Formal Logic: An Informal Overview
Inspired by the promise of our new approach, we set out two tasks necessary for a formal test of validity.
1. Get the form.
2. Test the form.
We’ll tackle these tasks, in that order. Here, we’ll start developing our method of ‘extracting’ an argument’s logical form; and that will occupy us for the next few sections. Once we’ve mastered the art of getting the form, we’ll be ready to move on to our second, and most glorious task: testing the form for validity.
We need to develop a method for isolating the logical skeleton of an argument, which lies beneath the flesh of irrelevant subject matter – that is, we need a kind of logical x-ray for arguments. This might fill the imagination with visions of fancy electronic equipment, logicians in protective lead vest, and so on; but we know by now to be a little skeptical where the imagination is concerned. Instead, our method of extracting logical form will be simple, involving no equipment and no risk to our health: we will extract the form by developing a language of pure form, and then translating arguments from English into that form language.
The idea is this: suppose we had a language of pure logical form – that is, a language that can’t talk about subject matter, only logical form. Obviously, it would be very different from English. English – and any other natural, human language, such as Italian, or Hebrew, or Cantonese – mostly talks about subject matter, and only occasionally throws in a word or phrase talking about logical form. Think, for example, of this argument, which we looked at in the last section.
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.
When we stripped away all the subject matter words, only a few words were left.
That’s generally how sentences in English go: mostly we talk about subject matter, and only occasionally, and in passing, about form.
A language of pure form, by contrast, would only be able to talk about logical form. It would have no words for tacos, chicken, birds, clouds, frogs – only form. A form language like that wouldn’t be of much use for everyday purposes, where we’re mostly concerned with tacos, chicken, birds, clouds, frogs, and so on. But a form language would be perfect for extracting the logical form of English language sentences. Here’s why: if we translated an English sentence into this language of form, which parts of the sentence would survive the translation process? The subject matter wouldn’t survive, obviously. There are no words or phrases in the form language to express the subject matter; so all the subject matter of the sentence would get “lost in translation,” as they say. Clearly, the only part of the original English sentence which would survive the translation would be the logical form of that English sentence.
And that suggests a strategy for extracting the logical form of an English sentence: by translating the sentence into the language of pure form, all of the subject matter of the sentence will get burned off in the process; the only part that will survive the translation will be the logical form. The final product, at the end of the translation process – that sentence in the language of form – just is the logical form of the original English sentence.
So that will be our technique for logically ‘x-raying’ arguments: build a language of pure logical form; translate the English-language argument into this language of form; and the end result of the translation will be the logical form of the English argument we started with.
(But wait, you say: you’ve explained how to get the logical form of a single English sentence this way – translate that sentence into the language of form. But how do we ge the form of an entire argument? That’s easy: we know, from way back, what an argument is: an argument is “a string of declarative sentences, intended to convince someone of something”. Now, since an argument is a string of sentences, once we’ve extracted the logical form of each of these sentences, we’ve extracted the form of the whole argument. So, if we want the logical form of some English argument, we translate the first premise into the language of form; then translate the second premise, and so on; and finally translate the conclusion into the language of form. Having extracted the form of every sentence in the argument, we will have extracted the form of the argument as a whole – because that argument is nothing more than that string of sentences.)
So we’ve taken our first task – “Get the form” – and broken down into two smaller tasks.
1. Get the form.
1a. Build a language of pure logical form.
1b. Translate from English into that form language.
(And then:
2. Test the form.)
So we set out to develop a language of pure logical form. What sorts of things will count as logical form, and be the sort of things talked about in form language?
We already have some examples of logical form, from the simple little arguments we examined in the last section. We ended up saying that any argument with one of these forms will be a valid argument.
In fact, just from these little argument-forms, we can pick out four examples of logical form – that is, four sorts of things that a language of form should be talking about.
Let’s start a list.
Four Examples of Logical Form:
1.
2.
3.
4.
And now all we need to do is fill out this list, drawing from the arguments above, and we’ll have an idea of what the language of form will be made of.
The thing to keep in mind is that these little examples are not complete arguments – they’re what’s left of an English language argument, after all the subject-matter words have been stripped away. For example, we started with an argument like this.
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.
Then we stripped out all the little subject matter sentences. (I’ve put the subject matter sentences in bold, for emphasis.) Once we stripped away all the subject matter words, all that’s left of this argument is this.
(The
and 
are just little blanks, marking the spots where the little subject
matter sentences used to be.) Now, notice that, even though all the
subject matter words are gone, a few words and phrases remain – “either…
or,” and “not” These are the (rare) sort of English words and phrases
that are talking about logical form, not subject matter. So these are
the sorts of things a language of form should talk about.Likewise, in the second argument example, even after all the subject matter sentences are tripped away, one word – “and” – remains.
It’s sunny and it’s warm.
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(So,) It’s warm.
“And” is another example of the sort of thing a form language should talk about.
So we add these three examples of logical form to our list.
Four Examples of Logical Form:
1.
2. Not
3. And
4. Either… or
But now you’re wondering: why is line 1 blank? What’s the remaining example of logical form?
This turns out to be perhaps the simplest example, but the most unexpected. It turns out that we need to include, as part of an argument’s logical form, some way of marking where the little subject matter sentences go, and which are the same as which. That might sound odd: didn’t we just emphasize, over and over, that subject matter is precisely what doesn’t show up in a language of pure form? Yes indeed! So then, why do we need little markers for each of these little subject matter sentences?
We need to be careful here, and distinguish between (i) showing the places where a particular little subject matter sentence shows up, and (ii) showing what the subject matter of that sentence is. What the subject matter is – (ii) – is definitely not part of the logical form. But remarking on whether or not a subject matter sentence in one place is the same as a subject matter sentence in another place – that will turn out to be an important ingredient in logical form.
Perhaps you’re not convinced that I’m right about this. Very well, then: I will make an argument, to convince you that this really is true.
Compare these two simple little English language arguments.
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.
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,) It’s snowing.
The first little argument we’ve already looked at several times; and we’ve decided that it’s a valid argument. The second argument, on the other hand, is nuts: it’s definitely invalid. Surely it’s possible to have either tacos or chicken, and for dinner to turn out not to be tacos – without it also being true that it’s snowing! Validity counterexamples for this second argument are easy to think up. The second argument is invalid.
But the two arguments have the same premises; the only difference is the conclusions. Yet that makes all the difference, as far as validity is concerned. And here’s why: in the first (valid) argument, the conclusion isn’t some entirely new sentence out of the blue – it’s a sentence we’ve seen before in the argument. In fact, the conclusion is just the “or” part of the first premise.
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.
Whereas in the second (invalid) argument, the conclusion is not the same sentence as that “or” part of the first premise.
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,) It’s snowing.
Do you see the point I’m getting at here? In the first (valid) argument, the conclusion is the same little subject matter sentence that is also the “or” part of the first premise; while in the second (invalid) argument, the conclusion is not the same little subject matter sentence that’s in the right half of the first premise. That’s the only difference between these two arguments. Yet it makes all the difference in the world, as far as validity is concerned. In the valid argument, the conclusion is the same little subject matter sentence as before; in the invalid argument, the conclusion is not the same little subject matter sentence as before. So it looks like when it’s the same subject matter sentence as before, and when it’s not, makes a big difference to the validity of arguments. But wait: we said that there’s only one thing that determines the validity of arguments, and that’s logical form. So we see that: when it’s the same subject matter sentence as before, and when it’s not, must be part of logical form.
Looking back, we now realize that our little diagram of the valid argument was already marking out when the same subject matter sentence shows up again.
We put the little
marker in both spots, to show that it was the same little subject matter
sentence in both places. This is an essential ingredient in the valid
form of this argument – as we see by contrasting it with its invalid cousin.The invalid argument must have an invalid logical form (since we’re assuming only logical form makes an argument valid or invalid). It must have a form different from the valid argument, above. We would diagram the logical form of the invalid argument more like this.
This logical form, where the conclusion is not the same as the “or” part of the first premise, is an invalid form – as we can also see from further examples of arguments with this bad form.
Either I live in Douglas County or I live in Champaign County.
I don’t live in Douglas County
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(So,) We’ll have a quiz on Thursday.
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,) Spinoza is the greatest post-Cartesian rationalist.
These arguments are glaringly invalid; and they all share the invalid logical form.
The moral of these examples is that when the same subject matter sentence shows up again, and when not, makes a difference to the logical form of an argument (and that, in turn, makes a difference to whether or not the argument is valid).
So we see that noting when it’s the same little subject matter sentence as before, and when it’s not, is one of the things that the language of logical form needs to talk about. (On the other hand, what that subject matter is, is not something the language of form needs to show. Notice that, in our little diagrams, the little marker,
, didn’t look anything like
chicken. It wasn’t important that the “or” part of the first premise is
talking about chicken; but it was
important that the “or” part of the first premise – whatever it’s about –
be the same little subject matter sentence as the conclusion.)That’s the last example of something that the language of logical form should talk about: when it’s the same little subject matter sentence as before, and when not.
Four Examples of Logical Form:
1. When it’s the same little subject matter sentence as before, when not.
2. Not
3. And
4. Either… or
These will be the four basic parts in our language of logical form. We now turn to setting out this language in official, technical form – right after a brief, edifying meditation on the nature of definitions.