logic as an exercise in style

Unlike the red sauce, this post falls in the cat­e­go­ry of “prob­a­bly use­less”.

Once upon a time I took cours­es in log­ic at Uni­ver­si­ty.  I did fine, because if you know how to do math and pro­gram, there isn’t much to it, at least at the under­grad­u­ate lev­el.  (It was a big dis­ap­point­ment to learn that Gödel’s incom­plete­ness the­o­rems look kind of like St. Anselm’s onto­log­i­cal proof for the exis­tence of god.)  I was tak­en in, I think, by the 19th cen­tu­ry idea that log­ic is some­how the most “fun­da­men­tal” form of think­ing, or the foun­da­tion upon which the sci­ences are built.  I’m not sure how this think­ing coex­ist­ed with my actu­al work in sci­ence, math, etc., where log­ic almost nev­er came up (except occa­sion­al­ly as a minor ele­ment in a proof).  No math­e­mati­cian or sci­en­tist I know makes much use of for­mal log­ic.  If it’s foun­da­tion­al, then this foun­da­tion is deeply buried indeed.

It’s true that one can begin with sec­ond-order log­ic, and get from there to num­bers and so on.  On the oth­er hand, one can begin with num­bers, and get from there to log­ic, a good deal more eas­i­ly.  One can also begin with geom­e­try, or with some­thing very non-reduc­tive, like an ani­mal with a ner­vous sys­tem that can learn asso­ci­a­tions between cor­re­lat­ed com­plex, noisy sen­so­ry inputs.  That’s how we do it in real life, after all.

My recent “insight”, if some­thing so triv­ial could be called that, is that log­ic not only lacks any spe­cial place in the scheme of the uni­verse, but is in fact just a the­o­ret­i­cal frame­work like any oth­er— and a bit of a back­wa­ter at that.  A the­o­ret­i­cal frame­work is some­thing that hangs togeth­er as a sys­tem for explain­ing or pre­dict­ing phe­nom­e­na, express­ing ideas, gen­er­al­iz­ing and mak­ing infer­ences, and iden­ti­fy­ing sur­pris­es or vio­la­tions.  To achieve all of that, a frame­work needs to abstract away, sim­pli­fy or approx­i­mate.  Log­ic relies on some fair­ly bru­tal approx­i­ma­tions.

In log­ic, as in pret­ty much any oth­er frame­work, the first approx­i­ma­tion comes with the assign­ment of sym­bols.  Many “log­ic puz­zles” play with assump­tions regard­ing the inter­pre­ta­tion of pred­i­cates, as did Bill Clin­ton (let’s say gen­er­ous­ly) when he said “I did not have sex with that woman”.  It’s exceed­ing­ly easy to get into trou­ble when one con­nects the real world, via lan­guage and abstrac­tion, with a log­i­cal pred­i­cate P, or a rule or asser­tion like P→Q or P&Q.  Log­ic in itself has noth­ing to say about the valid­i­ty or cor­rect­ness of an axiom, mean­ing a state­ment used as an input.  Worse, such a state­ment may be cor­rect in the sense orig­i­nal­ly intend­ed, but may break lat­er on due to shifts in con­text or exter­nal­i­ties.  It’s true, for exam­ple, that “bike” is short for bicy­cle; that a bicy­cle by def­i­n­i­tion has two wheels; and that an exer­cise bike has no wheels.  “You do the math”, as they say.  Luck­i­ly for us, our brains don’t go into a ker­nel pan­ic when we encounter such log­i­cal con­tra­dic­tions; in fact, only the most pedan­tic among us even notice.  Not even the pedan­tic then pro­ceed to con­clude that the moon is made of cheese, as clear­ly it must be:

  • B is the set of all bikes
  • e is an exer­cise bike
  • W is the set of all things that have two wheels
  • C is the set of all things that are made of cheese
  • m is the moon
  • giv­en:
  • e is in B
  • (x is in B) implies (x is in W)
  • e is not in W
  • then:
  • from (e is in B) we have (e is in W)
  • thus (m is in C) or (e is in W)
  • (e is not in W) implies (m is in C)

What’s inter­est­ing to notice about this kind of log­i­cal non­sense is that one can eas­i­ly con­struct exam­ples in which each pred­i­cate or rule on its own appears sound, but tak­en as a whole there’s an inbuilt con­tra­dic­tion.  That’s because the con­text with­in which each pred­i­cate must apply is defined by the domain of the prob­lem, and as one adds more and more pred­i­cates, rules and givens, one is often implic­it­ly expand­ing or redefin­ing the con­text.  Our asso­ci­a­tions and assump­tions about the mean­ings of sym­bols in real life are flu­id, allow­ing us to think about all sorts of com­plex cat­e­gories, duck­ing and div­ing as need­ed.  The price we pay is that we need to keep the big pic­ture in mind, think­ing glob­al­ly in order to ensure that our argu­ments con­tin­ue mak­ing sense.  With log­ic, on the oth­er hand, we can deal triv­ial­ly with huge sys­tems and rely on pure­ly local rules to gen­er­ate cart­loads of true state­ments, as in auto­mat­ed the­o­rem prov­ing; but now we have an exceed­ing­ly brit­tle sys­tem— with a sin­gle con­tra­dic­tion, the entire struc­ture fails.

But wait, it gets worse.  When log­ic is applied to any­thing which is not itself a very con­strained for­mal sys­tem— such as the world we actu­al­ly inhab­it— then we have uncer­tain­ty, so we must at a min­i­mum con­sid­er every pred­i­cate a ran­dom vari­able.  For P&Q we need to write the prod­uct of prob­a­bil­i­ties PQ; for P|Q we need to write P(1‑Q)+Q(1‑P)+PQ = 1-(1‑P)(1‑Q) = P+Q‑PQ; and so on.  There’s the some­what relat­ed, some­what unsat­is­fy­ing field of “fuzzy log­ic”, in which we take a con­tin­u­um of states for what are nor­mal­ly con­sid­ered Boolean vari­ables, such as “the ball is in the box”.  We can always split hairs, and say things like “so where exact­ly is the ball?  Does the box have a lid, and is the lid open?  What if the ball is half in and half out?” and so on.  One can then assign this vari­able 0 for ful­ly out of the box, 1 for ful­ly in, and 0.5 when the ball is halfway.  This makes my math friends gri­mace, because now there are all sorts of messy func­tions to con­sid­er, like whether the fuzzy ball-in-the-box mea­sure is by ball vol­ume frac­tion in the box vol­ume, or by Euclid­ean dis­tance, or (more usu­al­ly) by some cooked-up sig­moid with a rea­son­able length­scale.  Yuck!  Now add prob­a­bil­i­ty on top of that.  What about ensem­bles of sys­tems, and pri­ors on the prob­a­bil­i­ties?  What about exter­nal cor­re­la­tions?  What about uncer­tain­ty on the uncer­tain­ty, and so on to nth order?  Yes, dear friends, log­ic isn’t a fun­da­men­tal thing at all, but rather a very severe, very brit­tle approx­i­ma­tion scheme in which we neglect all of these effects of con­text, fuzzi­ness and uncer­tain­ty, and pre­tend that there are such things as Booleans, and ignore whether or not they car­ry mean­ing.  What we’re left with in this ster­ile Pla­ton­ic world is a sim­ple and not par­tic­u­lar­ly pow­er­ful frame­work for manip­u­lat­ing Boolean vari­ables.  Is this real­ly a sound foun­da­tion for life, the uni­verse, or any­thing?

And why does all this mat­ter?  Should we care that our cul­ture has iden­ti­fied ana­lyt­i­cal think­ing, skilled rea­son­ing and intel­lec­tu­al rig­or with this par­tic­u­lar rather under­pow­ered for­mal sys­tem?

Nowhere am I so des­per­ate­ly need­ed as among a shipload of illog­i­cal humans.” — Spock

I am designed to exceed human capac­i­ty, both men­tal­ly and phys­i­cal­ly.” — Data

I can think of at least two places where the log­ic fetish real­ly hurts us.  One is in the teach­ing of sci­ence and math in grade schools, where teach­ers and admin­is­tra­tors who aren’t them­selves sci­en­tists or math­e­mati­cians teach that these fields are ground­ed in for­mal meth­ods and log­i­cal deduc­tion— at best a very par­tial view, and cer­tain­ly not one that encour­ages the cre­ativ­i­ty, curios­i­ty and thought­ful explo­ration that under­lie these fields.

The oth­er one is law.  We set up increas­ing­ly elab­o­rate, qua­si-log­i­cal legal sys­tems osten­si­bly to ensure that laws are applied uni­form­ly and con­sis­tent­ly, mech­a­nis­ti­cal­ly, with­out the human judg­ment we’d call “cor­rup­tion” or “judi­cial activism”.  In court, we argue about whether or not a par­tic­u­lar pred­i­cate or axiom applies in a giv­en sit­u­a­tion; of course we’re real­ly argu­ing about what is or isn’t fair or sim­ply desir­able to us, but the argu­ment is always at a remove.  Mon­ey counts, as this sort of sock pup­petry requires pro­fes­sion­als “skilled in the art”, as they say.  Cas­es can hinge on nuances in the syn­tax of a rule.  Enor­mous com­plex­i­ty and expense goes into admin­is­ter­ing and apply­ing the rule­book.  It’s safe to assert that no state or nation­al legal “code” actu­al­ly “com­piles”, in the sense of being self-con­sis­tent even under care­ful treat­ment of the sets and pred­i­cates.  The rules are, after all, writ­ten over the course of cen­turies, by a parade of law­mak­ers with dif­fer­ing agen­das and pred­i­cate con­texts that mutate over time.  Para­dox­i­cal­ly, the more rules, the greater the need for “inter­pre­ta­tion”, which in turn com­pro­mis­es the intend­ed lev­el­ing effect.

Do we ben­e­fit from the exten­sive legal code­book, pre­sumed fixed at the time of judg­ment while the “inter­pre­ta­tion” is left to those “skilled in the art”?  (And does this sound famil­iar?)  Does the result­ing mix­ture of medieval scholas­ti­cism, Tal­mu­dic hair­split­ting and Roman ora­to­ry help us to be fair and just?  Giv­en sta­tis­ti­cal evi­dence, like the fact that after cor­rect­ing for crime sever­i­ty, black felons are over four times more like­ly to be giv­en the death sen­tence than white felons, I’m skep­ti­cal.  The judges are still human, still full of prej­u­dices and pri­ors, (and still white), but we now have an obfus­ca­tion mech­a­nism so that we can more eas­i­ly pre­tend it’s not so.  I don’t have a solu­tion, but I’d say that if we’re inter­est­ed in fair judg­ments, legal doc­u­ments mas­querad­ing as first-order log­ic— the blind fol­low­ing the one-eyed, as it were— may not be the best start­ing point.

Some­one I know, who spent sum­mers in Lagos as a teenag­er, enthus­es about the rule of law, because he has seen the hor­rors of oppor­tunis­tic law­less­ness. I agree, but it seems to me that by assert­ing that lengthy legal codes are the solu­tion, we com­mit the same error that Fun­da­men­tal­ists do when they claim that athe­ists have no moral com­pass. Yes, a belief that one is observed and judged all the time by a high­er pow­er hold­ing a book of laws will tend to con­strain one’s behav­ior; but does the choice to con­strain one’s behav­ior on moral grounds imply that one is reli­gious? Even log­ic is good enough to give us the answer: (A→B) ≠ (B→A).

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