The excluded middle

The law of the excluded middle may be succinctly stated as

A v -A

that is, either A or not A, where ‘A’ stands for some proposition such as ‘Socrates is mortal’. That should be in a nice chunky font with the correct symbols. The devil is in the ‘or’, and the one logicians usually mean is the exclusive or: you can’t have it both ways. Socrates took the hemlock, and died. QED.

Beginning students of logic commonly find this hard to digest. It is a poor representation of the way we usually think and argue. When it comes to mortal questions (to borrow from Thomas Nagel’s title), we tend to see in black and white. Either something is wrong or it is right; a person is a man or a woman; a person is black or white. Our world is made up of structural oppositions: the raw and the cooked. The black that excludes every shade of grey doesn’t even exist. It doesn’t help to say that it emits no photons. For the Chinese, this is a calligraphic axiom, or aphorism: black writing on white paper.

I have recently encountered my own philistine impatience with the dry preoccupations of analytic philosophy anew in Oswald Wiener, whose vituperative and obscene novel, with an English translation in the works, takes aim at Wittgenstein, both early and late. Perhaps the nub of the frustration is the sense that philosophy has become scholastic, turning away from all important problems. I’m no philosopher, but the other way to see this is that philosophy north of the English Channel, starting with Kant, is deflationary. That’s a term with a specific epistemological Sitz im Leben, but I take it as emblematic of a certain humility, expressed in Kant’s metaphor of the Wohnhaus, reason’s homely abode (I can’t remember whether this comes in the Preface or the Introduction to the first Critique). It’s no good answering those large questions if the foundations are unsound. That being recognised, there is an obligation to avoid pronouncing on them. There was nothing humble, though, about the tone of voice in which that was first asserted at Oxford a century ago.

Much of the basis on which we lead our lives is false, but we require those fictions to live at all, just as Hume said he needed faith to drink a glass of water. How, for example, can we ever trust another person? To do so relies on a presumption of good faith, or perhaps simply goodness, that invites refutation by experience, and calls for the blind eye. The world is grey, but we must pretend it is black and white to make it intelligible.

The origin of the idea of structural oppositions lies with Saussure: phonology is the logic of the sounds in a language according to the distinctions it deems to be salient, in order that man may speak intelligibly. This is especially clear when it comes to vowels, which are formed by positioning tongue and jaw across a continuum of available space. A is not A in proportion as it matches certain criteria (for example, if the tongue is a certain number of millimetres from the palate) but by virtue of not being E. The line between them is an indistinct border region that is in fact unfenced. The native speaker (barring interference from surrounding consonants, always present, but we must avert our gaze from it) aims for the middle of the correct region, but achieves idiomatic fluency not by hitting one spot, but by staying as far away from the edges as may be. This is different from playing the violin, but it feels the same: it’s very hard as a foreigner to get it right, but effortless for those born to it. The temptation (to take a different example) is to fudge the distinction between long and short by aiming at the border, but you must articulate it with conviction (as in Italian, or Finnish, or for that matter, Latin); this is hidden from English speakers because distinctions of length generally coexist with differences of quality.

Language spoken idiomatically gives an impression of well-tempered rightness, with everything in its place, like a familiar domestic setting. Moving to the higher level of (I suppose) syntax shows how much fiction is woven into that sense. If one attempts to accurately transcribe recorded speech, it disintegrates into a concatenation of false starts and mumbles. There is no single level of accurate transcription, as opposed to the tidied up version. When linguists make such transcriptions, the level of detail will depend on their purpose. Anyone who has tried it with a tape recorder knows just how hard that is.

A squirrel just raced across my lawn, and probably up a tree, a perfect sine wave rippling through it as its mode of locomotion in the horizontal plane. That is its nature, one thing visible to us that it knows superlatively well; as the spider weaves her web, and as we do the sort of thing I have been trying to write about. Dogs can see it too, and it commonly enrages them: that sinuosity cries out to be expunged, if only it could be caught before the tree. Sometimes when I am cycling I almost run over a squirrel, transfixed by frisky indecision.

But the world is not structured like a language; we are. There is blindness in that, and it cannot be cured by philosophy averting its gaze.


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