John Corcoran corcoran at buffalo.edu
Fri Jun 23 14:08:17 EDT 2006


mathematics can be learned only by observation and experience. In fact,
the ability to reason correctly and to understand correct reasoning is
itself a prerequisite to the study of formal logic. - Solomon Feferman,
The Number Systems. 1964.

Wednesday, June 28, 2006	12:00-1:30 P.M.	142 or 280 Park Hall
SPEAKER: John Corcoran Philosophy, University of Buffalo.
TITLE: Counterexample-Content Theorems.
ABSTRACT: This article culminates one strand of the discussion reported
in BULLETIN OF SYMBOLIC LOGIC 11(2005) p. 460 and 11(2005) pp. 554-5.
Two propositions are [logically] equivalent if each is a consequence of
the other.  A suitable first-order language is interpreted in the set N
of [natural] numbers.  A number n is a uniform [counterexample] number
if, for every false universal proposition Ax P(x), there exists an
equivalent universal proposition Ax Q(x) having n as a counterexample.
Let n* be a numeral for n.  Given any n, every proposition Ax P(x) is
equivalent to Ay (y = n*--> Ax P(x)).  Every n is uniform.  A set S is a
uniform [counterexample] set if, for every false Ax P(x), there is an
equivalent universal proposition Ax Q(x) having S as its set of
counterexamples.  Every singleton is uniform.  Cardinality
considerations show that not every non-empty set is uniform.  Every
proposition P is logically equivalent to Ax ((Ex Q(x)-->Q(x))-->P),
where Q(x) is arbitrary. With E(x) true and P false, the counterexamples
for Ax ((Ex Q(x)-->Q(x)--P) are exactly the numbers satisfying Q(x).
Thus, every non-empty definable set is uniform. It is easy to see that
this result cannot be improved. Thus we have the Numerical Uniform
Counterexample-Set Theorem: given any false proposition F and any
non-empty set S of numbers, there is a false universal proposition
logically equivalent to the given proposition F and having the given set
S as counterexample set if and only if the set S is definable. Analogous
results evidently apply with other languages interpreted in other
universes. Roughly speaking, this shows that there is essentially no
connection between the counterexamples for a given false universal
proposition and its logical information content. Special thanks to Mark
Brown (Syracuse) and Frango Nabrasa (Tampa).

Wednesday, July 5, 2006	12:00-1:30 P.M.	141 Park Hall

SPEAKER: John Corcoran, Philosophy, University of Buffalo.
TITLE: Existential-Import Theorems.
ABSTRACT: A universal proposition has general universal import (GEI) if
it implies the corresponding existential.  A universalized conditional
has conditional (or relative) universal import (CEI) if it implies the
corresponding existentialized conjunction.  
Some basic definitions and facts about existential import are given and
some open questions are asked in an April 2006 FOM Digest piece
http://www.cs.nyu.edu/pipermail/fom/2006-April/010387.html. For
classical first-order logic, every universal has GEI. Moreover, the
question of a necessary and sufficient condition for conditional
existential import is not open.  The (somewhat surprising) Basic
Existential Import Lemma BEIL is that a universalized conditional Ax
(S(x) --> P(x)) implies the corresponding existentialized conjunction Ex
(S(x) & P(x)) iff Ex S(x) tautological (logically true), where S(x) and
P(x) are arbitrary first-order formulas having only x free. Consider a
first-order language L with an interpretation I having a non-empty set U
as universe of discourse [or range of the individual variables]. Since
Ex x = x is tautological, Ax (x = x --> P(x)) has CEI (and GEI) and the
set U is "subject to existential import in I" - a property precisely
defined in the paper.  (From here it is easy to see that every universal
proposition is logically equivalent to a universalized conditional
having both GEI and CEI.) The main question treated in this paper is:
how extensive is the class of proper subsets of U that are subject to
existential import?  The rhetoric encountered in some textbooks would
suggest the hypothesis that very few if any sets other than U itself are
subject to existential import in I.  BEIT implies that S is subject to
existential import iff it is expressible (or definable) in I by a
formula S(x) where Ex S(x) tautological (logically true). Such sets are
said to be tautologically non-empty.  Thus, the above question amounts
to the question of which non-empty expressible sets are tautologically
non-empty. This paper proves that they all are.  More generally, we have
the Existential Import Distribution Theorem EIDT: Let L, I, and U be
arbitrary. Every non-empty subset of U definable under I by a formula of
L is both (1) definable by a formula whose existentialization is
tautological and also (2) definable by a formula whose
existentialization is not tautological. Thus, except for formulas
defining the null set, every formula that does not give rise to
conditional existential import is coextensive with one that does give
rise to conditional existential import and conversely. The holding of
existential import is as widely distributed as its failing. Thanks to
M.Brown, L. Compton, M. Davis, H. Enderton, J. Friedman, D. Kaplan, K.
Miettinen, J. Miller, M. Mulhern, and A. Urquhart.

Wednesday, July 12, 2006	12:00-1:30 P.M.	141 Park Hall

SPEAKER: John Kearns, Philosophy, University of Buffalo.
TITLE: The Epistemic Character of Deduction: A Speech-Act Approach.
ABSTRACT: Historically, the subject matter logic has had both an
epistemic and an ontic or ontological dimension. From the time of
Aristotle until the mid-nineteenth century, the focus was primarily
epistemic. Logic dealt with arguments, deductions, and proofs. Following
the work of Boole and Frege, logic took an ontic turn. The move toward
ontology was a genuine advance for logic, and both broadened and
deepened the subject. But this advance should not lead to the
abandonment of the epistemic dimension of logic. Frege and others may
have confused a concern for epistemology with the psychologism which
they hold in contempt. That is simply a mistake.
	Illocutionary logic, which is the logic of speech acts, or
language acts, comes closer to giving equal time, or equal
consideration, to both the ontic and the epistemic. This talk will
sketch a simple system of illocutionary logic, and comment on some of
the issues that are clarified, and some of the puzzles that are solved
or dissolved with the help of illocutionary logic. 

Wednesday, July 19, 2006	12:00-1:30 P.M.	141 Park Hall

TENTATIVE SPEAKER: Frango Nabrasa, Mathematical Sciences, Manatee
Institute, Coquina Beach, FL.
TITLE: Universal Import.
ABSTRACT: The phenomenon of existential import of universal propositions
has received much attention both before and after Keynes gave it its
name in the late 1800s.  However, the equally fundamental phenomenon of
universal import of existential propositions has been totally ignored -
perhaps because it does not occur in Aristotle's logic or in Boole's.
An existential proposition has general universal import (GUI) if it
implies the corresponding universal.  An existentialized conjunction has
conditional (or relative) universal import (CUI) if it implies the
corresponding universalized conditional.  It had been conjectured that,
aside from trivialities such as tautologies and contradictions, no
existentials have universal import of either kind.  But for standard
(one-sorted first-order) logic the conjecture was quickly refuted by
number-theoretic propositions: "some number x is such that every number
y is x" has general universal import and "some number x is such that x
is different from zero and every number is either x or zero", which
implies "every number x is such that if x is different from zero then
every number is either x or zero", has conditional universal import.
The question then arises as to which (standard) existentials have GUI
general universal import and which existentialized conjunctions have CUI
conditional universal import.  Instead of attacking this question
directly we asked the closely related question: which universal
propositions are logically equivalent (LE) to existentials having GUI
and which universalized conditionals are LE to existentialized
conjunctions having CUI.  The main results are that every universal
(resp. universalized conditional) is LE to an existential (resp.
existentialized conjunction) which has general (resp. conditional)
universal import.  This leads directly to the conclusion that every
existential (even those which are not existentialized conjunctions) is
LE to an existential (in fact to an existentialized conjunction) which
has both general universal import and conditional universal import. This
is part of a study of argument schemata: an argument schema is called
omnivalid if all of its instances are valid, omninvalid if none are, and
binivalid if some but not all are. The universal and existential import
schemata are two of many interesting binivalid schemata. Thanks to Ole
Anders, Mark Brown, John Corcoran, Todd Ernst, Leonard Jacuzzo, Linda S.
Lavida, and Mary M. Mulhern.

Wednesday, July 26 2006	12:00-1:30 P.M.	141 Park Hall
TENTATIVE PANEL: John Corcoran, John Kearns, Leonard Jacuzzo and others.

TITLE: Teaching Logic.
ABSTRACT: Each member of a panel of logic teachers will give a
ten-minute presentation of a message about teaching logic followed by
ten minutes of open discussion. Among the topics that are under
consideration are: the goals of logic, what to say the first day, the
role of paradigm cases, the role of fallacies, the role of history, the
best non-logical content to use in introductory courses, number theory,
alternative logics, existential import, identity logic, logic textbooks,
which system of logic should be taught first. 
Future Speakers: Daniel Merrill (Oberlin College), Stewart Shapiro (Ohio
State University), Barry Smith (University of Buffalo).


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