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Home , Principia Mathematica

Introduction Bernays Behmann Carnap Conclusion

Principia Mathematica and the Development of

Richard Zach

Department of Philosophy University of Calgary www.ucalgary.ca/∼rzach/

May 23, 2010 Principia Mathematica @ 100 Logic from 1910 to 1927 McMaster 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 : Metatheory of PM

3 Heinrich Behmann: PM and the Decision Problem

4 : 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 Göttingen Ackermann, Behmann, Bernays, Gentzen, Schönfinkel, (Hertz, Curry, Church, Weyl) The Polish School Le´sniewski, Łukasiewicz, Tarski The Carnap, Gödel, Hahn, Kaufmann The Theorists Fraenkel, 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 and Principia (Behmann) 1917 Axiomatic Thought (Hilbert) 1917–18 Principles of (Hilbert) 1918/26 Contributions to the Axiomatic Treatment of the 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 (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 analytic theory in Göttingen Assistant to Hilbert from 1917 onward Habilitation in 1918 on the propositional calculus of Principia Had to leave Germany in 1933; moved to Zurich Hilbert-Bernays, Foundations of Mathematics (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 → X 2)X → XY 3)XY → YX 4) X(Y Z) → (XY )Z 5) (X → Y) → (ZX → ZY) II. 1) (x)Z → Z 2) (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 and the Stufenkalkül

Version of : Level 1: propositions and functions of individuals with quantification over individuals Level 2: propositions and functions of individuals, level 1 functions with quantification over individuals and level 1 functions Level 3: . . . Assign indices to all variables. Index of an expression 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 and Independence in PM

When a proposition q is a consequence of a proposition p, we say that p implies q. Thus deduction relies upon the relation of implication, and every deductive system must contain among its premisses as many of the properties of implication as are necessary to legitimate the ordinary procedure of deduction. In the present section, certain propositions will be stated as premisses, and it will be shown that they are sufficient for all common forms of . It will not be shown that they are all necessary, and it is possible that the number of 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 system for logic rests on the following fact: If by a “provable” formula we mean a formula which can be shown to be correct according to the , and by a “valid” formula one that yields a true proposition according to the interpretation given for any arbitrary choice of propositions to substitute for the variables (for arbitrary “values” of the variables), then the following theorem holds:

Every provable formula is a valid formula and conversely.” (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 independent P. 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, the predominant tendency is to reduce the number of basic connectives and therewith the number of axioms. One can, on the other hand, sharply distinguish the various connectives; in particular, it would be of interest to investigate 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 → A A & B → B (A → B) → ((A → C) → (A → B & C)) III. A → A ∨ B B → 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) → A A → A A → A

13/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion Sheffer’s Stroke and Schönfinkel’s Combinators

What you told me in your letter about Scheffer’s symbol was completely new to me at the time and of course very intersting. I have reported to the mathematicians at Göttingen on the subject of this reduction of logical symbols, and it has led to further investigations in this direction. In particular, Mr. Schönfinkel has discovered that also in the field of the calculus wih variables all logical symbols can be reduced to a single one, φ(x) |x ψ(x), to which one can give the : “for no x do both φ(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 as one-place functions with functions as values: F(x, y) as (Fx)(y). Introduce combinators: Ix = x (T φ)xy = φyx Sφχx = (φx)(χx) (Cx)y = x Zφχx = φ(χx) Uφχ = φx |x χx Get rid of variables, e.g., (f )(∃ g)(x)∼f x & gx (f x |x gx) |g (f x |x gx)] |f [(f x |x gx) |g (f x |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 Theoretical Logic 1928

First “modern” logic textbook Essentially (in large part, literally) based on Hilbert’s 1917/18 lectures Principles of Mathematics; 1920 Logical Calculus Propositional and predicate logic on axiomatic basis Hilbertian symbolism, Hilbert/Bernays axioms Metalogical questions and results (, completeness) Type theory and paradoxes, criticism of (Ramsey?)

16/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion Heinrich Behmann, 1891–1970

Studied mathematics under Hilbert Dissertation in 1918 on Principia Mathematica Habilitation in 1921 on the decision problem Lectured on logic in Göttingen 1923 Moved to Halle-Wittenberg in 1925 Dismissed in 1945

17/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion The of Transfinite Number, 1918

Study of cardinal arithmetic, paradoxes in light of Principia 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 me a view of that wonderful province of human knowledge which ancient Aristotelian Logic has nowadays become by the use of an adequate symbolism. But, I daresay, it might be said of your work just as well what H. Weyl said of his own book, that “it offers the fruit of knowledge in a hard shell” [. . . ] Several years ago, I had therefore resolved to write something like an introduction or commentary to that work, providing a way by which the unavoidable difficulties of understanding are separately treated [. . . ] in order that the Principia Mathematica might become as well 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 Decision Problem

Proves decidability of monadic second-order logic Adopts modified symbolism of PM, but Uses transformation rules instead of axiomatic derivations Link between Schröder and PM

H. Behmann, “Beiträge zur Algebra der Logik, insbesondere zum 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 Logic Not axiomatic, but “calculational” Decision procedure via tables, normal forms Logic of Concepts (Begriffslogik) Simple types Extensionality Logic of Classes (Klassenlogik) Logic of Relations (Zuordnungslogik) Arrow diagrams Cardinal Arithmetic

H. Behmann, Mathematik und Logik. Leibzig: Teubner, 1927 21/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion Avoiding Paradoxes without Types

Russell’s : Define R(P) ≡df ∼P(P). Then R(R) ≡ ∼R(R) Behmann: definitions only admissible if definiendum can be replaced by definiens But in R(R), R cannot be so replaced Criticized by Bernays, Gödel, Ramsey: can be derived without definition Behmann proposes type-free solution with further restrictions

H. Behmann, “Zu den Widersprüchen der Logik und Mengenlehre,” 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, now Wuppertal University of Jena (1910–14, 1918–20), student of Frege Dissertation on philosophy of (Der Raum), 1922 Dozentur in Vienna under Schlick 1926 Professor at German University in Prague 1931–36 Professor at the University of Chicago 1936–1954 Professor at UCLA 1954–1970 23/41 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 the External World Aufbau an application of Russell’s logic to wider philosophical issues Abriss der Logistik, 1929 General Axiomatics Logical 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 the first Englishman to whom I may extend my hand in the scientific field, since already at the time of the War you have stood so openly against the intellectual enslavement resulting from hatred between peoples and in favor of a human and pure way of thinking. When I remember that Couturat, who unfortunately died too early, held the same convictions, I ask myself: Can it be mere coincidence that it is those who achieve the greatest clarity in the most abstract area of mathematical logic who then also fight clearly and forcefully against the narrowing of the human spirit though 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 Principia Propositional logic including truth-functions Examples, arrow diagrams, matrix representation of relations Simple theory of types Applications of relation theory to philosophy Constitution theory (Aufbau) Formalization of axiom systems (arithmetic, , geometry, space-time-topology) Beginnings of formal ( of sentences) 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 were yesterday. (∃ α, β) :. (α is a state of me) . (β is a state of you yesterday) . α h see i β :. (γ, δ) : (γ is a state of me in the past) . (δ is a state of some human) . γ h sehen i δ .⊃. (δ is not as agitated as β). (∃ α, β) : α ∈ h I ] . β ∈ h you ] ∩ h yesterday ] . α h see i β :. (γ, δ) : γ ∈ h I ] ∩ h past ] . δ ∈ | ∈ h humans ] . γ h see i δ .⊃. h agitation i ‘δ < h agitation i ‘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 corresponding concepts Uninterpreted: Basic symbols are variables, axioms are propositional functions (AS(P, Q)) Axiom system implicitly define corresponding concepts (as improper concepts) But also: axiom system defines an explicit concept, i.e., the of structures satisfying it, viz., PˆQˆ AS(P, Q)

30/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion The General Axiomatics Project

Writings Untersuchungen zur allgemeinen Axiomatik 1928 “Eigentliche und uneigentliche Begriffe” (1927) “Bericht über Untersuchungen zur allgemeinen Axiomatik” (1929) Synthesis of Frege’s and Russell’s approach to logic with Hilbert’s axiomatics Influence on Gödel, Fraenkel Criticized by Behmann, Tarski, Gödel, abandoned

R. Carnap, Untersuchungen zur allgemeinen Axiomatik. Bonk and 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 is consistent if and only if it is satisfied (”Gödel completeness”) decidable[ entscheidungsdefinit] if and only if it is non-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 constants R = P, Q, R a : 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)(f R → gR))

33/41 Richard Zach (Calgary) Principia Mathematica and the Development of Logic Introduction Bernays Behmann Carnap Conclusion Properties of Axiomatic Systems

f R is satisfied: (∃ R)f R, empty: ∼(∃ R)f (R) f R is consistent: ∼(∃ h)(R)(f R → (hR & ∼hR)) f R is monomorphic:

(∃ R)f R & (P, Q)((f P & f Q) → Ismq(P, Q))

f R is forkable[ gabelbar]:

(∃ g)[(∃ R)(f R & gR) & (∃ R)(f R & ∼gR))

f R is decidable[ entscheidungsdefinit]:

(∃ R)f R & (g)((f R → gR) ∨ (R)(f R → ∼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)(f R → (hR & ∼hR)) (1) (h)(∃ R)∼(f R → (hR & ∼hR)) (2) (h)(∃ R)(f R & ∼(hR & ∼hR)) (3) (∃ R)f R (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 itself But no definition of “provable”, “model”, “true in” Quantification over propositional functions, not sentences Can’t specify “language” of h “Truth in a model” inherits notion of truth from basic 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 () Right?

Consequence, satisfaction is the obvious way of expressing intuitive notion, also in Hilbert/Bernays Carnap proves that “g follows from f ” iff there is a Hilbert-style proof of G from F But: “G is not provable from F” not the same as PM ` ∼(R)(f R → f R), specifically: no contradiction is provable from F not equivalent to PM ` ∼(∃ h)(R)(f R → (hR & ∼hR)) Carnap distinguishes between a(bsolut) and k(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 fragments Propositional, functional, second-order logic Metatheory of logical calculus Truth-value semantics Extensionality Metatheory of axiomatic systems Satisfaction, categoricity, completeness Provability vs. consequence Clarification of issues in Type 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 results Endorsement of logicisim by Hilbert (until 1921), Carnap Propositional, first-order fragments, and axiomatizations (Hilbert) Adoption of notation by Carnap (until 1929) Applications of theory of relations Extensive 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 PM Independence of axioms (Bernays 1918) Decidability (Behmann 1922) Modification: extension, simplification Combination with notations and methods from (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, and categoricity. Erkenntnis 54 (2001) W. Goldfarb. On Gödel’s way in. Bull. Sym. Logic 11 (2005) P. Mancosu. The Russellian influence on Hilbert and his school. Synthèse 137 (2003) P. Mancosu, R. Zach, C. Badesa. The development of mathematical 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