|
Logic Colloquium
Tuesdays,
3:30 - 4:30 pm in BSB/340
Please
note the new day, time and location
Tuesday, April 8
Speaker: Dr. Richard Arthur
Department of Philosophy
McMaster
University
Title:
"Leibniz's Actual Infinite as an Alternative to Cantor's Transfinite"
Abstract:
It is often claimed (e.g. by Cantor) that Leibniz was inconsistent
in advocating the actual infinite in his metaphysics and physics and
yet rejecting infinite number. Here I defend Leibniz's conception of
the actual infinite as a fully consistent alternative to both the Aristotelian
potential infinite and the Cantorian transfinite, and show how the same
philosophy of the infinite informs both his natural philosophy and his
interpretation of the calculus.
Tuesday, March 25
Time:
3:30
pm
Location: BSB/340
Speaker: Bernhard
Banaschewski
McMaster
University
Title: "How
Boolean was Boole?"
Tuesday, February 11
Time:
3:30
pm
Location: BSB/340
Speaker: Dr.
Stan Burris
Department
of Pure Mathematics
University
of Waterloo
Title: "Boole's
Equational Treatment of 'Some'"
Abstract:
We will look at Boole's equational treatment of "Some" and
the criticism that followed. Then I will argue that the criticism was
seriously flawed.
Wednesday, November 20
Dr. Nick Griffin
Department of Philosophy and
Director of the Bertrand Russell Research Centre
McMaster University
Title: "How Russell Discovered
His Paradox"
Abstract:
The papers surveys what is known of the process by which Russell arrived
at his paradox, bringing together contributions by Ivor Grattan-Guinness,
Greg Moore, Alejandro Garciadiego, and Alberto Coffa. It also addresses
the question of why Russell was so disturbed by the result when previous
thinkers (Burali-Forti, Cantor), coming upon similar results,
had not been.
Wednesday, November 6
Dr.
Bradd Hart
Department of Mathematics and Statistics
McMaster University
Title: "The spectrum of
countable theories"
Time: 1:30 pm
Room: BSB/101
Abstract:
One of the simplest questions one can ask about a first order theory
is how many models does it have? Assume the theory is consistent, it
has at least one model and if there is an infinite model then there
is at one model of every cardinality greater than or equal to the cardinality
of the theory. On the other hand, for a countable theory, in any infinite
cardinal k, there are at most 2^k many models of cardinality k by simple
counting. This is a big gap between 1 and 2^k; I will explain by way
of example the different possibilities between these extremes.
Wednesday, October 23
Gregory
Moore
Department of Mathematics and Statistics
McMaster University
Title: "Russell,
Whitehead, and the Writing of Principia Mathematica"
Abstract:
Russell's two famous books on the foundations of logic and mathematics,
The Principles of Mathematics (1903), written by
himself alone, and Principia Mathematica (1910-1913), written jointly
with Whitehead, were intimately related. For most of the
time between 1903 and 1910, what was eventually Principia Mathematica
was known to both Russell and Whitehead as Volume 2 of The Principles
of Mathematics. This talk deals with the evolution of Russell's thinking
about logic in the period 1903-1910, particularly emphasizing the evolution
in his logical symbolism during that period. Much of that evolution
appears in previously unpublished manuscripts.
Wednesday, October 9
Dr.
D. Lippel
Department of Mathematics and Statistics
McMaster University
Title: "Models
of Complete Sentences"
Abstract:
By a "complete sentence" I mean a first order sentence
that has infinite models and axiomatizes a complete first order theory.
For example, the sentence expressing "dense linear order without
endpoints" is a complete sentence. The most obvious complete sentences
all have the "strict order property", which roughly means
that their models contain a definable partial order with an infinite
chain. Complete sentences without the strict order property are known
to exist, but there are many open questions about what sort of properties
they can have. For example, is there an omega-categorical complete sentence
without strict order property? I will survey the state of knowledge
about complete sentences, concentrating on the omega-categorical case.
Wednesday, September 25
Dr. David Hitchcock
Philosophy Department
Title: "The
peculiarities of Stoic propositional logic"
Abstract:
Between Aristotle, writing in the fourth century BCE, and Boole (1847),
writing more than two millennia later, only one logician published a
system of logic. That was Chrysippus (c. 280-207 BCE), the third head
of the Stoic school. The recent careful work of such scholars as Mates
(1953), Frede (1974), Hülser (1987-1988) and Bobzien (1996, 1999)
has allowed us to reconstruct Chrysippus’ system of propositional
logic and appreciate once again his achievement.
Despite its rigour and soundness, the system is oddly incomplete. One
can show that a conjunction follows from its conjuncts, but not that
either conjunct follows from the conjunction. One can detach the antecedent
from a conditional, but not put it back on; in other words, there is
no deduction theorem, no rule of "conditional proof" or "if
introduction". One can show what follows from an exclusive disjunction
and the affirmation or denial of one of its disjuncts, but not what
the exclusive disjunction follows from. Further, there is no evidence
that anybody ever tried to extend the system.
Why was Stoic propositional logic so incomplete? I shall argue that
many of its peculiarities can be explained by the rather restrictive
accounts of argument and of validity which Chrysippus adopted as the
foundation of his system. The omissions from the system were not accidental
oversights, or not just accidental oversights, but were dictated by
the requirement that everything demonstrable in the system be a valid
argument. With complex and restrictive pre-systematic conceptions of
argument and of validity, Chrysippus was forced into a much more restrictive
formal system than contemporary classical propositional logic.
The
seminars below were held in 2001/02
Bill Farmer
Computing and Software
McMaster University
Date: Wednesday April 10, 2002
Time: 2:30 pm
Room: BSB/101
Title: "Partial First-Order
Logic"
Abstract:
First-order logic, and most other traditional logics, are designed for
use in theory; they are not well suited for doing mathematics in practice.
Partial First-Order Logic (PFOL) is a version of first-order logic that
admits partial functions, undefined terms, and definite descriptions.
As a result, it is much more convenient for formalizing mathematics
than first-order logic. The ideas of PFOL have been successfully employed
in LUTINS, the logic of the IMPS Interactive Mathematical Proof System.
In this talk, we will present the syntax and semantics of PFOL as well
as an axiomatization of PFOL.
Mai Gehrke
Mathematics Department
New Mexico State University
Date: Wednesday March 27, 2002
Time: 2:30 pm
Place: BSB/101
Title: "Canonical extensions, duality,
and Kripke semantics"
Abstract
Canonical extensions were first introduced by Jonsson and Tarski in
1951 for Boolean algebras with operators. For Boolean algebras, the
canonical extension is an algebraic formulation of Stone duality. The
advantage of the algebraic perspective is that additional operations
and many of their equational properties are easy to understand in this
setting. In this talk we will introduce canonical extensions for lattice
ordered algebras in general and explain how these extensions relate
to topological duality and to Kripke semantics. Finally we will survey
some of the recent results in the area and discuss their impact in various
fields.
Alasdair Urquhart
Philosophy Department
University of Toronto
Date: Wednesday, March 6, 2002
Time: 2:30 pm
Place: BSB/101
Title: "Research frontiers in propositional
logic"
Abstract:
Lower bounds for Frege systems. This topic is quite accessible, since
virtually nothing has been proved, and almost all interesting questions
are open.
Salma Kuhlmann
Department of Mathematics
University of Saskatchewan, Saskatoon
Title: "On the Arithmetic
of Lexicographic Exponentiation"
Date: Wednesday February 13, 2002
Time: 2:30
Place: BSB-101
Abstract:
(In memory of Felix Hausdorff on the 60th
anniversary of his death.)
In 1908, Hausdorff developed several arithmetic operations on totally
ordered sets, generalizing many aspects of Cantor's ordinal arithmetic.
In two subsequent publications, he investigated the basic properties
of this arithmetic. Many open questions arise naturally.
In [K], we studied lexicographic powers of the form $\R^{\Gamma}$,
and investigated whether the exponent is an
isomorphism invariant. After establishing further arithmetic rules,
we provide in [HKM] examples of nonisomorphic
chains $\Gamma$ and $\Gamma'$ such that the lexicographic powers $\R^{\Gamma}$
and $\R ^{\Gamma '}$ are
isomorphic. Moreover, for a countable infinite ordinal $\alpha$, we
show that $\R ^{\alpha ^* +\alpha}$ and $\R ^{\alpha}$ are isomorphic.
We encountered further related open questions while studying the question
of defining an exponential function on a power series field. In [KKS1]
this result applies to prove that power series fields never admit an
exponential function. For a fixed nonempty chain $\Delta$, we derive
necessary and sufficient conditions for the existence of nontrivial
solutions of several lexicographic functional equations.
References
[HKM] Holland, W. C.\ -- Kuhlmann, S.\ -- McCleary, S.$\,$: On the Arithmetic
of Lexicographic Exponentiation, preprint 2002
[K] Kuhlmann, S.$\,$: Isomorphisms of Lexicographic Powers of the Reals,
Proc.\ Amer.\ Math.\ Soc.\ {\bf 123}, Number 9, September 1995
[KKS1] Kuhlmann, F.-V.\ -- Kuhlmann, S.\ -- Shelah, S.$\,$: Exponentiation
in power series fields, Proc.\ Amer.\ Math.\ Soc.\ {\bf 125}, Number
11, November 1997
David Hitchcock
Philosophy Department, McMaster University
Title: "Tarski's Polish
paper on logical consequence"
Date: Wednesday January 30, 2002
Time: 2:30 pm
Room: BSB/101
Abstract:
Tarski's classic 1936 paper on the concept of logical consequence appeared
both in Polish and in German. The only published
English translation of this paper is a rather inexact translation of
the German version. My colleague Magda Stroinska and I have produced
an exact translation of the Polish version, copies of which I will be
distributing. Although the two versions
are basically identical, there are more than 400 small differences,
which we record in footnotes to our translation. I argue that, where
these differences are substantive, the Polish version is the more authoritative
one, and that English-speaking scholars should therefore rely on our
translation if they cannot read the Polish original.
Alfred Tarski was born on 14 January 1902.
This talk can therefore be regarded as McMaster's celebration of the
centenary of his birth.
Michael Soltys
Department of Computing and Software
McMaster University
Title: "A Logical Theory for
Feasible Reasoning"
Time: 2:30 pm
Date: Wednesday, January 16
Place: BSB-101
Abstract:
Since the 1960s, feasible problems are understood to be problems which
can be solved with a polynomial time algorithm. Stated more mathematically,
we say that a function f is feasible, if there exists a Turing Machine
M and a polynomial p such that on input x, M computes f(x) in p(length
of x) many steps.
On the other hand, a proof is feasible, if all the objects constructed
in the proof are "polynomial time objects". A logical theory
that captures polynomial time reasoning nicely is Buss's theory S12,
which he proposed in his PhD thesis in 1986. S12 is a fragment of Peano's
arithmetic, where induction has been severely restricted.
In this talk, I want to present S12 and explain what is meant by the
statement "S12 captures polytime reasoning", avoiding technical
definitions and proofs (as much as possible). Matching complexity classes
and proof systems is a growing area of theoretical computer science,
so I will conclude with a table of more examples of logical theories
"capturing" complexity classes.
Professor Juris Steprans
Department of Mathematics and Statistics
York University
Title: "A field guide to cardinal
invariants of the continuum"
Date: Wednesday, November 21, 2001
Time: 2:30 pm
Room: BSB/101
Professor Phil Kremer
Philosophy Department
McMaster University
Title: "Some results on
truth and the liar's paradox"
Date: Wednesday, November 7, 2001
Time: 2:30 pm
Room: BSB/101
Speaker: David Lippel, University of California
at Berkeley
Title: "Finitely
axiomatizable, omega-categorical theories"
Room: BSB/229
Date: Friday November
2 , 2001
Time: 2:00 pm
Speaker: Dr. Matt Valeriote, McMaster
University
Title:
"On Algebraic Classes having Decidable Theories"
Date: Wednesday,
October 24
Time: 2:30 pm
Room: BSB/101
Speaker: Professor Bill Farmer, McMaster University
Title: "Logic
and mathematical knowledge management"
Date: Wednesday
October 10
Time: 2:30pm
Room: BSB/101
|
