## Wednesday, August 24, 2011

### Tracking a silly syllogism

Though I did not read the hundred page paper (by Mercier and Sperber) on argumentative theory that I mentioned earlier, I did spot this:
Categorical syllogisms are one of the most studied types of reasoning. Here is a typical example: “No C are B; All B are A; therefore some A are not C.” Although they are solvable by very simple programs (e.g., see Geurts 2003), syllogisms can be very hard to figure out – the one just offered by way of illustration, for instance, is solved by less than 10% of participants (Chater & Oaksford 1999).
(Mercier and Sperber, pg 30)
The syllogism is being used as an example of how people are poorly motivated to falsify their own conclusions.  The problem is that the syllogism they give is false.  We need an additional premise: "There exist some B."

It's somewhat surprising to spot such a straightforward logical error.  It's as if I found a misspelled word in the title.  I was curious if the paper they cited, by Chater and Oaksford, made the same mistake.  It turns out that the paper, "The probability heuristics model of syllogistic reasoning," is not only correct, but interesting in itself.

The paper is about formally correct syllogistic inferences vs the inferences that people actually make in practice.  The paper proposes a new model for how people draw inferences.
People may not be trying and failing to do logical inference, but rather succeeding in applying probabilistic reasoning strategies.
(Chater and Oaksford, pg 193)
It's another long paper--70 pages--which I'm not willing to read because I am not that invested in it.  And yet already, it seems that Mercier and Sperber seem to have missed the point of the paper.  It is not about how people are poorly motivated to falsify their own conclusions, but how people use probabilistic reasoning rather than formal logic.

I noticed that in Table 1, they show three systems of syllogistic reasoning.  The table is full of cryptic letters and codes, but I took the time to figure out what it all meant.
1. Aristotle's logic:* You are allowed to assume that there exist at least one object of A, B, and C.  However, in Aristotle's logic, the order of the syllogisms matters!  If the first premise uses A and B, and the second premise uses B and C, then the conclusion must be of the form C-A.  That is, the conclusion is either "All C are A," "No C are A," "Some C are A," or "Some C are not A."  We may not conclude "All A are C," because the premises are in the wrong order.
2. Johnson-Laird's logic: This is the same as Aristotle, except that the order of the premises does not matter.
3. Frege's logic: We are not allowed to assume the existence of objects in every category.  If I say "All A are B," that just means that given an object in A, it must also be in B.  However, it could be the case that there are no objects in A, thus it would also be true that no A are B.  The ordering of premises does not matter.
*Just because they naming this after Aristotle, I would not assume that they are actually trying to attribute it to Aristotle.  After all, this is a cognitive psychology paper, not a classics paper.

Chater and Oaksford are essentially constructing a fourth system of syllogistic reasoning, using additional quantifiers of "most" and "few".  It's quite complicated, but when compared with survey data, this fourth system mostly agrees.  (A caveat: just because this model has been proposed in a paper does not mean it is correct.  That simply means it has been proposed, and some evidence put forward.  The paper has 144 citations, which suggests that experts consider it a serious contender.)

The syllogism at the top of this post, “No C are B; All B are A; therefore some A are not C,” is valid according to Aristotle and Johnson-Laird, but not Frege.  It is unsurprising that I use Frege's logic, because I have mathematical training, and that's the appropriate reasoning for mathematics.  But it was premature of me to label the syllogism as simply incorrect; it simply uses different syllogistic reasoning.

In any case, it seems that most people draw a conclusion that is incorrect by any formal logic system. 60% conclude "No A are C" or "No C are A".  10% draw the conclusion which is correct according to Aristotle and Johnson-Laird.  25% draw no conclusion, which is correct according to Frege.  The other 5% draw other conclusions entirely.  While that's still pretty bad, as far as formal logic goes, I still feel that Mercier and Sperber misrepresented the results by only citing the 10% figure.

#### 1 comment:

Anonymous said...

When Aristotle talked of A, B and C, he maybe didn't bother about existence.

Why ? .. some say Descartes was one of the first ever to propose existence could be logically derived from mere logic (which at all put existence in doubt).

That's said to be a genuine secular move - which means that otherwise existence was always already guaranteed theologicallcy through god.

So when Aristotle talked of A, B and C, he maybe didn't bother about existence - it's just us who do that.