The Decision Problem and the Model Theory of First-order Logicpeople.ucalgary.ca/~rzach/static/cshpm.pdf · Proc. LMS, ser. 2, vol. 30 (1929), submitted November 1928 Deals with satisﬁability

Algebraic Manipulation Semantic Methods Development of Semantic Reasoning

The Decision Problem and theModel Theory of First-order Logic

Richard Zach

University of Calgarywww.ucalgary.ca/rzach/

CSHPMMay 29, 2015

Richard Zach The Decision Problem May 29, 2015 1 / 17


Outline

1 Algebraic Approaches to Decidability

2 Decidability by Semantic Methods

3 Development of Semantic Reasoning



Introduction

Decision Problem: find a procedure which decides afterfinitely many steps if a given formula is derivable/valid ornot

One of the fundamental problems for the development ofmodern logic in the 1920s and 1930s

Connected to:ñ Hilbert’s philosophical projectñ development of notion of decision procedure/computationñ development of model-theoretic semantics for first-order

logic



Early Work on Propositional Logic

Attempt at a decidability prooffor propositional logic inHilbert’s 1905 lectures on logic.

Hilbert’s 1917/18 lecture coursePrinciples of Mathematicscontain a completeness prooffor propositional logic.

Proof by algebraicmanipulation; uses normalforms in essential way.

Decidability follows andsemantic completeness follows(Bernays 1918).



The General Decision Problem

“[We require] not only the individual operations but also thepath of calculation as a whole should be specified by rules, inother words, an elimination of thinking in favor of mechanicalcalculation.

If a logical or mathematical assertion is given, the requiredprocedure should give complete instructions for determiningwhether the assertion is correct or false by a determinatecalculation after finitely many steps.

The problem thus formulated I want to call the general decisionproblem.”

Behmann, “Entscheidungsproblem und Algebra der Logik”, Göttingen, May

10, 1921



Behmann’s Algebraic-Syntactic Proof

Result: decidability of monadicsecond-order logic

Procedure modelled onalgebraic elimination problem

Successively removesquantifiers from a given (closed)formula

Reduces truth of formula tocondition on the size of thedomain



Hilbert and Bernays on the Decision Problem

Lectures Logische Grundlagen der Mathematik, co-taught byHilbert and Bernays, Winter 1922/23

Statement of decision problem

Discussion of importance, examples from geometry

New proof of decidability of first-order monadic logic(without =)

Proof is not algebraic, but semi-semantical:ñ If not valid, not valid in some finite individual domain with

at most 2n elementsñ Validity in a finite domain reducible to propositional

calculus



Schönfinkel’s Original Proof

Talk given by Bernays andSchönfinkel in December 1921

Manuscript (in BernaysNachlass) from Winter term1922/23

Treats case of validity offirst-order formulas of the form(∃x)(∀y)A where A onlycontains one binary predicatesymbol

Uses infinite sums and products(like inSchröder/Löwenheim/Skolem)



Bernays and Schönfinkel 1927

“Zum Entscheidungsproblem der mathematischen Logik”, Math.Ann. 99 (1928), submitted March 1927.

Discussion of decision problem in general, cardinalityquestions in particular

Duality of satisfiability and validity made explicit

Discussion of satisfiability and finite satisfiability(Löwenheim and Skolem)

Interpretation of predicates extensional

Proof of finite controllability of monadic class with bound(from 1922/23 lectures)

Discussion of prefix classes



The Bernays-Schönfinkel Class

Class ∃∗∀∗A now called“Bernays-Schönfinkel class”

The class ∀∗∃∗A is shown to bedecidable for validity in Bernaysand Schönfinkel 1928.

It is the preliminary “trivial”case before Bernays goes on tothe actual contribution of thepaper.

The Bernays-Schönfinkel class isnot dealt with in Schönfinkel’smanuscript.



The Schönfinkel Class

(∃x)(∀y)A with A quantifier-free and containing n binarypredicate symbols decidable for validity

Proof is by giving bound on countermodel: 2n

Schönfinkel’s manuscript gives algebraic argument

Semantic proof thus due to Bernays



The Ackermann Class

“Über die Erfüllbarkeit gewisserZählausdrücke”, Math. Ann. 100(1929), sumitted February 1928

Deals with formulas of the form∃∗∀∃∗A and satisfiability(i.e., ∀∗∃∀∗A and validity)

Emphasizes satisfiability andupper bound on model



The Ramsey Class

“On a problem of formal logic”,Proc. LMS, ser. 2, vol. 30 (1929),submitted November 1928

Deals with satisfiability offormulas of the form ∀∗A and=; sketch for ∃∗∀∗A.

Requires finite Ramsey theorem(and thus sparked Ramseytheory)

Like Behmann, focuses onspectra: sizes of domain whereformula satisfiable



The Development of Semantics

No clear semantics in H 1917/18

Issues:ñ Domain of quantificationñ Interpretation of predicates and predicate variablesñ Range of quantified variablesñ Open formulas

“correct [richtig]” vs. “valid [allgemeingültig]”

validity vs. satisfiability



Behmann on Semantics: Correctness

Distinguishes predicate constants and variables

Formulas with (free) predicate variables are notpropositions [Aussagen]

Basic notion: correct (applies to 2nd order sentences only)

Russellian notion, but domain is variable

Decision problem: decide if a 2nd order sentence is correctfor every domain of individuals



Bernays on Semantics: Notions

Formulas are schemata for propositions

They contain two variable elements:ñ domain of individuals (values of x’s)ñ predicates (values of P ’s—considered as functions from

objects to truth values)

Validity defined for a specific domain

Decision problem: Is A valid for every domain?

Dual notion: satisfiability (taken most likely from Skolem1920)

Empty domain excluded: trivial for decision problem(∀ always valid, ∃ always unsatisfiable)



Bernays on Semantics: Results

Validity and satisfiability are dual

If A satisfiable in domain X, it is satisfiable in any domain Ywhere Y and X have same cardinality (X finite)(Essentially a proof that isomorphism implies elementaryequivalence)

If A is satisfiable in X, then A satisfiable in any Y ⊇ X.


Documents

The Decision Problem and the Model Theory of First-order Logicpeople.ucalgary.ca/~rzach/static/cshpm.pdf · Proc. LMS, ser. 2, vol. 30 (1929), submitted November 1928 Deals with satisﬁability