Modern interpretations of Aristotle's logic
waveletter
I'd like to discuss modern interpretations of Aristotle's logic. The
two core ancient texts are the "Analytics". I don't think the issue
has been settled as of this date (Spring 2007). But there are two
counterposed strands of thought on how we should view Aristotle's
contribution to the analysis of reason.
There is a standard interpretation of The Philosopher's logic that has
been largely accepted since the mid-1950s or thereabouts--that of
Lukasiewicz ("Aristotle's Syllogistic From the Standpoint of Modern
Formal Logic," Oxford: Clarendon, 1951). Lukasiewicz contended that
Aristotle's "Prior Analytics" put forward, in effect, a deficient
axiomatic system, and the Polish logician offered a replacement
axiomatization of Aristotle's logic in terms of modern predicate
logic. Aristotle's theory of perfecting a syllogism had to be
abandoned, in Lukasiewicz's account, as misguided. The first
successful logicians were, evidently, the Megarians, a school of Stoic
logic thriving in late antiquity.
A generation or so after Lukasiewicz, in the 1970s, some logicians--
principally T. Smiley and J. Corcoran--offered a counter-theory. Their
interpretation preserved almost all of Aristotle's original teaching
from the "Analytics." In addition to rebutting the mistakes attributed
to Aristotle by Lukasiewicz, the alternative interpretation revealed
logical soundness and completeness results that had not been
previously noticed. With a natural axiomatic system laid down in the
"Posterior Analytics," and a supporting sound, complete deductive
system in the "Prior Analytics," Aristotle began to look extremely
modern: Carnap could be envious! The contributions of Corcoran,
Smiley, and their students established a new perspective on
Aristotle's deductive and axiomatic systems--and this is what I will
be exploring in detail on this list for the next few months.
Thanks!
--Ron
waveletter
Let me compose a bibliography for the upcoming discussion of modern
interpretations of Aristotle's "Prior Analytics" and "Posterior
Analytics."
1. J. Łukasiewicz, "Aristotle's Syllogistic From the Standpoint of
Modern Formal Logic," Oxford: Clarendon, 1951).
2. There's a 2nd edn. to the above, which I don't have. I think it has
some remarks in the preface on Łukasiewicz's motivation and
justification for his interpretation. It's often quoted. Sorry, I'm
not the best scholar here. Maybe over to the state college library
sometime to see what they've got.
3. Poof! Now I'm a scholar. In opposition to Łukasiewicz, you can just
read, J. Corcoran, 'Completeness of an ancient logic,' in "Journal of
Symbolic Logic," vol. 37, no. 4, pp. 696-702, Dec. 1972.
4. There's another article, the research being developed in complete
independence from Ref. 3, that follows Corcoran's line of thought: T.
Smiley, 'What is a syllogism?,' in "Journal of Philosophical Logic,"
vol. 2, pp. 136-154, 1973.
5. See as well the articles in the volume J. Corcoran, ed., "Ancient
Logic and its Modern Interpretations," Dordrecht-Holland: D. Reidel,
1974.
6. In particular, check out Prof. Corcoran's article 'Aristotle's
Natural Deduction System,' ibid., pp. 85ff.
One remark that Corcoran makes is as follows: "...we attempt to show
that Aristotle's syllogistic is an *underlying logic* which includes a
natural deductive system and that it is not an axiomatic theory as had
previously been thought." [Previously thought by Łukasiewicz, for
instance!]
I'll add some more standard references for this problem as I progress
in my critique. Thanks! Just getting started, amidst all the local
surrounding chaos....
--Ron
Thanks!
--Ron
waveletter
Yow! For some reason, I got about 8 copies of this message in my
email. I see that it's only posted once on the Google Groups site,
which is good, but I hope the rest of you did not get plastered with
this post. Oh, I noticed as well that the "L with stroke" character
that begins Prof. Lukasiewicz's name in Polish did not come through in
the email. 8-bit ASCII problem I suppose. It seems to be working well
here, through the browsers.
By the way, the "L with stroke" character is pronounced as a short
English 'w' sound, as in the words 'were', 'with', 'well', and
'willful'. But not like 'few' or 'stew', where there is a distinct 'u'
vowel sound signaled by the 'w'. So we would pronounce
'Lukasiewicz' (I took out the L-with-stroke character) like "Wu-kah-
see-witz". I'm not sure where the stress accent lies. Does anyone on
the list know?
Let me list the bibliography again, without any unicode characters (I
hope). I'll add some other items that you might want to collect over
the discussion. Comments in [brackets].
1-2. J. Lukasiewicz, "Aristotle's Syllogistic From the Standpoint of
Modern Formal Logic," Oxford: Clarendon, 1951). [Marks the beginning
of modern logic's interpretation of Aristotle.]
3. J. Corcoran, 'Completeness of an ancient logic,' in "Journal of
Symbolic Logic," vol. 37, no. 4, pp. 696-702, Dec. 1972. [Alternative
interpretation of Aristotle, counterposed to Lukasiewicz, Refs. 1-2].
4. Smiley, 'What is a syllogism?,' in "Journal of Philosophical
Logic," vol. 2, pp. 136-154, 1973. [Agreement with Corcoran, Ref. 3]
5. J. Corcoran, ed., "Ancient Logic and its Modern Interpretations,"
Dordrecht-Holland: D. Reidel, 1974.
6. J. Corcoran, 'Aristotle's Natural Deduction System,' ibid., pp.
85ff.
Other references:
7. R. Smith, trans., "Aristotle: Prior Analytics," Indianapolis, IN:
Hackett, 1989. [Outstanding introduction to Aristotle's deductive
system, followed by new translation taking into account Corcoran &
Smiley's interpretations of Aristotle's contribution. You must buy
this book. You can afford it.]
7. M. Frede, 'Stoic vs. Aristotelian Syllogistic,' in "Essays in
Ancient Philosophy," Minneapolis, MN: Univ. of MN Press, 1987.
8. Gabbay & Woods, eds., "Handbook of the History of Logic," vol. 1,
Amsterdam: Elsevier, 2004. [Four essays, some 300 pages on
Aristotelian logic. If you can afford it, you must buy this book. If
you can't afford it, you should get a 2nd job and then buy this book!]
9. B. Mates, "Stoic Logic," Berkeley: Univ. of CA Press, 1961.
[Classic breakthrough study of Stoic propositional logic.]
10. G. Patzig, "Aristotle's Theory of the Syllogism," Dordrecht,
Holland: D. Reidel, 1968. [In general, supports Prof. Lukasiewicz's
interpretation.]
11. I.M. Bochenski, "Ancient Formal Logic," Amsterdam: North-Holland,
1951. [Interpretation of Aristotle similar to Lukasiewicz.]
Comprehensive general histories:
12. Kneale & Kneale, "The Development of Logic," Oxford: Clarendon
Press, 1962. [The standard ought to be read again and again.]
13. I.M. Bocheński, "A History of Formal Logic," Notre Dame, IN: Univ.
Notre Dame Press, 1961. [Almost like an annotated bibliography
compared to the Kneales' book, a little more mathematical notation and
results, extended quotations from the sources, more figures, and
numerous bits of additional information. Oh yeah, let me try to add
the acute accent mark on the 'n' in 'Bochenski' and see how it mutates
in ASCII with the email. I believe it is pronounced like the Spanish
'ñ', as in 'año,' 'year'.]
Next time, why don't we start reconstructing The Philosopher's
syllogistic?
--Ron
Ron Allen
Let's get started by reconstructing the syntax of
Aristotle's syllogistic. I'll be using Prof. Robin
Smith's translation and introduction of the "Prior
Analytics" as well as Prof. John Corcoran's seminal
article from the "Journal of Symbolic Logic" (refs. 3
and 7 in the previous, attached message). The other
parts to a modern logic are its semantics and its
schema for deductions.
So, I'm taking The Philosopher's "Prior Analytics"
apart, using the tools of modern philosophical and
mathematical logic, and I'm putting it back together
to see just how strong a system it is. Actually, as
Corcoran (ref. 3) and Timothy Smiley (ref. 4) show,
it's really strong and quite an impressive feat for 25
centuries ago!
1. The symbols. The first part of a logical syntax is
its set of symbols. In the case of syllogistic, there
are variables, represented by upper-case letters: A,
B, C, ..., X_1, X_2, X_3, ... etc. And there are
logical constants, which, following a tradition that
arose in medieval times, we will denote by a, e, i,
and o.
2. The language is (finite) strings of symbols. Thus,
the language of Aristotle's syllogistic is all
(ordered) strings that consist of zero or more
variables or logical constants. The order matters.
Thus, the string AaB is different from the string aAB.
Oh, there's also an empty string, consisting of no
symbols. It's not something that Aristotle would have
been pleased to entertain, but it's usually included
for mathematical elegance in modern treatments. We are
modern, but not quite post-modern here.
3. Most of the strings in the language are
meaningless. The ones that are meaningful are the
sentences. A sentence is a string <variable><logical
constant><different variable>. Thus, AaB is a
sentence, but AeA is not a sentence, nor is oAB.
Well, let me see if this posts of Google Groups...the
last one didn't. And if it works, then we'll keep
reconstructing away at Aristotle's syllogistic.
Thanks!
--Ron
--- waveletter <wave...@pacbell.net> wrote:
=== message truncated ===
waveletter
It's cool and still here on the US west coast, I've walked the dogs,
and I'd like to continue with my modern inspection of Aristotle's
deductive system in the "Prior Analytics."
I'm having a little browser problem with the Google Groups editor, so
I'll keep this short. Seems that my sentences run on forever, without
wrapping around. I'm forced to continue to the right forever or hit
<cr><lf>, the 'Return' key, to come back to someplace where I started
typing.
Well, OK. To summarize from last time:
1. Symbols: Variables (A, B, ..., P, Q, X_1, X_2, ...) + Logical
constants (just four-- a, e, i, o).
2. Language: Finite strings of symbols. The order is significant. Most
are meaningless.
3. Sentences are finite strings of symbols that begin with a variable,
which is followed by a logical constant, which is followed by a
*different* variable.
Remarks. We recognize English without it being a sentence. One place
where you can find lots of recognizably English language non-sentences
is in e.e. cummings poetry.
"anyone lived in a pretty how town
(with up so floating many bells down)
spring summer autumn winter
he sang his didn't he danced his did."
We covered this poem in high school (US, California, age 16, sophomore
year). You can spot a non-sentence here, but it's still English. It
means something, too. But we want to isolate out certain meaningful
strings as sentences that play a role in the deductive system. It's
hard to deduce things from "spring summer autumn winter". So, it makes
some sense to say that finite strings are the language, and the define
the sentences as strings with a specific form.
4. How the sentences read.
4a. PaQ: "P belongs to all Q"
4b. PeQ: "P belongs to no Q"
4c. PiQ: "P belongs to some Q"
4d. PoQ: "P belongs to some not-Q"
5. There are contradictories of sentences in the syntax.
5a. The contradiction of PaQ is PoQ and vice-versa.
5b. The contradiction of PiQ is PeQ and vice-versa.
We could have a theorem here, by the way.
Theorem. For any sentence in the language of Aristotelian syllogistic,
the contradiction of the contradiction of the sentence is the
sentence itself.
This is a meta-theorem, a result shown about the logic, but not shown
within the logic.
The next component part of the deductive system is the schema for
deductions. I'm departing a bit from Corcoran's presentation in his
JSL article (1972) as I lay this out.
Let's see how this posts. I wish the words would wrap. Alas.
Thanks!
--Ron
waveletter
I'm continuing to jot down formal elements of Aristotle's natural
deduction system, as given in his "Prior Analytics."
We've defined sentences to finite strings of symbols of the form PxQ,
where P and Q are distinct variables and x is a logical constant (a,
e, i, o).
6. We also say that the first variable in a sentence string is the
*predicate* and the second is the *subject* of the sentence. Note that
this is, generally speaking, backwards from the way subjects and
predicates occur in English sentences. The movie "300" just came out
of Hollywood (http://300themovie.warnerbros.com/), so maybe I can make
a sentence as follows: "All Spartans are brave." The subject is
'Spartans' and the predicate is 'brave'. But in the formal syntax of
Aristotelian syllogistic, the predicate comes first. "Brave belongs to
all Spartans."
7. One final syntactical notion is that of an *argument* or an
*account*. In ancient Greek, the word was 'logos'. Now this is a very
overworked noun in ancient Greek. It means, among other things,
"word", "speech", "account", "reason", "mind", "rational number
(fraction)". An *argument* is a sequence of sentences. The order is
important, and that is why I said that an argument is a "sequence" and
not a "set" or "collection" of sentences. Two arguments may contain
the same sentences, but they differ because of the order of the
sentences.
Next time, we'll pick up the juicy topic of deductions.
Thanks!
--Ron
> which is followed by a logical constant, which is followed by a
> *different* variable.
waveletter
There are three parts of a formal logic:
(1) The syntax (from the Greek verb 'suntasso', to put into order
together), which I covered in the last few posts (attached below);
(2) The deduction scheme, which we're going to start with next;
(3) The semantics, how the language is related to a model, an abstract
or concrete world where there are things to which the language
elements are related.
A deduction is an argument that has a special form. That is, it is a
sequence of sentences in the language of syllogistic, Aristotle's
logic, that has specific properties.
We first need to explain *conversion*. There are three types of
conversion:
1a. PeQ converts to QeP ("P belongs to no Q converts to Q belongs to
no P");
1b. PiQ converts to QiP ("P belongs to some Q converts to Q belongs to
some P");
1c. PaQ converts to QiP ("P belongs to all Q converts to Q belongs to
some P").
I'll use the following notations for conversion: -> or |-. Thus, PeQ -
> QeP, AaB |- BiA, and so forth.
Next time, I'll cover the complete deductions from Aristotle's First
Figure, or Schema.
Thanks!
--Ron