THE SOUL OF THE ELITE LOGICIAN: The look and feel of professional logic!

TUESDAY, DECEMBER 11, 2018

The chains of Marley's ghost: Where have all the logicians gone? We asked that question last week.

Our society has badly needed their help over the course of the past thirty years. But every time we waited for rescue, no logician arrived on the scene.

"Hold on!" the voices angrily cry. Academic logicians, those in the academy, don't waste their time on the trifling disputes which form the public discourse!

People die all over the world when our "public logic" fails. But our elite logicians are occupied with loftier concerns, such as the ones which emerge in a passage from Professor Hart's 2010 book, The Evolution of Logic.

We're basically quoting this passage at random. It appears at the start of Chapter 3, Expeditions: Which Sets Exist?

One more point should be made clear. We've selected Hart's book, not because we think it's lousy work, but because we assume it's technically competent, good:

HART (page 59): Frege layered functions. A first-level function assigns objects to objects: doubling is a first-level function that assigns six to three; and the concept:green is a first-level function that assigns truth to all and only the green things. The derivative of the square function is the doubling function, while that of the sine is the cosine, so differentiation is a second-order function. In another example, Frege reads "There are carrots," so its subject is the concept:carrot and its predicate is the concept:existence. Existence is thus a second-level concept whose value is truth at all and only the first-level concepts under which something falls. This allows Frege to refine Kant's criticism of the ontological argument for the existence of God. Kant said that existence is not a predicate, which is heroic, or even quixotic, grammar. Frege could say that since existence is a second-level predicate, it is at the wrong level to be a defining feature of an object like God.

There! That's a taste of what "logic" is like on the modern professional level. Involved in lofty discussions like this, the elite logician has little time for the minor errors which may decide presidential elections and with them the use of deadly force against children all over the world.

Professor Hart's book is part of a six-volume series, The Evolution of Modern Philosophy. The series is published by the Cambridge University Press, the world's oldest publishing house.

As noted, we chose Hart's book because we assume it represents a competent example of modern high logic. The publisher describes the book like this:

CAMBRIDGE UNIVERSITY PRESS: Examines the relations between logic and philosophy over the last 150 years. Logic underwent a major renaissance beginning in the nineteenth century. Cantor almost tamed the infinite, and Frege aimed to undercut Kant by reducing mathematics to logic. These achievements were threatened by the paradoxes, like Russell's. This ferment generated excellent philosophy (and mathematics) by excellent philosophers (and mathematicians) up to World War II. This book provides a selective, critical history of the collaboration between logic and philosophy during this period. After World War II, mathematical logic became a recognized subdiscipline in mathematics departments, and consequently but unfortunately philosophers have lost touch with its monuments. This book aims to make four of them (consistency and independence of the continuum hypothesis, Post's problem, and Morley's theorem) more accessible to philosophers, making available the tools necessary for modern scholars of philosophy to renew a productive dialogue between logic and philosophy.

According to that blurb, mathematical logic became a recognized subdiscipline in mathematics departments. As an unexplained consequence, philosophers lost touch with its monuments.

That said:

Especially at this time of year, we typically try to stress the distinction between Morley's theorem and Marley's ghost. The distinction tends to be lost on those who lack interest in logic as well as literature.

Is something gained from modern high logic as practiced down through all those years? We're not sure how to answer your thoughtful question.

Plainly, something is lost when a society's logicians ignore the failed logic of daily life, focusing solely on loftier topics. Then too, a question arises when we review the sweep of the upper-end work the Cambridge University Press blurb describes:

Is it possible, just perhaps ever so slenderly possible, that our greatest logicians have never been all that sharp? We can't stop ourselves from asking such questions when we read a book like Rebecca Goldstein's favorably-blurbed 2005 volume, Incompleteness: The Proof and Paradox of Kurt Godel.

As we've noted in the past, Goldstein is a highly-regarded novelist, but she's also a ranking philosophy professor. Her book was favorably blurbed by a trio of ranking figures—by Stephen Pinker, by Brian Greene, and also by Alan Lightman.

After that, the book was favorably reviewed by Jim Holt in The New Yorker.

Writing with a novelist's flair, Goldstein discusses the work of Godel, who she describes as "the greatest logician since Aristotle." She also describes Godel's interactions with Ludwig Wittgenstein, whose later work was very hot until, according to Professor Horwich, the professional world whose work he critiqued decided to turn him out.

How sharp have our greatest logicians been? We'll scan Goldstein's portraits all week.

Tomorrow: Aping Wittgenstein's tics

Also this: A footnote is appended to the part of Hart's text which we've highlighted. The footnote reads like this:

Perhaps more generally a quantifier is a second-level function whose value at an (n + 1)-ary first-level concept is an n-ary concept, unless n is zero, in which case its value is a truth value, an object. In that case, quantifiers would be second-level functions sometimes having first-level concepts as values and sometimes objects as values. When the value of a first-level concept at an object is truth, Frege says the object falls under the concept. Perhaps the concept:falls-under is a binary second-level concept whose first argument is an object and whose second is a first-level concept. In that case, second-level concepts could also have arguments of different levels.

We selected this book because Hart's fully competent. This is high order "logic" at work.