Principia Mathematica and the Development of Logicrzach/files/rzach/pm100.pdf · Introduction Bernays Behmann Carnap Conclusion Outline 1 The Development of Logic in the 1920s 2 Paul

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


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



Influence of Principia Mathematica

Adoption of symbolism and results

Metatheoretical investigations

Applications

Modification: extension, simplification



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



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)



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)



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)



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



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)



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)



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


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)



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



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)



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)



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?)



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



The Antinomy of Transfinite Number, 1918

Study of cardinal arithmetic, paradoxes in light ofPrincipia

Mainly non-technical

Remained unpublished



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)



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)



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


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)



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


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

. . .



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)





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)



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



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.



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)



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



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)



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))



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))



. . . 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)



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)



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



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




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




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)



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


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Principia Mathematica and the Development of Logicrzach/files/rzach/pm100.pdf · Introduction Bernays Behmann Carnap Conclusion Outline 1 The Development of Logic in the 1920s 2 Paul