Andrzej Mostowski and Foundational Studies 289 A. Ehrenfeucht, V.W. Marek and M. Srebrny (Eds.) IOS Press, 2008 ©2008TheauthorsandIOSPress.Allrightsreserved. The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical Dana S. SCOTT Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic (Emeritus) Computer Science Department, Carnegie Mellon University Pittsburgh, Pennsylvania, USA [email protected] Keywords. Algebraic logic, completeness, Boolean and Heyting algebras “Die Mathematiker sind eine Art Franzosen: redet man zu ihnen, so übersetzen sie es in ihre Sprache und dann ist es alsobald ganz etwas Anderes.” J.W. von Goethe 1. Introduction In their 1963 book, The Mathematics of Metamathematics [Rasiowa and Sikorski, 1963], the authors present an approach to completeness theorems of various logics using alge- braic methods. Perhaps they put the quotation from Goethe over their preface to suggest that mathematicians can be forgiven if they introduce new interpretations of old ideas. The idea of an algebra of logic can of course be traced back to Boole, but it was revived and generalized by Stone and Tarski in the 1930s; however, the most di- rect influence on the work of Rasiowa and Sikorski came from their well-known col- league, Andrzej Mostowski, after WW II. Mostowski’s interpretation of quantification [Mostowski, 1948] can as well be given for intuitionistic as classical logic. The present expository paper will briefly review the history and content of these ideas and then raise the question of why there was at the time no generalization made to higher-order logic and set theory. Entirely new light on this kind of algebraic semantics has more recently been thrown by the development of topos theory in category theory. Some reasons for pursuing this generalization will also be discussed in Section 12. In order to dispel any confusion that might arise about the title of this paper, the author would like to point out that there are other meanings to “algebraic” as relating to the semantics of Logic. Perhaps the first interpretation of quantifiers in an algebraic style goes back to consideration of early work in Descriptive Set Theory. Quantifier manipula- tion was explained in this regard in the well known book on Topology by K. Kuratowski (also quoting Tarski). In 1946 C.J. Everett and S. Ulam published their paper on Pro- 290 D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical jective Algebras. A. Tarski, building on his famous paper on Truth in formalized lan- guages and on his studies of Relation Algebras (along with J.C.C. McKinsey, L. Henkin and many collaborators) developed an elaborate equational theory of Cylindric Algebras (duly quoting Everett and Ulam). In a related but different style, P.R. Halmos and collab- orators studied Polyadic Algebras. A generalization of Relation Algebras was proposed by P.C. Bernays and that research has been continued over many years by W. Craig. Most recently, F.W. Lawvere unified many of these ideas – along with the lattice-theoretic se- mantics explained in the present paper – by showing how quantification can be construed in suitable categories by using the idea of adjoint functors. None of these works are cited in our bibliography here, because the literature is too vast to be explained in such a short paper. 2. Preliminaries, Terminology, Notation The following standard terminology and notation will be used throughout the paper. Acompletelattice(cLa)isapartiallyorderedset(poset) A, where every subset ! !" has a least upper bound (lub) under !.(Duallywecansay,everysubsethasagreatest lower bound (glb).) A broader notion of a lattice requires only every finite subset to have alubandaglb.Alubisalsocalledasup;andaglbaninf. Theorem 2.1. If a poset has all lubs, it has all glbs; and conversely. The proof is well known: the greatest lower bound of a family is the least upper bound of all the lower bounds of the family. Theorem 2.2. Using the following notation, lubs and glbs can be uniquely characterized as follows: (LUB) y X z[ x X. x z = y z] ! ⇐⇒ ∀ ∀ ∈ ! ⇒ ! (GLB) X y z[ x X. z x = z y] ! ⇐⇒ ∀ ∀ ∈ ! ⇒ ! (TOP) 1=!A = ∅ (BOT) 0=" A = ! " ∅ Theorem 2.3. Lattices" (not! necessarily complete) are characterized by these axioms: • ! is a partial order • x z & y z x y z ! ! ⇐⇒ ∨ ! • z x & z y z x y ! ! ⇐⇒ ! ∧ • 0 ! x ! 1 Examples of complete lattices abound in set theory. Here is an example of a direct consequence of Theorem 2.1. Theorem 2.4. The powerset (A) is always a cLa; hence, if a family (A) is P C⊆P closed under all intersections (unions), then , is also a cLa. !C ⊆" We need to remember that in the cLa (A) the intersection of the empty family is P taken to be A. D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical 291 3. Closures on a Power Set (A) P Closure operations C : (A) (A) give interesting examples of complete lattices. There are several kinds ofP closures→P axiomatized as follows. General: • X C(X)=C(C(X)) • X ⊆ Y = C(X) C(Y ) ⊆ ⇒ ⊆ Topological: • X C(X)=C(C(X)) • C(⊆)= • C(∅X Y∅)=C(X) C(Y ) ∪ ∪ Algebraic: • X C(X)=C(C(X)) • C(⊆ X)= C(X) for directed (A) X X W⊆P #∈W #∈W For any closure operation we say that a set X A is closed provided C(X) X. ⊆ ⊆ Theorem 3.1. The family of closed sets of a general closure operation is closed under arbitrary intersections and so forms under inclusion a complete lattice. Moreover, an isomorph of any complete lattice can be found in this way. For the easy proof see [Rasiowa and Sikorski, 1963]. The complete lattices formed from topological or algebraic closure operations have additional properties. We will dis- cuss the topological case further below. 4. Implication and Distribution Definition 4.1. Alatticeoperation is called an implication if and only if it satisfies this axiom: → • x y z x y z ∧ ! ⇐⇒ ! → Note. The axiom uniquely determines an implication on a lattice if it exists. There is an equational axiomatization where we can define the partial ordering in several ways. • x ! y x y = y ⇐⇒ x ∨ y = x ⇐⇒ x ∧ y =1 ⇐⇒ → See [Rasiowa and Sikorski, 1963] for details. A lattice with implication is called a Heyt- ing algebra.(Haisusedforshort.) Theorem 4.1. All Ha’s are distributive lattices. 292 D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical Proof. From basic lattice properties we have x y (x y) (x z).Theny ∧ ! ∧ ∨ ∧ ! x ((x y) (x z)).Similarlywehavex z (x y) (x z).Thusz → ∧ ∨ ∧ ∧ ! ∧ ∨ ∧ ! x ((x y) (x z)).Itfollowsthaty z x ((x y) (x z)).Butthen → ∧ ∨ ∧ ∨ ! → ∧ ∨ ∧ x (y z) ! (x y) (x z).ThisishalfoftheDistributiveLaw.Theotherhalfholds in∧ all lattices.∨ ∧ ∨ ∧ Definition 4.2. AHeytingalgebrawhichhasallglbsandlubs(= complete lattice) is called a complete Heyting algebra (or cHa, for short). Theorem 4.2. AcompletelatticeisacHaif,andonlyif,itsatisfies: x y = (x y ). ∧ i ∧ i i I i I $∈ $∈ The proof of the above equation is a direct generalization of the proof of Theorem 4.1. For the converse we can use the following definition of implication: y z = x x y z . → { | ∧ ! } $ The results about distributivity of Heyting algebras were proved independently by Stone [Stone, 1937] and Tarski [Tarski, 1938]. 5. Boolean Algebras The difference between classical and intuitionistic logic lies in the fact that the former satisfies the law of the Excluded Middle. In algebraic terms we have the following. Definition 5.1. AHa(resp.cHa)isaBooleanalgebra(orBa)(resp.completeBoolean algebra or cBa) iff it satisfies x (x 0) = 1. ∨ → The element x 0 is called the negation of x and is denoted by x. → ¬ Perhaps we should pause to note this passage from [Rasiowa and Sikorski, 1963] on pp. 8–9: The inclusion of two chapters on intuitionism is not an indication of the authors’ positive atti- tude towards intuitionistic ideas. Intuitionism, like other non-classical logics, has no practical application in mathematics. Nevertheless many authors devote their works to intuitionistic logic. On the other hand, the mathematical mechanism of intuitionistic logic is interesting: it is amazing that vaguely defined philosophical ideas concerning the notion of existence in mathematics have lead to the creation of formalized logical systems which, from the math- ematical point of view, proved to be equivalent to the theory of lattices of open subsets of topological spaces. Finally, the formalization of intuitionistic logic achieved by Heyting and adopted in this book is not in agreement with the philosophical views of the founder of in- tuitionism, Brouwer, who opposed formalism in mathematics. Since in treating intuitionistic logic we have limited ourselves to problems which are directly connected with general alge- braic, lattice-theoretical and topological methods employed in the book, we have not included the latest results of Beth and Kreisel concerning other notions of satisfiability which we have adopted. D.S. Scott / The Algebraic Interpretation of Quantifiers: Intuitionistic and Classical 293 As a matter of fact the topological modeling of intuitionistic formal rules is not really to be regarded as “amazing”. For, in view of Theorem 4.2, we conclude at once this theorem and corollary. Theorem 5.1. Every sublattice of a cBa (also cHa) closed under and is a cHa. ∧ Corollary 5.1. The lattice of open subsets of a topological space forms a! cHa. There is also a reverse connection between Heyting algebras and Boolean algebras. An element x of a Heyting algebra is called stable (sometimes regular) if it satisfies x = x. ¬¬ Theorem 5.2. The stable elements of a Ha (cHa) form a Ba (cBa). Aproofisgivenin[RasiowaandSikorski,1963],pp.134–135.Anotherconnection is an embedding theorem. Theorem 5.3. Every cHa can be embedded in a cBa so as to preserve and .