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LESSON: SETS, MULTI-SETS, AND SEQUENCES: The letters occurring in the
word ‘LETTER’ form a set of four, namely {L, E, T, R} in order of
occurrence or {E, L, R, T} in alphabetical order, thus using two of the
many list-names of that set.  Other names of the same set are {L, E, T,
T, R}, {L, E, T, T, E, R}, and {L, L, E, E, T, T, R, R}.  In naming
sets, both order and number of member-name occurrences are irrelevant ­
the set is determined completely by its members; repeating or
rearranging member-names does not change the set named.  Many of the key
ideas of a theory of sets in this sense are found in Boole 1847.  The
occurrences of letters in ‘LETTER’ form a multi-set of six, namely [L,
E, E, T, T, R], using one of several list-names of that multi-set.
Other names of the same multi-set are [E, L, E, T, T, R], [L, E, T, T,
E, R], and [E, E, L, R, T, T].  In naming multi-sets, order is
irrelevant but repetition of a name signifies repetition of the thing
named ­ the multi-set is determined completely by its members together
with their “multiplicities”.  I do not know who is to be credited for
the key ideas, but a categorical axiomatization of a theory of
multi-sets is given in §3.1 “Repetition Theory” of Corcoran 1980,
“Categoricity”, HPL 1:187-207.  The order of letters occurring in the
word ‘LETTER’ forms a sequence of length six, namely < L, E, T, T, E, R
 >, using one of the list-names of that sequence.  In naming sequences,
both order and number of member-name occurrences are crucial ­ the
sequence is determined completely by its members in their order.  There
was no axiomatization of a theory of letter sequences before Tarski
published his categorical string theory in the 1933 truth-definition
paper. See pages xxi and 173-4 of Logic, Semantics, Metamathematics.
John Corcoran, 15 July 2004.


Wednesday, July 21, 2004
12:00-1:30 P.M.
141 Park Hall
ml> David Hitchcock, Philosophy, McMaster University.
TITLE: Stoic Propositional Logic: a New Reconstruction.
ABSTRACT: The only system of logic published in the 2,200 years between
Aristotle and Boole was the system of propositional logic invented by
Chrysippus (c. 280-207 BCE), the third head of the Stoic school.
Chrysippus’ system was dominant for 400 years, until it came under
attack from advocates of Aristotle’s categorical syllogistic; bits of it
were eventually absorbed into a confused amalgamation with Aristotle’s
system, and the system itself was forgotten. Scholars of the past 50
years have proposed 10 different reconstructions of the system from the
scattered, incomplete and often hostile ancient testimonies. Standing on
the shoulders of these 10 predecessors, I propose an eleventh. I aim to
show that we can infer with reasonable confidence the parts of the
system for which we have no direct testimony. I shall draw attention to
some often ignored features of the system which remain of logical or
philosophical interest: its punctuation-free notation, the status of the
premisses of an argument as a multi-set rather than a set or a sequence
(i.e. repetition counts but order does not), Chrysippus’ unusual
conception of validity, the correct translation of the Greek label for
the primitives of the system (“undemonstrated arguments”, not
“indemonstrable arguments” (Latin indemonstrabilia), the almost
universal translation), the great generality of the descriptions of
these primitives in our sources (as compared to the moods and examples),
the probable existence of an extended set of primitives which
accommodates conjunctions with more than two conjuncts and disjunctions
with more than two disjuncts, how Chrysippus avoided both invalidity and
fudging in the latter case, the absence of a contraction rule, the basis
for the system’s exclusion of redundant premisses, the reason why
hypothetical syllogisms (arguments of the form if p then q; if q then r;
therefore if p then r) are not derivable in the system (even though
Aristotle’s pupil Theophrastus had already pointed out the validity of
this form of argument). Finally, I shall argue that, even if we use the
criterion of completeness which Chrysippus himself would have accepted
(and could have formulated), the system is not complete. It is an open
problem, I shall conclude, what one needs to add to the system of
Chrysippus in order to make the system Chrysippean-complete, or even
whether it is possible to do so without making it Chrysippean-unsound.


Wednesday, July,28 2004
12:00-1:30 P.M.
141 Park Hall
ml> John Corcoran, Philosophy, University at Buffalo
TITLE: Meanings of Argument


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