Logic: finding valid conclusions

Introduction

This is not a general post about all methods of extracting valid conclusions, but I will be looking at more than just syllogistic reasoning.

I want to talk about the four logic questions I asked in my most recent happenings post.

You may, but don’t need to, go back to that post… here are four pairs of premises. (And because we have two premises for each question, we may consider syllogistic reasoning, as well as other methods.)

some A are B
no B are C

no B are A
some C are B

no A are B
some B are C

some B are A
no C are B

What valid conclusion or conclusions, if any, may we draw?

Follow me….

Question 1: intuitive and modern approaches

I will spend most of my time on question one. Here is some, but not quite all, of the information which Dodd & White provided. I have shown the answer, that 17/20 people got the right answer, I have labeled the premises and the conclusion as IEO, and I have written out the symbolic logic.

1. 17/20

I: some A are B \exists x Ax \land  Bx
E: no B are C \forall x Bx \Rightarrow  \neg Cx
O: \therefore\ some A are not C. \therefore  \exists x Ax \land  \neg Cx

Let’s not worry about Aristotle yet.

First of all, let’s just think about it. I believe that my thought process went:

some A are B, no B are C…
so those A’s that are B’s are not C’s…
so some A are not C.

So, I got the answer they gave. That’s nice.

But what if we want to try other things?

We could draw some Venn diagrams. We have that B and C do not intersect (their intersection is the null set); and that A and B do intersect. I would not limit myself to just one possibility.

As you can see, in fact, we can draw the disc for the set A in ways such that

no C is A
some C is A
all A are B
all C are A.

All we need to do is make sure that A and B intersect. A variety of drawings gives us some idea of all the things that might or might not be true — some of the potential conclusions which are not valid inferences.

For example, from the fourth drawing I see that I cannot say that “some C are not A”.

I have to say that the Venn diagrams alone would almost convince me that the only valid conclusion is “some A are not C” — and its ugly commutative version, “some not-C are A” (which is not the same as “some C are not A”).

OK, let’s look at the symbolic logic. I can get Mathematica to confirm the conclusion, but not very nicely. I can define two premises and the conclusion…

Fine, why not just prove it?

In what follows, ES stands for existential specification; US for universal specification; EG for existential generalization: I have included a more detailed comment with each one. At some level, this is just about all we need to know about dropping quantifiers (“specialization”) and inserting them (“generalization”). I will be talking about this more, but not in this post. Think of them as fancy names for such statements as “let y be such an x”.

That is, “some A are not C”. QED. Great.

Question 1: syllogism

Finally, let’s look at it as a syllogism. More importantly, let’s start by looking only at the two premises:

I: some A are B
E: no B are C

The middle term is “B”, the one that is common to both premises. The subject S is the other term in the 2nd premise, so we seek a valid conclusion of the form

{all or some} C {are or are not} A.

Isn’t that interesting? First off, we already know that the answer is not of that form. But that’s the only form of conclusion allowed by these two premises in this order. Hmm.

Second, the Venn diagrams showed cases where no C was A, and where some C was A, and some C was not A, and all C was A. Those are the four possibilities, and we cannot assert that any one of them is true. Each of them can be true — but none is guaranteed to be true.

It would appear that — as written — this is not one of the 15 (or 19) valid syllogisms. “Hmm”, indeed.

Let’s recall the four figures and the inversion:

And, also for reference, here’s the medieval pseudo-Latin:

Barbara, Celarent, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundae;
tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet; quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The two premises are in figure 4, so we extract the pseudo-Latin for figure 4: Bramantip, Camenes, Dimaris, Fesapo, Fresison.

“Fresison” has the right vowels, but not in the right order. Gee, IE is not part of any valid syllogism in figure 4. Sure enough, Aristotle has it right: there is no valid conclusion with C as the subject and A as the predicate.

The point is simply that we are looking at one of the other 256 – 15 = 241 syllogisms — one of the 241 invalid syllogisms. That reckoning of them by figures 1-4 (not inversions) always got the subject from the second (the minor) premise.

Aristotle (modified by the medievalists) didn’t organize it as: what valid conclusion(s) may I draw? He organized it by: having defined S and P, when can I reach a conclusion of the form S P?

And we can’t get one of those here.

To put that another way, Aristotle didn’t lay out sets containing two premises; he laid out ordered pairs of two premises. We have the right set but the wrong ordered pair.

OK, now it’s time to interchange the premises — to check the other order.

E: no B are C \forall x Bx \Rightarrow  \neg Cx
I: some A are B \exists x Ax \land  Bx

Now we are in figure 1 instead of figure 4, and the subject is A instead of C. Is there a valid syllogism? We certainly hope so, since we’ve worked out a proof that some A are not C.

We extract the pseudo-Latin for figure 1: Barbara, Celarent, Darii, Ferioque… and we see that Ferio works (we can ignore the -que, which just means “and”), and it says we have a valid conclusion in O:

some A are not C.

That’s it.

There is a valid conclusion, but it’s for a different syllogism than we were given.

That’s one reason I asked what valid conclusions we could draw — not whether the syllogisms were valid. On the other hand, perhaps I should not have said that all four questions were syllogisms. (I knew there was a problem, but I hadn’t sorted it all out.)

And yet, given what we just saw, we now know that if we want to use Aristotle, we must consider both of the possible orderings of the two premises.

For reference, let me also write this, rewritten example 1, as

E(B,C)
I(A,B)
O(A,C).

What we are seeing is that the organization is not as transparent as we might have thought at the beginning. If we look for valid syllogisms in figures 1-4, then we are restricted to conclusions of the form S P.

Yet another way to look at this is: the idea of extracting all possible conclusions from a set of premises is not the same as organizing syllogisms. The valid conclusion “some A is not C” is a possible conclusion from the set of given premises, or from the re-ordered pair of premises, but it is not even a conceivable syllogistic inference from the given ordered pair.

That said, 17 out of 20 people got it right. I did, too, in my head.

Johnson-Laird and Steedman, who did the test, decided on the basis of many additional questions that what seems to matter is that the given forms

A, B
B, C

encourage us to look for a conclusion of the form

A, C

and that’s right (albeit not Aristotelian).

Question 2

Here is the second question all laid out with additional information.

2. 14/20

Figure 1 “Ferioque”

E: no B are A \forall x Bx \Rightarrow  \neg Ax
I: some C are B \exists x Cx \land  Bx
O: \therefore\ some C are not A. \therefore  \exists x Cx \land  \neg Ax

I’m not going to punch through the Venn diagram, or the symbolic logic proof: using modern techniques, this is the same as the rewritten question one… There we rewrote the question as:

E(B,C)
I(A,B)
O(A,C).

and here we have

E(B,A)
I(C,B)
O(C,A)

which is of the very same form, with C and A interchanged.

Consequently, there is no problem with Arisitotle: this question is posed as Ferio in figure 1 — just as the rewritten question 1 was.

Incidentally, we already know that if we interchange the the premises, we will be in figure 4 … but this is the same as question 1 (with A and C interchanged), so there is no valid figure 4 syllogism.

That is, we have the only right answer. (Again, except for the equivalent, but ugly, “some not C are A”.)

Note that 14 out of 20 got it right; and I did, too, in my head.

Note also that we have the sequences

B A
C B

in the premises, and the investigators decided that “C A” is the kind of conclusion people look for, and in this case also, there is a valid conclusion of that form.

Question 3

3. 8/20

Figure 4 “Fresison”

E: no A are B \forall x Ax \Rightarrow  \neg Bx
I: some B are C \exists x Bx \land  Cx
O: \therefore\ some C are not A. \therefore  \exists x Cx \land  \neg Ax

This time we have a valid figure 4 syllogism. Recall the pseudo-Latin: Bramantip, Camenes, Dimaris, Fesapo, Fresison.

This is Fresison. The subject S is C, from the second premise.

I could do Venn diagrams and symbolic logic… but I’m convinced by Aristotle.

Well, we know there is one thing we must check: what if we interchange the premises?

I: some B are C \exists x Bx \land  Cx
E: no A are B \forall x Ax \Rightarrow  \neg Bx
\therefore\ {some or all} A {are or are not} C.

Look back at the figures… this is figure 1. Is there a valid figure 1 syllogism whose first two vowels are IE?

Barbara, Celarent, Darii, Ferioque…

No. We already found our answer, and it’s unique.

Note, however, that only 8 out of 20 people got this one right.

Even I got it wrong — in my head. A Venn diagram showed me that my answer was wrong, and then I located it in Aristotle (well, in the medievalists, since Aristotle would have used the inversion instead of figure 4).

Why did most people get it wrong?

The investigators observed that we have the sequences

A B
B C

in the premises, so people tend to look for, and state, a conclusion with the sequence

A C.

Wrong.

Ding! Thanks for playing.

Question 4

As I did for questions 2 and 3, let me present all the data I was given or inferred.

4. 5/20

Figure 1?

I: some B are A \exists x Bx \land  Ax
E: no C are B \forall x Cx \Rightarrow  \neg Bx
O: \therefore\ some A are not C. \therefore  \exists x Ax \land  \neg Cx

Once again, Dodd & White have made the very same mistake as in question 1: they said this syllogism was figure 1. Let me recall the figures.

No. The subject is C — it always comes from the second premise — so “some A are not C” is not a even a possible conclusion, never mind valid or invalid.

Interchange the premises. Then we have made A the subject…

E: no C are B \forall x Cx \Rightarrow  \neg Bx
I: some B are A \exists x Bx \land  Ax
O: \therefore\ some A are not C. \therefore  \exists x Ax \land  \neg Cx

and the rewritten syllogism is figure 4, and we’re looking at Fresison again. It is a legitimate syllogism, and a valid syllogism.

(And because it’s Fresison, as was question 3, we’ve already confirmed that there is no valid conclusion if we interchange the premises again, thereby using them in their original order.)

Note that only 5 out of 20 got this right — and I’m on the losing side again, at least until I draw a Venn diagram. My reasoning was faulty — oops, I hope you didn’t think I was perfect — but I knew how to check it, and catch the mistake.

And this too follows the pattern which the investigators conjectured: the sequences of terms in the premises are

C B
B A

so most people look for, and often state, a conclusion with the sequence C A.

Wrong again, because the valid conclusion has the sequence A C.

Summary

What the investigators had was the four sequences

A B
B C
A C

B A
C B
C A

A B
B C
C A

B A
C B
A C

and they decided that people had troubling reaching conclusions when the A and C in the conclusion were not in the same columns as in the premises. (That’s my phrasing of it.) This is actually the key point about the four questions, that we have these four distinct and exhaustive sequences.

As a set of sequences, those four are exemplary. Furthermore, because one of the premises was a universal negative (“no B is A”), the conclusions were best phrased in one way (“some A is not C”); the equivalent valid “some not C is A” is rather awkward to champion.

(If the conclusion had been positive (“some A is B”) then I would immediately add “and some B is A”. And the experiment would have been all mixed up, with some people saying one or the other, and some people saying both — as well as, perhaps, some people getting none of the right answers.)

I was rather shocked to see that Dodd & White had messed up. The modern approach to logic (“extract all possible valid conclusions”) works fine — it’s what we did back in question 1. But if you want to phrase those 4 questions as syllogisms, you can’t phrase the conclusions the way they did. (Given, for example, question 1 as a syllogism to be investigated, there is no valid conclusion.)

All this forced me to clarify the Aristotelian and medieval organization of syllogisms. It’s crucial that the subject is in the second premise, and that we look for conclusions of the form S P. To find the valid “some A are not C”, we had to interchange the premises — thus making A the subject — and then all conclusions of the form

{all or some} A {are or are not} C

are open to consideration, and one of them is valid.

If we want to use the list of valid syllogisms, we have to consider both possible orderings of the two premises. To put that another way, we need to consider the premises as a set .

As I said, this was an unscheduled post, but it gave me more insight into the use of syllogistic reasoning. I had never said to myself that interchanging the premises could move me between valid & invalid syllogisms. Oh, of course, it’s blindingly obvious once I imagine doing it.

But it’s only when I try to work out the details that I imagine doing a lot of things.

About these ads

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 44 other followers

%d bloggers like this: