Friday, November 07, 2008

A new logical fallacy

Those of us who have taught logic to students are familiar with some of the fallacies they make: (1) confusing OR with XOR (reasonable given English Lang use), (2) thinking that (a--> b) --> (b-->a) and (3) thinking that from

A1 AND A2 AND A3 --> B

you deduce something about the writer's opinion of A3.

This recently happened, though it wasn't a student. It was a commenter on this blog. In a recent blog I wrote about McCain's concession speech:
If he has that way the entire time he might have won (if he also didn't pick Palin and we didn't have the economic crisis and we were more clearly winning the Iraq War).
This is an IF-THEN statement. There is no logical way to deduce what I think of any of the premises. One of the commenters committed error (3) above:
A country is destroyed and half million people are killed, and yet the only thing you feel regret about is not "more clearly winning". Excuse me, Professor Gasarch, I never held hope for the humanity of US, but a comment like this from an intellectual in this country, just taught me how coldblooed the americans can be."
The commenter raised an interesting question: If a writer says A-->B then what can you deduce about the writers opinion of A?
  1. If the work is in Large Cardinals then likely the writer thinks that the Large Cardinal hypothesis is true. Note that we do not know this logically.
  2. In papers that prove things contingent on P\ne NP or that factoring is hard or the usual derand assumptions the authors thing the assumption is true. Note that we know this by sociology, not by logic.
  3. (I may be off on this one- if so please correct.) Non-Euclidean Geometry was started by assuming the Parallel Post was false, hoping to prove that that assumption was FALSE, and seeing what can be derived from it. Let A be For a line L and a point p not on that line there are an infinite number of lines through p that do not intersect L. Let B be The sum of the angles of a triangle are LESS THAN pi. When someone proved A implies B they may have thought that A was false.
  4. Pat Buchanan said (I am paraphrasing)
    If McCain had presented more ties linking Obama to Ayers and Wright then he would have won.
    While this could be a simple IF-THEn statement, given who he is we know that he things these ties are relevant. Keith Oberman may have expressed a similar sentiment differently:
    If McCain had presented more lies linking Obama to Ayers and Wright then he would have won.
    In this case we can tell what Keith Oberman thinks of the assumption.
  5. SO- if someone says A-->B then you can't really deduce what the speaker thinks of A LOGICALLY, but you can use other things he has said and his reputation to discern what he thinks of A. Reasoining from context and personality can be useful, though it is not as rigorous as we are used to. Students in a logic course should not use it.


  1. If people view A as offensive, then A => B may be offensive also.

    Example: If group X is generally bad at Y, then doing Z would not help much.

  2. English semantics is not logic.

  3. I agree with arnab. You can almost always infer something about someone's opinion of what they're saying by their choice of words and phrasing and other contextual clues (e.g. if that person is Keith Olbermann, as you say). Even if they go out of their way to phrase something neutrally, you then gain the non-trivial information that they felt that this statement was something that should be made with neutral language, at least to their current audience.

    The real response to the commenter isn't that their conclusion isn't supported by formal logic, it's that it isn't supported by your actual language or context. There are cues that can be pointed to as evidence, but ultimately it will always be subjective since there are countless connotations in everyday language that will be interpreted very differently depending on the hearer. So we just have to settle for the heuristic argument of "people who agree with anonymous's politics still disagree with their interpretation of Gasarch's post", which is something I suspect is empirically true in a great number of cases.

  4. Gasarch and Lance have quite different posts in terms of content. Why not split the blog?

  5. I find the diversity of posts refreshing.

  6. Please don't mix mathematical logic when you're discussing your commenter's point, which is based on a murky English sentence.

    You said something like:
    "if [A (oh, and btw, AND B and C and D)] then maybe E".

    Since we know E didn't happen, and your primary comment was your lamenting Not(A), together with the fact that Not(B) and Not(C) are so obviously true, it stands to SIMPLE HUMAN REASONING, without any knowledge about your biases and prejudices, that you believe Not(D) as well.

    No math here, no mathematical logic here, just making a standard social inference. Your comment wouldn't have passed any test of mathematical rigor, any way, and you almost certainly didn't intend to give it a mathematical rigor that you now seem to claim you did.

    The commenter correctly called your bluff, and you're now hiding behind obscurities.

    By the way, I fully grant that even if you think that we (the U.S.) "aren't more clearly winning the war", it does not mean you *wish* we were more clearly winning the war. Making that mis-inference would be bad, and is quite common.

  7. Like aranb said, English can give you much more information than logic. Even when writing about math, as noted by Terry Tao

  8. should we discuss the over-generalization of the commenter? they are proving a universal statement with a single instance

    "...just taught me how coldblooed the americans can be..."

    no, he just taught you how one person (you) can interpret the words of one American (gasarch) as being coldblooded

    the commenter could be wrong about gasarch, and certainly can't generalize to the other 300+ million Americans

    I read this as someone who is anti-America and anti-Americans looking to justify their feelings.


  10. It is a convenient simplifying approximation to assume that mathematical reasoning is strictly about logic. But of course, the real-world context of mathematics (and logic too) much larger.

    Especially in politics, but also in politics, business, romance, and gambling, mathematical logic is a too for constructing narratives. With these broad, narrative-driven venues, the assertion "a->b" reliably allows one to assume that "a" will play a role in the narrative.

    In playwriting, this is know as Checkhov's Gun Principle: "A gun on the mantelpiece in Act 1 must be fired by Act 3".

  11. Postscript to the above ...

    On checking, I found not only numerous web pages (including a Wikipedia page) devoted wholly to discussions of "Checkhov's Gun", there is even a 1997 movie titled Chekhov's Gun whose summary is as follows:

    In 1897, Anton Chekhov first articulated the most famous axiom of story structure: if a gun appears on stage in a play, someone must be shot by the final curtain. Chekhov's Gun imagines what might happen if the characters in a film somehow discovered this "rule" and and then set about avoiding their fate.

    In the immortal words of James T. Kirk, "Sounds like fun!" :)

  12. I think the very terminology of "if [...] we were more clearly winning the Iraq War" is offensive, no matter what the author's opinion is about what is happening in Iraq.
    (Both the choice of the words "winning" and "War", for example.)

  13. Anonymous, why choose a loaded word word "offensive" rather than a less-loaded word like "imprecise"?

    Nothing has been more dismaying (to me) in the present election, than the prevalence of factions whose members are incurious, uncompromising, readily offended ... and proud of it.