we have a random deductively valid argument:
- s (random valid deductive argument)
If the argument is deductively valid–the premises logically lead to the conclusion such that it’s logically impossible for the premises be true and the conclusion false at the same time–then it is always possible to conditionalize by moving any of the premises, joining it to the original conclusion to form a conditional:
- if q then s
Again, I need to emphasize–the move works only if the initial argument is deductively valid. And you can do this for any and all premises. In fact, you can even move all of the premises down at once, for instance:
- if (p and q and r) then s
What just happened? There’s nothing left in the premises and yet we have a conclusion?!
What just happened is that for every valid deductive argument (remember that the initial argument is supposed to be deductively valid), we can always form a corresponding conditional (whether the antecedent is the conjunction of all the premises, and the consequent is the conclusion) that is not just true, but logically necessarily true. It’s as if we have a new valid deductive argument where from ‘nothing’ the conditional conclusion follows logically.