"CALCULEMUS" DOMAIN
"Forum" 2000 Contents
"CALCULEMUS" DOMAIN
Komitet Nauk Filozoficznych PAN
Polskie Towarzystwo Logiki i Filozofii Nauki
Katedra Logiki, Informatyki i Filozofii Nauki UwB
invite to 30.IX - 3.X. 2000, Zakopane
Polish Academy of Sciences: Committee for Philosophy
Polish Associaton for Logic and Philosophy of
Science
Białystok University: Logic, Informatics and Philosophy of
Science Dep.
IV Workshop for Logic, Informatics and Philosophy of Science
or
Semantic Consequence vs Formal Derivability
(Wynikanie Sematyczne a Wyprowadzalność Formalna)
in the Centenary of Hilbert's
Second Problem
This Workshop is also meant as preparatory to the International Conference
to be held in the Centenary of Alfred Tarski's Birth
Organizational Messages (in Polish)
1. The last is not least, since terminology provides us with a thread of thought (filum cogitationis, according to Leibniz) to conduct our thoughts (Cantor, for example, carefully devised terms and symbols when building his set theory). Especially, in teaching logic we should apply a language nicely coherent, and not disregarding common usages.
There is a bit of confusion with the term "logical consequence" as introduced by Tarski. Though termed "logical", this relation involves inferences which transcend systems of logic (see 2 below). Should we employ, instead, "semantic consequence" or, after Beth, "semantic entailment" (in Polish "wynikanie semantyczne")?
2. Here is Tarski's celebrated definition of logical consequence. The sentence X follows logically from the sentences of the class K if and only if every model of the class K is a model of the sentence X. This definition has been tailored mainly to catch the infinite induction rule. However, that rule is valid in the domain of natural numbers alone. What, then, about the universal range of logical rules? This is a brainteaser for philosophers of logic to be addressed before one decides which adjective is best suited for the consequence named "logical" by Tarski himself.
3. There are two historical processes relevant to the opposition of semantic consequence and formal derivability.
One of them consists in the evolution of the very concept of logic. In the thirties it was usual to include Russell type theory into logic (so did Tarski and Lindenbaum as well as Gödel, Post, Church etc). May have Tarski's terminological proposals (see 2 above) been influenced by that idea of logic? Why has that idea of logic been abandoned? Did it happen for theoretical reasons or for a historical accident?
Moreover, there was that pregnant process set in motion by Hilbert's Second Problem, as stated at the Paris Congress of Mathematicians in 1900, within the set of Mathematische Probleme. It was the task to prove the consistency of arithmetic.
That may be compared with another great event of the same year -- the origin of quantum theory with Max Planck's discovery of the law of heat radiation (published in December 1900). Though it was only the posing of a question with Hilbert, and a definite discovery with Planck, either initiated an enormous stream of ideas and discoveries throughout the whole century. At its very end these streams astonishingly meet in the project of quantum computer. In fact, the idea of digital computer has grown from Hilbert's Enscheidungsproblem in Turing's 1936 study, and that dicision problem is latent in the Second Problem of 1900.
Hilbert not only stated the task but also devised a method to cope with it. In the current terminology, that method may be briefly called algorithmic; the algorithm in question depended on formal derivability. A thorough analysis of that algorithm, due to Tarski, Gödel, Turing, Post and Church, resulted in surprising discoveries. There is among them Tarski's concept of logical consequence to provide inference rules different from those algorithmic "old rules of inference" (Tarski's phrase) which should have sufficed "to reproduce in the shape of formalized proofs all the exact reasonings which had ever been carried out in mathematics" (Hilbert's point as articulated by Tarski).
As a corollary to such a discussion one may obtain some elucidation of the term "philosophical logic" which is rather carelessly used by a lot of authors. To wit, the question of appropriateness of calling a consequence logical is closely related to what Tarski himself [1936, 1986, 1987] vividly discusses, and what Simons [1992a] calls the issue of the limits of logic (commented by Maciaszek [1997]).
Suppose, one decides to retain the term "logical consequence" for the relation referred to in Tarski's definition (above, item 2). Then one should test the inference rules in the system in question (of what he calls philosophical logic) against Tarski's definition of logical consequence; if they pass the examination, the system will win the status of logic (being, in addition, philosophical because of a philosophical meaning of its logical constants).
Now suppose the opposite: one gives up the adjective "logical", replacing it by "semantic" or something else, in order to reserve the term "logical" for rules of predicate logic as ones being valid in every non-empty domain. Then there disappears any title to apply the term "logic" to those formalized philosophical theories which are restricted to such domains as knowledge, belief, preference, duty, time, etc. (as presented in Rescher [1968], Gabbay and Guenthner [1974]).
Thus a theoretical discussion on the limits of logic as well as that on semantic consequence should advance our terminology to make it a more reliable thread of thought.
Literature referred to and some auxiliary items
Hilbert, David 1900 Hilbert, David, and Wilhelm Ackermann 1928 Hilbert, David, and Paul Bernays 1934-1939 Tarski, Alfred 1956 Tarski, Alfred 1995 Tarski, Alfred 1936 Tarski, Alfred 1969 Tarski, Alfred 1986 Tarski, Alfred 1987 Tarski, Alfred, and Adolf Lindenbaum 1956
Church, Alonzo 1936 Gödel, Kurt 1931 Post, Emil Leon 1936 Turing, Alan 1936
Gabbay, D. and F.Guentner (eds.) 1994 [first published 1983] Maciaszek, Janusz 1997 Marciszewski, Witold and Roman Murawski 1995 Murawski, Roman 1994 Rescher, Nicolas 1968 Simons, Peter 1992 Simons, Peter 1992a Woleński, Jan 1985 Woleński, Jan (ed.) 1985
Mathematische Probleme. Vortrag, gehalten auf dem internationalem
Mathematiker-Kongress zu Paris 1900. Archiv der Mathematik und
Physik, 3rd series, 1, 1901; pp.44-63, 213-237.
Grundzüge der theoretischen Logik. Springer, Berlin. [A crucial
text due to the stating of the Entscheidungsproblem, referred to by Church
1936 and Turing 1936 (see below).]
Grundlagen der Mathematik. Springer, Berlin; vol.1 in 1934, vol.2 in
1939.
Logic, Semantics, Metamathematics. Papers from 1923 to 1928
translated by J.H.Woodger. Clarendon Press, Oxford.
Pisma logiczno-filozoficzne. Tom I: Prawda ed.by Jan Zygmunt.
Wydawnictwo Naukowe PWN, Warszawa.
O pojęciu wynikania logicznego, Przegląd Filozoficzny,vol.39,
pp.58-68. [Polish original reprinted in Tarski 1995, English version in
Tarski 1956].
Truth and proof, Scientific American, vol.220, no.6, pp.63-77.
Polish translation by Jan Zygmunt in Tarski [1995].
What are logical notions?", History and Philosophy of Logic, 7,
pp.143-154.
A philosophical letter of Alfred Tarski [to Morton White], The Journal of
Philosophy, vol.84, no.1, pp.28-32. Polish translation by Jan Zygmunt in
Tarski [1995].
On the liminations of the means of expression of deductive theories [a final
version in Tarski 1956, a partial version presented first in 1927].
A note on the Entscheidungsproblem, The Journal of Symbolic Logic,
vol.1, pp.40-41 [contains references to a preceding fundamental study].
Über formal unentscheibare Sätze der Principia Mathematica
und verwandter Systeme I, Monatshefte für Mathematik und Physik,
vol.38, pp.173-198.
Finite combinatory process, formulation I, The Journal of Symbolic Logic,
vol.1, pp.103-105.
On computable numbers with an application to the Entscheidungsproblem,
Proceedings of the London Mathematical Society, 2nd series, vol.44, 1937
[submitted in 1936], pp.230-265; correction pp.544-546.
Handbook of Philosophical Logic, vols.1-4, Kluwer, Dordrecht etc.
Tarski on logical entities, Logica Trianguli, No.1.
Mechanization of Reasoning in a Historical Perspective.
Rodopi, Amsterdam.
Hilbert's Program: incompleteness theorems vs. partial realizations, in:
Woleński [1994].
Topics in Philosophical Logic. Reidel, Dordrecht.
Philosophy and Logic in Central Europe from Bolzano to Tarski,
Kluwer, Dordrecht, etc. (editorial collaboration by W.Marciszewski).
Bolzano, Tarski, and the limits of logic" in: Simons [1992].
Filozoficzna Szkoła Lwowsko-Warszawska. PWN, Warszawa.
Philosophical Logic in Poland. Kluwer, Dordrecht, etc.
Zgłoszenia prosimy wysyłać, podając
dane o pozycji naukowej i afiliacji (z dokładnością do zakładu lub katedry),
do 25.VI.2000 e-mailem jednocześnie na dwa adresy:
Organizator:
Witold Marciszewski: witmar@calculemus.org
Menedżer: Roman
Matuszewski: romat@mizar.org
Koszt udziału wynosi 500 zł., gdy
zostanie uiszczony do 20.VI.1999, a 600 zł. po tej dacie. Ostateczny termin
wpłat -- 20.VIII. Należności prosimy regulować przelewem na konto: Fundacja
na rzecz Informatyki, Logiki i Matematyki, Warszawa; PKO BP, XV O. w
Warszawie, rach. nr 10201156-6334-270-1-111.
Miejsce Warsztatów: Zakopane
ul. Makuszyńskiego 12 (róg Piłsudskiego)
Pensjonat "Rzemieślnik" - oddalony o kilka minut drogi od wejścia do doliny
Białego i tyleż od terenu Krokwi (początek Drogi pod Reglami).
Dojazd: bezpośredni z W-wy ekspresem Tatry lub dowolnym pociągiem do Krakowa, stąd autobusem do Zakopanego (dwie godziny jazdy, autobusy co 20 min. sprzed dworca PKP).
Ramowy porządek zajęć. Zjazd 30.IX. (sobota) do godz.18 (kolacja). Pierwsza sesja - 30.IX. po kolacji, ostatnia 3.X. do godz.14 (do obiadu). Rozkład sesji i tytuły odczytów - w następnym komunikacie.
Zaproszenie do wygłoszenia wykładów w wymiarze kilku godzin przyjęli koledzy Roman Murawski i Jan Woleński - niekwestionowani znawcy obu członów problemu (Murawski z akcentem na Hilberta, Woleński z akcentem na Tarskiego). Inne zaproszenia są w toku ustalania.