Formal Logic: The Basic Philosophy
Disillusioned as we’ve become with using just the bare imagination to search for Validity Counterexamples, we are looking for another approach to testing arguments for validity – an approach that is not susceptible to the vagueness and limitations of something like the human imagination.
Our inspiration for this new and improved test comes in noticing a striking feature of a group of arguments. First, consider this argument:
Either we’re having tacos for dinner, or we’re having chicken for dinner.
We aren’t having tacos for dinner.
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(So,) We’re having chicken for dinner.
Now, just informally and intuitively, we would say (with quite a bit of confidence) that this argument is valid. Think about it: suppose you were in a situation where I’d promised we’d have either tacos or chicken for dinner, and I kept that promise, but not by us having tacos for dinner. But in a situation where I keep that promise (so we have either tacos or chicken), and we don’t have tacos – would that be a situation where we have chicken? It would have to be a situation where we have chicken. So that would be a situation where the conclusion is true as well. Just informally, it certainly seems safe to say: in a situation where the premises of this argument are true, the conclusion would have to be true as well. It seems fairly obvious that this argument is valid.
VALID!
(T) Either we’re having tacos for dinner, or we’re having chicken for dinner.
(T) We aren’t having tacos for dinner.
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(T) We’re having chicken for dinner.
But now consider a second argument:
Either I live in Douglas County or I live in Champaign County.
I don’t live in Douglas County
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(So,) I live in Champaign County
It seems equally obvious that this argument is valid – that is, that if the premises are all true, then the conclusion must be true as well. (If it’s true that I live in one county or the other, and it’s not the first one, then it must be the second one.)
VALID!
Either I live in Douglas County or I live in Champaign County.
I don’t live in Douglas County
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(So,) I live in Champaign County
How about a third example?
Either we’ll have the quiz on Tuesday or we’ll have the quiz on Thursday.
We won’t have the quiz on Tuesday.
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(So,) We’ll have the quiz on Thursday.
Do you suppose that this argument is valid as well?
By now you probably think that question is pretty silly. Of course this argument is valid, you’ll say – it’s just the same argument as in the last two cases. Any argument like this is going to be valid!
But stop and think about that: any argument like what? How are all these examples ‘the same argument’?
They’re certainly not exactly the same argument: the first one is about dinner options; the second one is about where I live; and the third one is about quiz days. They’re arguing about entirely different topics.
Fair enough, you might say: each of these arguments has a different subject matter from the others; but they all share exactly the same argument pattern.
That does seem right, doesn’t it? Moreover, it seems like a pretty obvious and uninteresting point to make about these three arguments. But in fact, by noticing this one little point, we’ve put our finger on some profound facts about arguments, that will point to a whole new test of validity.
First, observe that we’re now splitting an argument up into two features, or components – not the two parts of the argument we’re already familiar with (premises and conclusions), but something different: subject matter; and argument pattern, or argument structure. While these three arguments have different subject matters, they share a common logical structure, or logical skeleton – or, as we’ll call it, a common logical form. We could even sketch out this common form, by simply taking away the different subject matters words. It would be something like this.
That’s the first important observation we’ve made: arguments have both subject matter and logical form.
Second, we noted that all three of these arguments are valid. Even though they have very different subject matters, they are all equally valid. And we ended up saying something stronger: any argument with this logical form is bound to be valid. That’s important: that’s saying that we can change the subject matter as much as we like, but – as long as the logical form stays the same – the argument we end with is sure to be valid. And that means that the subject matter of the argument is making no difference to whether or not the argument is valid. As long as an argument has this logical form, it’s going to be valid, regardless of the subject matter.
So: it is only the logical form of the argument that makes the argument valid; the subject matter makes no difference to its validity.
Think about it: we could even make an argument with this logical form, and give it a subject matter we don’t even understand, or know anything about.
Either Leibniz is the greatest post-Cartesian rationalist, or Spinoza is the greatest post-Cartesian rationalist.
Leibniz is not the greatest post-Cartesian rationalist.
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(So,) Spinoza is the greatest post-Cartesian rationalist.
Now, you may have no clue what this argument is talking about: you may not know who Spinoza or Leibniz are, or what a post-Cartesian rationalist is. And so you may have no ability at all to imagine a situation where all the premises are true. Still, you would conclude with confidence that any time these premises are true (whatever that would be like), the conclusion would also have to be true. So it’s not the subject matter that’s telling you the argument is valid – you may have no grasp of the subject matter at all. What you do see is the logical form of this argument – specifically, you can see that it has the structure sketched out above, the same form that the previous three arguments have. And it looks like that’s all you need to understand, to see that the argument is valid.
That’s the important point here: only the logical form of the argument makes the argument valid; the subject matter makes no difference to its validity.
Of course, we only looked at one example of logical form making an argument valid. But other examples are easy to think up.
It’s sunny and it’s warm.
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(So,) It’s warm.
I don’t say this is an interesting argument; as a matter of fact, it’s pretty dull, and maybe even silly. But, say what you want about the argument, you’d still have to admit one thing: if all (one) of the premises of this argument are true, then the conclusion must be true as well. That is: any possible situation where it’s warm and sunny has got to be a warm situation. This argument is valid.
And it’s perhaps more boring than instructive to spin out other examples of arguments with the same structure.
It’s Thursday and we’re having class.
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(So,) We’re having class.
Any possibly situation where it’s Thursday and we’re having class (i.e., where the premise is true) must be a day when we’re having class (so the conclusion is true as well).
And so on. We can skip further examples, and just sketch out the logical form which these (and many other) arguments share.
It seems clear that any argument with this logical form is bound to be valid – as we can illustrate further, by way of examples where we’re don’t understand anything except the logical form of the argument.
NP’s must be case-checked, and the FI principles holds at LF.
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(So,) The FI principle holds at LF.
Don’t know what the premise and conclusion are talking about? Don’t feel bad, because that won’t interfere with your ability to see that this argument is valid. Any argument with this form has got to be valid.
And what holds for valid arguments also applies to invalid arguments.
Either we’ll have cake, or we’ll have ice cream.
We’ll have cake.
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(So,) We’ll have ice cream.
Wait a minute, you say: that conclusion doesn’t follow! You’re right: this argument is clearly invalid. For consider a situation where I’ve promised you we’ll have either cake or ice cream, and I keep my promise by providing a cake that is entirely uncontaminated by ice cream. Certainly that’s possible. In that situation, have I kept my promise of either cake or ice cream? Yes I have; so the first premise is true in that possible situation. And in that situation, do we end up having cake? Yes, we do; so the second premise is also true in that situation. But in that situation, do we have ice cream? No! So the conclusion is false in that situation. So we’ve found a possible situation where all the premises are true, yet the conclusion is false. In other words, we’ve found a validity counterexample. And that shows us – what we expected all along – that this argument is invalid. The conclusion doesn’t follow from the premises at all. (If you were the person who made the promise and brought the cake, and your friend complained that, without ice cream, you had broken your promise – well, you would think your friend unreasonable, wouldn’t you? Sure: your friend would be drawing a conclusion – that there’d be ice cream – which doesn’t follow logically from the premises – your promise + your cake).
And then some more examples:
Either today is Sunday or today is Monday.
Today is Sunday.
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(So,) Today is Monday.
Either Ace is home or Rex is at home.
Ace is at home.
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(So,) Rex is at home.
For none of these arguments could we say that, if the premises are true, then the conclusion must be true. That is, none of these arguments are valid; they’re all invalid.
And in general, the pattern that these arguments share seems like an invalid form of reasoning.
We could even make up mind-boggling arguments with this form, whose subject matter is a mystery to us.
Either the Johnson rod is misflanged, or the implicate reactor is positively ventilating.
The Johnson rod is misflanged.
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(So,) The implicate reactor is positively ventilating.
Even without understanding the subject matter of this argument, we can say with confidence that: the premises being true doesn’t guarantee that the conclusion is also true. (Of course, it might just so happen, in certain situations, that the premises and conclusion are all true. But that means nothing as far as validity is concerned. It still doesn't show us that if the premises are true, then the conclusion must also be true; and that’s what’s required for validity.) This argument pattern – this logical form – is invalid.
All of these cases point out what we will take as a general rule about validity: that the validity of an argument depends only on the logical form of that argument.
Now, admittedly, we’ve only looked at a few examples of logical form. That wouldn’t automatically make us confident that it’s always just the form of an argument that determines its validity.
Fair enough. As a matter of fact, we will go on now to look at lots of other forms; and we’ll find the same connection between form and validity, over and over again. And anyhow, my aim here was not really to prove this point about form and validity, from lots of cases, but merely to illustrate why it seems like a reasonable approach to validity. Taking these cases as simply instructive examples points us toward a hypothesis about validity – namely, that validity only depends on logical form. This is, in fact, the fundamental hypothesis, or principle, at the foundation of an entire approach to arguments and validity – an approach which (for obvious reasons) is called formal logic. Taking this approach to arguments will allow us to build new, and very much improved, tests of validity; and it will be the great successes of these new tests that will really sell us on formal logic as an approach to arguments.
So we set out the basic principle at the heart of our new approach to arguments and validity:
Basic Principle of Formal Logic: whether or not an argument is valid depends only on the logical form of the argument, not on its subject matter.
And though we haven’t yet seen, in any detail, how a new and improved test of validity will work, this basic principle already tells us the general nature of a formal test of validity. A formal test of validity will (naturally) look only at the logical form of the argument, and test that form to see if it’s valid or not. So we can already set out the two basic steps in a formal test of an argument’s validity.
Formally Testing an Argument for Validity:
1. Get the form.
2. Test the form.
We now proceed to fill in the details of these two steps: a general method for isolating, or ‘extracting,’ the logical form of an argument; and a general test of validity that looks only at that logical form.