What is Formal Logic?

I found this talk by accident, on William Byrd's web page. It was given at Stanford in March 2013 whilst I was living in Rurrenabaque trying in vain to sell my logic books to get some money to buy food. See the book Surfaces and Essences Analogy as the Fuel and Fire of Thinking.

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There used to be a distinction between symbolic logic and formal logic. If you ask Google AI what this distinction is it says it is no longer, since formal logic before 1900 was found to be wanting in rigour and thanks to the efforts of Frege, Russell et al it has been superseded by symbolic logic. Well, if Google AI says that then it must be true.

What if Frege, Russell et al had actually not properly understood what formal logic was? Then there would be a distinction between Formal Logic and modern symbolic logic which is unknowable to anyone who only recognises modern symbolic logic.

In fact you don't have to look very far to see that this is clearly the case. See Aristotle's De Interpretatione Chapter 1:

First we must settle what a name is and what a verb is, and then what a negation, an affirmation, a statement and a sentence are.

Now spoken sounds are symbols of affections in the soul, and written marks symbols of spoken sounds. And just as written marks are not the same for all men, neither are spoken sounds. But what these are in the first place signs of—affections of the soul—are the same for all; and what these affections are likenesses of-—actual things—are also the same. These matters have been discussed in the work on the soul and do not belong to the present subject.

Just as some thoughts in the soul are neither true nor false while some are necessarily one or the other, so also with spoken sounds. For falsity and truth have to do with combination and separation. Thus names and verbs by themselves—for instance ‘man’ or ‘white’ when nothing further is added—are like the thoughts that are without combination and separation; for so far they are neither true nor false. A sign of this is that even ‘goat-stag’ signifies something but not, as yet, anything true or false—unless ‘is’ or ‘is not’ is added (either simply or with reference to time). 

So we are directed to the work on the soul, De Anima, if we want to learn more about the denotation of spoken sounds and written marks and the meaning of truth and falsity. Perhaps it is because De Anima was not considered part of 'The Organon' that this point has been missed.

Therefore to understand what may have been lost in the "logical whirlpool" of the 1900s we need to understand what is the difference, if any, between truth as a judgement in a formal system of symbolic logic and truth as affections of the soul. If there is a difference then we will have made a step towards understanding what it was that was that we lost, what it is called, and whether and how we may be able to recover it.

De Interpretatione Chapter 3 

A verb is what additionally signifies time, no part of it being significant separately; and it is a sign of things said of something else.

It additionally signifies time: ‘recovery’ is a name, but ‘recovers’ is a verb, because it additionally signifies something’s holding now. And it is always a sign of what holds, that is, holds of a subject.

‘Does not recover’ and ‘does not ail’ I do not call verbs. For though they additionally signify time and always hold of something, yet there is a difference—for which there is no name. Let us call them indefinite verbs, because they hold indifferently of anything whether existent or non-existent.

The difference between symbolic and formal logic, then, is that in symbolic logic the meaning of the statements has been reduced to symbols on a page and these are not even symbols which signify sounds any more. What has been lost is the meaning of logical propositions, because it has been divorced completely from any and all affections of the soul. Formal logic on the other hand was all about what could be meaningfully said concerning these affections of the soul, and all of that is impossible in symbolic logic as handed down from Frege, Russell et al.

In formal logic, the deductive system is the whole Universe, including the individual substantial beings within.  See Szymon Kaliski - "Programmable Ink". What Hofstadter is talking about is really just the bootstrapping process.