principia mathematica and the development of logicrzach/files/rzach/pm100.pdf · introduction...
TRANSCRIPT
Introduction Bernays Behmann Carnap Conclusion
Principia Mathematica and theDevelopment of Logic
Richard Zach
Department of PhilosophyUniversity of Calgary
www.ucalgary.ca/∼rzach/
May 23, 2010Principia Mathematica @ 100
Logic from 1910 to 1927McMaster University
http://www.ucalgary.ca/rzach/files/rzach/pm100.pdf
1/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Outline
1 The Development of Logic in the 1920s
2 Paul Bernays: Metatheory of PM
3 Heinrich Behmann: PM and the Decision Problem
4 Rudolf Carnap: Bringing Logic to Philosophy
5 Conclusion
2/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Influence of Principia Mathematica
Adoption of symbolism and results
Metatheoretical investigations
Applications
Modification: extension, simplification
3/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
The Development of Logic in the 1920s
Hilbert’s GöttingenAckermann, Behmann, Bernays, Gentzen,Schönfinkel, (Hertz, Curry, Church, Weyl)
The Polish SchoolLesniewski, Łukasiewicz, Tarski
The Vienna CircleCarnap, Gödel, Hahn, Kaufmann
The Set TheoristsFraenkel, Skolem, von Neumann
4/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Logic in Hilbert’s School
1914–1918 Antinomies and Principia (Behmann)1917 Axiomatic Thought (Hilbert)
1917–18 Principles of Mathematics (Hilbert)1918/26 Contributions to the Axiomatic Treatment of the
Propositional Calculus of PM (Bernays)1922 Algebra of Logic and the Decision Problem
(Behmann)1922/24 The Basic Building Blocks of Logic (Schönfinkel)
1928 Principles of Theoretical Logic (Hilbert–Ackermann)1922/28 On the Decision Problem for Mathematical Logic
(Bernays and Schönfinkel)1928 Satisfiability of Certain Counting Expressions
(Ackermann)1928/29 Problems of the Foundations of Mathematics
(Hilbert)
5/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Paul Bernays, 1888–1977
Dissertation in 1912 on analyticnumber theory in Göttingen
Assistant to Hilbert from 1917onward
Habilitation in 1918 on thepropositional calculus ofPrincipia
Had to leave Germany in 1933;moved to Zurich
Hilbert-Bernays, Foundations ofMathematics (1934, 1939)
6/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
The Functional Calculus in 1918
Principles of Mathematical Logic, 1917/18
I. 1) XX → X2) X → XY3) XY → YX4) X(YZ)→ (XY)Z5) (X → Y)→ (ZX → ZY)
II. 1) (x)Z → Z2) (x)F(x)→ (Ex)F(x)3) (x)(Z × F(x))→ (x)(Z × (x)F(x))4) (x)(F(x)→ G(x))→ ((x)F(x)→ (x)G(x))5) (x)(y)F(x,y)→ (y)(x)F(x,y)6) (x)(y)F(x,y)→ (x)F(x,x)
7/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Paradoxes and the Stufenkalkül
Version of type theory:
Level 1: propositions and functions of individualswith quantification over individualsLevel 2: propositions and functions of individuals,level 1 functions with quantification over individualsand level 1 functionsLevel 3: . . .
Assign indices to all variables. Index of anexpression is max of indices occurring in it, + 1
8/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Completeness and Independence in PM
When a proposition q is a consequence of a propositionp, we say that p implies q. Thus deduction relies uponthe relation of implication, and every deductive systemmust contain among its premisses as many of theproperties of implication as are necessary to legitimatethe ordinary procedure of deduction. In the presentsection, certain propositions will be stated as premisses,and it will be shown that they are sufficient for allcommon forms of inference. It will not be shown thatthey are all necessary, and it is possible that the numberof them might be diminished. (PM, p. 90)
9/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Completeness of Propositional Logic
“The importance of our axiom system for logic rests onthe following fact: If by a “provable” formula we mean aformula which can be shown to be correct according tothe axioms, and by a “valid” formula one that yields atrue proposition according to the interpretation givenfor any arbitrary choice of propositions to substitute forthe variables (for arbitrary “values” of the variables),then the following theorem holds:
Every provable formula is a valid formula andconversely.”
(Bernays, 1918)
10/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Dependence and Independence
Propositional axioms of PM:
(Taut) p ∨ p ⊃ p,(Add) q ⊃ p ∨ q,(Perm) p ∨ q ⊃ q ∨ p,(Assoc) p ∨ ·q ∨ r ⊃ q ∨ ·p ∨ r ,(Sum) q ⊃ r · ⊃ ·p ∨ q ⊃ p ∨ r .
Bernays showed:
Assoc can be derived from the other four
remaining four axioms independentP. Bernays, “Axiomatische Untersuchungen des
Aussagen-Kalkuls der “Principia Mathematica”. Math. Z. 25 (1926)11/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Axiomatizations of Propositional Logic
In axiomatising the propositional calculus, thepredominant tendency is to reduce the number of basicconnectives and therewith the number of axioms. Onecan, on the other hand, sharply distinguish the variousconnectives; in particular, it would be of interest toinvestigate the role of negation.
(Bernays, 1923)
12/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
The Hilbert-Bernays Axiom System (1923)I. A→ (B → A)
(A→ (A→ B))→ (A→ B)(A→ (B → C))→ (B → (A→ C))(B → C)→ ((A→ B)→ (A→ C))
II. A & B → AA & B → B(A→ B)→ ((A→ C)→ (A→ B & C))
III. A→ A∨ BB → A∨ B(B → A)→ ((C → A)→ (B ∨ C → A))
IV. (A ∼ B)→ (A→ B)(A ∼ B)→ (B → A)(A→ B)→ ((B → A)→ (A ∼ B))
V. (A→ B)→ (B → A)(A→ A)→ AA→ AA→ A
13/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Sheffer’s Stroke and Schönfinkel’sCombinators
What you told me in your letter about Scheffer’s symbolwas completely new to me at the time and of course veryintersting. I have reported to the mathematicians atGöttingen on the subject of this reduction of logicalsymbols, and it has led to further investigations in thisdirection.
In particular, Mr. Schönfinkel has discovered that alsoin the field of the calculus wih variables all logicalsymbols can be reduced to a single one, φ(x) |x ψ(x),to which one can give the meaning: “for no x do bothφ(x) and ψ(x) hold together,” in symbols:(x).∼φ(x)∨∼ψ(x).
(Bernays to Russell, March 19, 1921)
14/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Sheffer’s Stroke and Combinatory Logic
“Currying”: Consider many-place functions asone-place functions with functions as values:F(x,y) as (Fx)(y).Introduce combinators:
Ix = x (Tφ)xy = φyx Sφχx = (φx)(χx)(Cx)y = x Zφχx = φ(χx) Uφχ = φx |x χx
Get rid of variables, e.g.,
(f )(∃g)(x)∼fx & gx
(fx |x gx) |g (fx |x gx)] |f [(fx |x gx) |g (fx |x gx)][U(Uf)(Uf)] |f [U(Uf)(Uf)]U[S(ZUU)U][S(ZUU)U] (by U(Uf)(Uf) = S(ZUU)Uf)
M. Schönfinkel, “Bausteine der mathematischen Logik.” Math.Ann. 92 (1924)
15/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Hilbert–Ackermann, Principles of TheoreticalLogic 1928
First “modern” logic textbook
Essentially (in large part, literally) based on Hilbert’s1917/18 lectures Principles of Mathematics; 1920Logical Calculus
Propositional and predicate logic on axiomatic basis
Hilbertian symbolism, Hilbert/Bernays axioms
Metalogical questions and results (consistency,completeness)
Type theory and paradoxes, criticism of axiom ofreducibility (Ramsey?)
16/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Heinrich Behmann, 1891–1970
Studied mathematics underHilbert
Dissertation in 1918 onPrincipia Mathematica
Habilitation in 1921 on thedecision problem
Lectured on logic in Göttingen1923
Moved to Halle-Wittenberg in1925
Dismissed in 1945
17/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
The Antinomy of Transfinite Number, 1918
Study of cardinal arithmetic, paradoxes in light ofPrincipia
Mainly non-technical
Remained unpublished
18/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Behmann and Russell
It was, in fact, that work of yours [PM] that first gave mea view of that wonderful province of human knowledgewhich ancient Aristotelian Logic has nowadays becomeby the use of an adequate symbolism. But, I daresay, itmight be said of your work just as well what H. Weyl saidof his own book, that “it offers the fruit of knowledge ina hard shell” [. . . ]
Several years ago, I had therefore resolved to writesomething like an introduction or commentary to thatwork, providing a way by which the unavoidabledifficulties of understanding are separately treated [. . . ]in order that the Principia Mathematica might become aswell known as both the work and the topic deserve.
(Behmann to Russell, August 8, 1922)
19/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Behmann’s Habilitation: The DecisionProblem
Proves decidability of monadic second-order logic
Adopts modified symbolism of PM, but
Uses transformation rules instead of axiomaticderivations
Link between Schröder and PM
H. Behmann, “Beiträge zur Algebra der Logik, insbesonderezum Entscheidungsproblem,” Mathematische Annalen 86 (1922)
20/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Mathematics and Logic (1927)
Propositional LogicNot axiomatic, but “calculational”Decision procedure via truth tables, normal forms
Logic of Concepts (Begriffslogik)Simple typesExtensionality
Logic of Classes (Klassenlogik)
Logic of Relations (Zuordnungslogik)Arrow diagrams
Cardinal Arithmetic
H. Behmann, Mathematik und Logik. Leibzig: Teubner, 192721/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Avoiding Paradoxes without Types
Russell’s Paradox: Define R(P) ≡df ∼P(P).Then R(R) ≡ ∼R(R)Behmann: definitions only admissible ifdefiniendum can be replaced by definiens
But in R(R), R cannot be so replaced
Criticized by Bernays, Gödel, Ramsey:Contradiction can be derived without definition
Behmann proposes type-free solution with furtherrestrictions
H. Behmann, “Zu den Widersprüchen der Logik undMengenlehre,” J. DMV 40 (1931)
22/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Rudolf Carnap, 1891–1970
Born 1891 in Ronsdorf, nowWuppertal
University of Jena (1910–14,1918–20), student of Frege
Dissertation on philosophy ofgeometry (Der Raum), 1922
Dozentur in Vienna underSchlick 1926
Professor at German Universityin Prague 1931–36
Professor at the University ofChicago 1936–1954
Professor at UCLA 1954–197023/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Carnap and Logic
Student of Frege in Jena, 1911–14
Studied Frege’s works as well as Principia
Influenced by Russell, esp. Our Knowledge of theExternal World
Aufbau an application of Russell’s logic to widerphilosophical issues
Abriss der Logistik, 1929
General Axiomatics
Logical Syntax of Language, 1934
. . .
24/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Carnap and Russell
I am particularly happy that it you in particular are thefirst Englishman to whom I may extend my hand in thescientific field, since already at the time of the War youhave stood so openly against the intellectualenslavement resulting from hatred between peoples andin favor of a human and pure way of thinking. When Iremember that Couturat, who unfortunately died tooearly, held the same convictions, I ask myself: Can it bemere coincidence that it is those who achieve thegreatest clarity in the most abstract area ofmathematical logic who then also fight clearly andforcefully against the narrowing of the human spiritthough emotional reactions and prejudices?
(Carnap to Russell, November 17, 1921)
25/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
26/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Outline of Logistic, 1929
Presentation of logic (relation theory) as in PrincipiaPropositional logic including truth-functionsExamples, arrow diagrams, matrix representation ofrelationsSimple theory of typesApplications of relation theory to philosophy
Constitution theory (Aufbau)Formalization of axiom systems (arithmetic, settheory, geometry, space-time-topology)Beginnings of formal semantics (logical form ofsentences)
R. Carnap, Abriss der Logistik, mit besonderer Berück-sichtigung der Relationstheorie und ihrer Anwendungen, Vienna:Springer, 1929 (Schriften zur wissenschaftlichen Weltauffassung,vol. 2)
27/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Simple Types
Individuals: t 0
Relation with arguments of types t ξ1, . . . , t ξn:t (ξ1, . . . , ξn).E.g.: ∈ t (0(0)); ⊂ t ((0)(0))
Relations can be methodically ambiguous
28/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Logical Form
I have never seen a person so agitated as you wereyesterday.
(∃α,β) :. (α is a state of me) . (β is a state of youyesterday) . α 〈 see 〉 β :. (γ, δ) : (γ is a state ofme in the past) . (δ is a state of some human). γ 〈 sehen 〉 δ .⊃. (δ is not as agitated as β).(∃α,β) : α ∈ 〈 I ] . β ∈ 〈 you ] ∩ 〈 yesterday ] .α 〈 see 〉 β :. (γ, δ) : γ ∈ 〈 I ] ∩ 〈past ] . δ ∈ | ∈〈humans ] . γ 〈 see 〉 δ .⊃. 〈 agitation 〉 ‘δ <〈 agitation 〉 ‘β.
29/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Axiomatic Systems
Interpreted: Basic symbols are nonlogical constants,axioms are propositions about the correspondingconcepts
Uninterpreted: Basic symbols are variables, axiomsare propositional functions (AS(P,Q))
Axiom system implicitly define correspondingconcepts (as improper concepts)But also: axiom system defines an explicit concept,i.e., the class of structures satisfying it, viz.,P QAS(P,Q)
30/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
The General Axiomatics Project
WritingsUntersuchungen zur allgemeinen Axiomatik 1928“Eigentliche und uneigentliche Begriffe” (1927)“Bericht über Untersuchungen zur allgemeinenAxiomatik” (1929)
Synthesis of Frege’s and Russell’s approach to logicwith Hilbert’s axiomatics
Influence on Gödel, Fraenkel
Criticized by Behmann, Tarski, Gödel, abandoned
R. Carnap, Untersuchungen zur allgemeinen Axiomatik. Bonkand Mosterin, eds. Darmstadt: Wiss. Buchgesellschaft, 2000
31/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Carnap’s “Results” . . .
An axiomatic system is
consistent if and only if it is satisfied (”Gödelcompleteness”)
decidable [entscheidungsdefinit] if and only if it isnon-forkable (semantically complete)
non-forkable if and only if it is monomorphic(categorical)
32/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Metatheory of Axiomatics in Principia
Axiom system with non-logical constantsR = P,Q,R a propositional function: f(P,Q,R).A (putative) theorem of the axiom system:g(P,Q,R).g is a consequence of f :
(P)(Q)(R)(f(P,Q,R)→ g(P,Q,R))
(in short (R)(fR→ gR))
33/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Properties of Axiomatic Systems
fR is satisfied: (∃R)fR, empty: ∼(∃R)f (R)fR is consistent: ∼(∃h)(R)(fR→ (hR &∼hR))fR is monomorphic:
(∃R)fR & (P,Q)((fP & fQ)→ Ismq(P,Q))
fR is forkable [gabelbar]:
(∃g)[(∃R)(fR & gR) & (∃R)(fR &∼gR))
fR is decidable [entscheidungsdefinit]:
(∃R)fR & (g)((fR→ gR)∨ (R)(fR→ ∼gR))
34/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
. . . Trivial or False
“Gödel completeness” simple logical proof:
∼(∃h)(R)(fR→ (hR &∼hR)) (1)
(h)(∃R)∼(fR→ (hR &∼hR)) (2)
(h)(∃R)(fR &∼(hR &∼hR)) (3)
(∃R)fR (4)
35/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
What Went Wrong?
Trying to do metatheory in the theory of PM itselfBut no definition of “provable”, “model”, “true in”
Quantification over propositional functions, notsentences
Can’t specify “language” of h
“Truth in a model” inherits notion of truth frombasic discipline (PM)
“follows from” and “is provable from” not the same(even though intended to be)
36/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
What Went (Almost) Right?
Consequence, satisfaction is the obvious way ofexpressing intuitive notion, also in Hilbert/Bernays
Carnap proves that “g follows from f ” iff there is aHilbert-style proof of G from F
But: “G is not provable from F” not the same asPM ` ∼(R)(fR→ fR), specifically:no contradiction is provable from F not equivalentto PM ` ∼(∃h)(R)(fR→ (hR &∼hR))
Carnap distinguishes between a(bsolut) andk(onstruktiv) versions of concepts
37/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
From Principia to Modern Logic
Restriction of interest to fragmentsPropositional, functional, second-order logic
Metatheory of logical calculusTruth-value semanticsExtensionality
Metatheory of axiomatic systemsSatisfaction, categoricity, completenessProvability vs. consequence
Clarification of issues in philosophy of logicType hierachies, avoidance of paradoxes,extensionality
38/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Influence of Principia Mathematica
Adoption of symbolism and resultsEndorsement of logicisim by Hilbert (until 1921),CarnapPropositional, first-order fragments, andaxiomatizations (Hilbert)Adoption of notation by Carnap (until 1929)
Applications of theory of relationsExtensive use of relation theory by Carnap (Aufbau)Metatheory of axiomatic systems in PM
39/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Influence of Principia Mathematica
Metatheoretical investigation of PMIndependence of axioms (Bernays 1918)Decidability (Behmann 1922)
Modification: extension, simplificationCombination with notations and methods fromalgebraic logic (Hilbert)Variations on theory of types (Behmann, Bernays,Carnap)Sheffer stroke and combinatory logic (Bernays,Schönfinkel)Getting rid of types (Behmann, Curry, Church)
40/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic
Introduction Bernays Behmann Carnap Conclusion
Reading List
S. Awodey, A. W. Carus. Carnap, completeness, andcategoricity. Erkenntnis 54 (2001)
W. Goldfarb. On Gödel’s way in. Bull. Sym. Logic 11 (2005)
P. Mancosu. The Russellian influence on Hilbert and hisschool. Synthèse 137 (2003)
P. Mancosu, R. Zach, C. Badesa. The development ofmathematical logic from Russell to Tarski. Haaparanta, ed.,The Development of Modern Logic (2009)
E. Reck. From Frege and Russell to Carnap. Awodey and Klein,eds., Carnap Brought Home, 2004
R. Zach. Completeness before Post. Bull. Symb. Logic 5 (1999).
http://www.ucalgary.ca/rzach/files/rzach/pm100.pdf
41/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic