SUO: Re: Peirce's MS 514
James Piat wrote:
>.... But what else? My guess is that the division between
>the two [axioms and rules of inference]
>also reflects some fundamental unrecognized meta-axioms and or
>meta-rules of inference upon which the proposed logic rests.
>Unrecognized assumptions that may come back to haunt the project.
Yes, indeed. Logicians usually make a distinction between
"provability" and "semantic entailment". Suppose you have
a set of statements A and another statement p. Then you can
define the two notions as follows:
1. Provability: p is said to be _provable_ from A in some
logical system L iff there are rules of inference that
allow p to be derived by performing some operations on
the statements of A to generate the statement p.
2. Entailment: p is said to be _semantically entailed_ by
A iff in any state of affairs (or universe of discourse,
or possible world) for which all the statements of A are
true, the statement p is also true.
This distinction is as old as Aristotle, but it was most
clearly formulated by the medieval Scholastics. In particular,
William of Ockham's _Summa Totius Logicae_ is a very clear
statement of this distinction. Aristotle, Ockham, and everybody
who read them, including Peirce, took semantic entailment as
the more fundamental notion and used it to justify their
rules of inference.
The situation was muddied at the end of the 19th century by
two prominent logicians who didn't do their homework: Gottlob
Frege and Bertrand Russell. Frege was an independent thinker,
who happened to invent the first complete system of inference
for first-order logic. But he hadn't read Ockham or the other
logicians, and his approach was highly idiosyncratic.
Instead of taking semantic entailment as the fundamental notion,
Frege took provability as fundamental and even claimed that
"truth" was too vague a notion to be used as a foundation
for logic. Bertrand Russell fell hook, line, and sinker for
Frege's point of view, which he preached to the masses.
In 1935-36, Alfred Tarski introduced (or actually reintroduced)
semantic entailment as the more fundamental notion. Tarski's
approach, which is usually called "model-theoretic semantics"
is equivalent to what Ockham and Peirce used as the foundation
for logic. Tarski, in fact, did quote Aristotle as a precedent
for his approach, because he was fighting an uphill battle
against the Frege-Russell propaganda, which had taken firm root
in the early 20th century.
For an indication of how hard Tarski had to argue for his
position, see the "polemical remarks" in a paper that Tarski
wrote in 1944 to support his position:
http://www.bestweb.net/~sowa/misc/tarski.htm
I also summarize some of those points in my commentary:
http://www.bestweb.net/~sowa/peirce/ms514.htm
JP>.... I'm not so sure rules of inference (for example) are
>operations. Maybe they are simply definitions of what constitute things
>--the properties of things. Something like that. Does any of this make
>sense? I guess I'm just questioning the notion that logic is (or should
>be) an account of how the transformation of things must necessarily (in
>all possible worlds) be. Maybe logic should be an account of what the
>notions of "transformation" and "things" presuppose.
Aristotle, Ockham, Peirce, and Tarski would be sympathetic
to your concerns. They developed their systems of logic
for the purpose of characterizing the more fundamental notion
of semantic entailment, which is based on the way the world
itself happens to be.
Frege and Russell, who had not read the literature, claimed
that logic was more fundamental and that no independent notion
of truth or semantic entailment was necessary.
Today, logicians have gone back to the Aristotelian-Scholastic
point of view, but they usually credit it to Tarski, who waged
the hardest battles against the mistakes of Frege and Russell.
John Sowa