Aristotle' Relations: An Interpretation in Combinatory Logic
aa r X i v : . [ m a t h . HO ] J u l ARISTOTLE’S RELATIONS: AN INTERPRETATION INCOMBINATORY LOGIC
ERWIN ENGELER
Abstract
The usual modelling of the syllogisms of the Organon by a calculus of classes doesnot include relations. Aristotle may however have envisioned them in the first twobooks as the category of relatives, where he allowed them to compose with them-selves. Composition is the main operation in combinatory logic, which thereforeoffers itself to logicians for a new kind of modelling. The resulting calculus includesalso composition of predicates by the logical connectives.
Introduction
Relations turn up at birthdays, congratulating. Even logicians have them; of thefirst one, Aristotle, we even know some of their names. But it is a question amonghistorians of axiomatic geometry whether he had the other kind of relations, theones that modern logicians are concerned with. Indeed, many hold that the Stagiritedid not have this concept, and that Greek mathematics, in particular Euclid showsthis: the relation of betweenness does not enter his axioms for geometry. Thetradition of Aristotelian Logic is often blamed for this serious lacuna. In factEuclidean axiomatics reached completion only in the 19th century, [4]. Missingthe concept of relation is perceived as making it unfit as an adequate logic fordeveloping formal axiomatic mathematics; that had to wait till the Boole-Peano-Russell disruption which eclipsed traditional logic. This, I think, is overstating thecase. The Organon itself gives a picture of Aristotle’s understanding of judgementsother than those that are formulated by the syllogistics and included relations andfunctionals. My argument comes in two parts, motivation and formal development.The first part experiments with concepts of definitions on an example that Aristotlehimself could have handled. These are discussed in terms, motives, that I discernin the Organon, in particular the composition of predicates called “relatives”, theuse of logical connectives and forms of recursion. Modalities, also an importantingredient of the Organon are not included here, which while feasible are irrelevantto the present theme. In the second part, the compositional aspect of predicatesis brought forward and made into the basis for interpreting the Organon . Thisresults in the establishment of a logical calculus E Λ of judgements. This provides amodel of syllogisms which include relations. The conclusion is, that Aristotle hadthe means to treat relations but chose not to do so for his syllogistics. Date : July 9, 2020. The first part of his little essay was written as a gift for my 90th birthday, (see dedication).
If you’ll bear with me, let us see how Aristotle would, and could speak of hisrelatives, formally, and within the framework of his toolkit, the
Organon — in mynaive and quite ahistoric reading, using an ad hoc formalism that we shall laterturn into a formal calculus.
The main grammatical operation is applying a predicate to a subject: [ red ] · [ blood ]is a statement which predicates that blood is red. All kinds of thought objectsare admitted as predicates; Aristotle divides them into “categories”, distinguishingfor example between predicates about quantity (“big”), and quality (“red”) andrelatedness, (the category of “relatives”).Looking at example of relatives, let [ mother ] predicate of a subject that it is enjoy-ing motherhood. Thus [ mother ] · [ Phastia ] states that Phastia is a mother. Nothingprevents us from using this thought object as a predicate:([ mother ] · [ Phastia ]) · [ Aristotle ]tells us that Phastia is the mother of Aristotle. This turn is what Aristotle (Cate-gories, Chapt.7) really meant with the category of relatives; he in fact called thiscategory “things pointing towards something”. The above predicate [ mother ] ofmotherhood predicates of a subject a that the subject b is her child, “pointing a to b ”. Thus, a relative predicate in fact introduces a binary relation by composition ofpredicates. The compositional nature of relatives is an early showing of Currying,a device that became one of the central aspects of combinatory logic.For the moment, we don’t concern ourselves with the question as to what categorysome predicate or subject might belong, all are treated as relatives; one of the usesthat Aristotle gets out of such prescriptions is to avoid predicating nonsense bydisqualifying predications between certain categories.Using the “relative” predications of motherhood and fatherhood, of marriage, andof being male or female applied to family members, we can easily envision the ge-nealogy of Aristotle as a list of such statements. He himself would use many morepredicates to talk about his relations: he would use “son”, “sister”, “grandfather”,“sister-in-law”, or even “male descendent” etc. as predicative concepts. Let us seehow that could fit in. One way of introducing a new predicative concept is explicitly, as compositepredicates. Using variables a, b, c, . . . for subjects, the definition of b being a childof a is simply([ child ] · b ) · a = ([ mother ] · a ) · b .For some relationships we need logical connectives such as “and” and “or”. Theseare denoted by ∧ and ∨ , and used on predicates P and Q to obtain P ∧ Q and P ∨ Q . Thus, the predicate of being a “son” is([ son ] · b ) · a = ([ mother ] · a ) · b ∧ [ male ] b ,Similarly, constituting the predicative concept of a “family”: For a, b, c, d to form acore family, predicated by the predicate [ f amily ] on some individuals a, b, c, d set[ family ] abcd = [ mother ] ac ∧ [ mother ] ad ∧ [ father ] bc ∧ [ father ] bd ∧ [ female ] a ∧ [ male ] b . Notation : we have dropped the center-dot that denotes application and adhere to
RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 3 the convention that sequences of applications are to be understood as parenthesisedto the left: uvw is read as ( u · v ) · w , etc.The idea of “pointing to something” has just been applied again: if P a is a relativepredication then
P a and
P ab may be too. P in the context P abc , for example,would introduce a ternary relation. This leads to a more general notion of explicitdefinitions:Formally, an explicit definition concerns an expression ϕ ( x , x , . . . x n ) built up fromvariables, predicates (introduced earlier as definienda) and the logical connectives.It defines a new predicate P by the defining equation P x x . . . x n = ϕ ( x , x , . . . , x n ).This is the Principle of Comprehension . It comprehends the connective structureof ϕ into a single predicate, an idea that goes back to Sch¨onfinkel and is a basicconcept for Curry’s combinatory logic. As ethnologists tell us, all, (even “primitive”) cultures allow definitions of famil-ial relatedness, some quite elaborate. The simplest ones are of the explicit kind asabove, but this is far from sufficient. Consider the notion of being (maternal) sib-lings. The desired predicate [ sibling ] cd hides a mother somewhere in its definition.Aristotle resolves this by introducing the construction “some P ”.We denote the construct “some P ” by ε x ( P x ). It implies a sort of existential ref-erent. The assertion “some P are Q ” would then transcribe to Q · ( ε x ( P x )). Thisdevice was introduced by Hilbert to be used in his foundational program as a toolfor the formal elimination of quantifiers. With it we can define the predicate ofbeing a sibling:[ sibling ] yz = [ mother ]( ε x ([ mother ] xy )) z ,stating that y ’s mother x is also mother of z . We now can have uncles, cousins of all kinds, marriages between them, etc.,enough to tell the story of Aristotle’s relations, and formulate things like “Aristotleis the father-in-law of his niece”. However, the concept “male descendent” whichwas very important at the time is still open. This is a typical case for using adefinition by recursion:[ mdesc ] xy = ([ male ] y ∧ [ father ] xy ) ∨ ( ε z ([ father ] zy ∧ [ mdesc ] xz ))stating that a male descendent y of x either is a son of x or there is some maledescendent who is his father.Are such definitions admissible? What it amounts to, is that it allows the use of the ε -operator also for predicates as variables: The definiendum [ mdesc ] is representedby a variable U .[ mdesc ] xy = ε U (([ male ] y ∧ [ f ather ] xy ) ∨ ( ε z ([ f ather ] zy ∧ U xz )))You may have noticed that the ε operator allows to understand the Aristotelean“some P are Q ” as Q · ( ε x P x ) in the present ad hoc formalism. We have not usedtwo other basic ingredients of Aristotle’s formalism; there was no need of negationand of the companion to “some x are y ”, namely “all x are y ”. This will be be taskof a later chapter. ERWIN ENGELER
Organon
For all I know, Aristotle would have accepted each definiens in the above def-initions as a statement in his sense. But he would have hesitated to call them“categorical” statements. The distinction arises in the course of his developmentof the
Organon .The Organon , the way I read it, has the character of a manual, a texbook that in-structs the reader in the art of preparing a conclusive argument using well-formedand immediately understandable statements. The
Organon consists of three parts,“The Categories”, “On Interpretation” and “Prior Analytics” (in two books). The first two introduce the notion of well-formed-ness by a discourse of examplesand grammatical distinctions. The question is how to combine predicates whileevading non-sensical, ambiguous or misunderstandable compositions as far as pos-sible: How do you compose [ good ], [ man ] and [ shoemaker ] (Int 11) ?You may take conjunction and application to form([ good ] ∧ [ shoemaker ])[ man ] or ([ good ][ shoemaker ])[ man ] or [ shoemaker ]([ good ][ man ])or ([ good ][ man ])[ shoemaker ],each with a valid and different meaning,.Disjunction can enter with predicates of the category of qualities such as colour;([ black ] ∨ [ white ])[ man ], (Cat 8), or([ green ] ∨ [ blue ])[ turquoise ], or [ green ]([ blue ][ turquoise ]) or ([ blue ][ green ])[ turquoise ],as you may judge the quality of the stone.Negation enters in two pairs of opposites (Cat 10, Int 6-10):“some P are Q ” as against “no P is Q ”, and“all P are Q ” as against “some P are not Q ”.These are the four kinds of statements that Aristotle calls categorical statements .In Prior Analytics, Aristotle gets down to the business of constructing conclu-sive sequences of arguments. Already required for categorical statements, eachargument should be terse and immediately understandable. He chose to restrictthe form of statements to the four categorical ones above, which for conveniencewe denote in the form ϕ ( x, y ) , ψ ( x, y ), etc.And then he shows unguem leonis : he develops a formal system of logic based onlogical arguments, called syllogism , of the form ϕ ( P, Q ) , ψ ( Q, R ) ⊢ χ ( P, R ),expressing that from the categorical statements ϕ ( P, Q ) and ψ ( Q, R ) you may in-fer χ ( P, R ), (PA A 1 - 14). Since there are four of these, there is a plethora ofsuch deductive patterns. Aristotle proceeds to eliminate all but fourteen of themby showing, using counterexamples, which of them preserve the truth of the state-ments.The high point of Prior Analytics is the proof of a metatheorem: all fourteen canbe deduced from just two of them, (PA A 23). Consulted translations: Owen [7] and Smith [10]. These are cited as (Cat), (Int), (PA A), (PA B) with only the chapters indicated, e.g. (Cat8) is Chapter 8 in Categories. “Writing with the the claws of a lion.” RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 5
Organon
By this metatheorem, Aristotle establishes a sort of completeness for his syllo-gistic proof system. This is different from the present notion of completeness inmathematical logic which involves models.The story of models for Aristotle’s logic can be traced from Boole’s Laws ofThoughts, to Lukasiewicz [6], Shepherdson [9] and Corcoran [1] in the 20th century.The view developed that this logic dealt with unary truth-functions which could beunderstood as classes. To accomodate the interpretation of categorical sentences,the classes had to be non-empty. Corcoran, for example, interpreted these as:“all A are B ” is: A ⊆ B ,“no A is B ” is: A ∩ B = ∅ ,“some A are B ” is: A ∩ B = ∅ ,“some A are not B ” is: A B .With such interpretations, syllogistic may be treated as a calculus of equations [9],something that Boole seems to have had in mind. It reflects a rather impoverished Organon . But if people considered this sort of models as the true interpretationof Aristotle, there is no place for relations, and the opinion that he therefore isresponsible for the lacunae of Euclidean axiomatics gets some support.Scholars of Euclidean axiomatics such as de Risi [3], do not share this opinion:Aristotle, an alumnus of Plato’s academy, where famously nobody entered withoutit, did know geometry. Prior Analytics contains a geometric proof of the equalityof base angles in an equilateral triangle,(PA A 26), and “the principle of Aristotle”of Euclidean tradition is related to parallelity.Posterior Analytics (Chapter 19), reflects Aristotle’s understanding of recursion as a mentally completed inductive definition of a concept. “Mental completion”is hard to understand without a set-theoretic mindset, and it was a controversialissue for many commentators of Aristotle. Logic definitely turned away from the Aristotelian tradition only at the turn tothe 20th century. Bertrand Russell was an important mover in this. He had learnedclassical logic as a student but also had read up on Leibniz’s attempts at reforminglogic, critically. He asked himself several questions inspired by this reading, inparticular one that is relevant here:“(1) Are all propositions reducible to the subject-predicate form?” [ 8, p.13 ] On the following pages of this book he proceeds to demonstrate by examples, thata logic adequate for mathematics cannot dispense with relations. Indeed later, in
Principia Mathematica they are a central ingredient. It was therefore only settled after the creation of set theory. I’m grateful to Prof. V. de Risi for pointing me to this book in recent correspondence.
ERWIN ENGELER
Organon in Combinatory logic
An adequate mathematical model for more of Aristotle’s logic seems to be miss-ing. This section describes my attempt to construct one.Logicians who have followed me to this place have long noticed that the ad hoc formalism that I introduced in the discussion of Aristotle’s family is in fact anextension of combinatory logic. Predicates, logical concepts and operations wereadded to the language by defining equations in these terms, combined with thecombinatory application operation. This will now be turned into a formal calculusbased on a language E which extends the language of combinatory logic.Before introducing this language, we choose a mathematical structure into whichthe language will be interpreted. Because it includes combinatory logic, we need amodel for that. The Language E of the modelling extends that of combinatory logic.We first consider a fragment E of E . It consists of expressions, built from variablesand constant predicates by the operations of application, the epsilon operator ε x and the alpha operator α x for variables x . These operations bind the variable x .We shall distinguish between objects that are called predicates and predications. A predication may be obtained by applying a predicate P to a variable x , “predicatingsomething about x ”, written P · x .The basis for our interpretation of the language E is the graph-model of com-binatory logic, Scott (1969), Plotkin (1972), Engeler (1981).Let A be a non-empty set and define recursively G ( A ) = A, G n +1 ( A ) = A ∪ { α → a : α finite or empty , α ⊆ G n ( A ) , a ∈ G ( A ) } ,where α → a is a notation for the pair h α, a i .The union of these G n ( A ) is denoted by G ( A ).The combinatory application operation is defined on subsets M and N of G ( A ) as M · N = { x : α → x ∈ M, α ⊆ N } .With this interpretation of the application operation, the set G ( A ) can be shownto be a model of combinatory logic. The elements of the model are the subsets of G ( A ). We shall show below that the model satisfies the Comprehension Axiom ofCombinatory Logic :For every expression ϕ ( x , . . . , x n ) built up by the application operation from theconstants and variables (interpreted as subsets of G ( A )), there exists an element M of the model such that M · x · x · · · · x n = ϕ ( x , . . . , x n ).The proof of this theorem actually produces an algorithm of comprehension toobtain M . Observe that all elements of M have the form α → ( α → . . . ( α n → a )) with α i ⊆ G ( A ) finite or empty, and a ∈ G ( A ).For the next steps we shall rely on Sch¨onfinkel’s comprehension theorem . Heshows that if we have the “combinators” S and K for which the equations S P QR = P Q ( P R ) and K P Q = Q hold for all predicates P, Q, R , then there is the followingconversion:
Comprehension Theorem of Combinatory Logic
RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 7
For every combinatory expression ϕ ( x , . . . x n ) built up from constants and thevariables x i there is a purely applicative expression ψ ( S , K ) such that(((( ψ ( S , K ) · x ) · x ) · · · ) x n = ϕ ( x , . . . , x n ).For completing the proof we need only produce interpretations of the two constantsand show, by inspection, that these conform to the equations, and thereby verifythat by our interpretation we have in fact a model of combinatory logic. This isdone in this author’s 1981 paper on graph models, [5]. Here are the interpretationsof the two combinators:[ K ] = {{ a } → ( ∅ → a ) : a ∈ G ( A ) } ,[ S ] = { ( { τ → ( { r , . . . , r n } → s ) } → ( { σ → r , . . . σ n → r n } → ( σ → s ))) : n ≥ , r , . . . , r n ∈ G ( A ) , τ ∪ σ ∪ · · · ∪ σ n = σ ⊆ G ( A ) , σ finite } . Combinatory Predicates are composed by the operation of application from pred-icate constants C j and variables x i to form expressions ϕ ( C , . . . , C m , x , . . . x n ).The constants C j are interpreted as subsets [ C j ] of G ( A ), each variable x i rangesover a specific subset of G ( A ). Their mention in ϕ is usually suppressed.Some predicates can be used for predications : If the predicate P is interpretedas [ P ], a subset of G ( A ), and [ P ] is a set of elements of G ( A ) of the form( α → ( α → . . . ( α n → a )) with a ∈ G ( A ) , α i ⊆ G ( A ) finite or empty , i = 1 , . . . , n ,then [ P ] can act as a predication [ P ] · [ x ] · · · [ x n ] on these variables, interpreted assubsets [ x i ] of G ( A ). Notation : Where no ambiguity results we may omit the brackets on interpretedvariables in the future.The intuition behind this interpretation of predication is that [ P ] as a predicationexpresses some facts about each subject-variable x i . These facts are the extentto which x i conforms to the predicate P , the conformity being expressed by thecorresponding sets α i . We call these facts “ attributes ”.An interpretation of a predication is perhaps best illustrated by an example whichwe take from the family context of section 1. The interpretation of the parentpredicate [ parent ( x, y )] is a set of expressions ( α → ( α → a )) with α , α ⊆ G ( A ) , a ∈ A , and where A is a set of people, each is present with the individualattributes. – In distinction to section 1 we added the variables inside the thebrackets for clarity, they relate the variables x, y to the sets α , α in that order.Each set α i is understood as a set of attributes: α of being a parent, α of beinga child.The meaning of the predication [ parent ( x, y )] · [ x ] · [ y ] therefore is: “[ y ] is the setof people in A for whom [ x ] is a set of parents”. Specifically: α is the set ofexpressions { x } → ( { z } → z ) for x male, { y } → ( { z } → z ) for female, and α consists of all { z } → z for the children z, with x, y, z ∈ A . The predication producesthe children of x and y if x is male and y female. Categorical Predicates , the analog to the categorical statements in the
Organon ,arise from combinatory predicates by using “ for some ” and “ for all ”, referring tothe variables of a predicate. They constitute our language E . We use ε x ϕ ( x ) todenote “some x has ϕ ( x )” and α x ϕ ( x ) to denote “all x have ϕ ( x )”. ERWIN ENGELER
Extending the modelling to the ε -operator is a bit subtle. The term ε x ϕ ( x ) is tobe interpreted as the result of a recursion in the sense of “completed induction”.Recall that the modelling of the language E is a process of finding denotations forelements of the language in a combinatory model. The modelling of ε x ϕ ( x ) involvesthe determination of an object F , a subset of G ( A ), which has the property ϕ ( F ).“Recursion” means that such an object is already determined by an object F , thebasis of recursion. This implies that the process of interpretation calls here for thechoice of a particular object F , which, as the case may be, is a challenge for theingenuity of the modeller.Given a unary predicate P , the object [ ε x ( P x )] is therefore determined by the inter-pretation, which proposes an initial set F ⊆ G ( A ) with the property F ⊆ [ P ] · F ,and yields[ ε x ( P x )] = S n [ P ] n F = F , [ P ] n denoting the n -th iteration.Then F is a fixpoint of [ P ], noting [ P ] · F = F .The finding of an appropriate F is the cardinal point on which it turns whether ornot the interpretation of the predication becomes vacuous, (see e.g. 4.1 below onthe existence of a model for projective geometry). F always exists, determined bythe interpretation, in the worst case it is F = F = ∅ .The α -operator is interpreted as[ α x ( P x )] = [ P ] · ext [ P ] ([ x ]), where ext [ P ] ([ x ]) = { a : ∃ α → a ∈ [ P ] } is the set ofpossible values for [ P ] x .The Aristotelian “some P are Q ” thus translates into [ Q ] · [ ε x ( P x )] and “all P are Q ” into [ Q ] · [ α x ( P x )].To extend these operations to n -ary predications we make another use of compre-hension to separate out a specific variable in an expression ϕ ( x , . . . , x n ):( ϕ j ( x , . . . , x n ) · x · · · x j − · x j +1 · · · x n ) · x j = ϕ ( x , . . . , x n ) · x · · · x n The ε -operator and α -operators for n -ary predicates [ ϕ ] are defined accordingly :[ ε x j ( ϕ j ( x , . . . , x n ))] · x · · · x j − · x j +1 · · · x n = F , where, for F given by the inter-pretation, F = S m ([ ϕ j ( x , . . . , x n )] · x · · · x j − · x j +1 · · · x n ) m · F . F is a set function with n − α x j ([ ϕ j ( x , . . . , x n ))] · x · · · x j − · x j +1 · · · x n = [ ϕ j ( x , . . . , x n )] · x · · · x j − · x j +1 · · · x n · ext [ ϕ j ] ([ x j ]), where ext [ ϕ j ] ([ x j ]) = { a : ∃ ( α → · · · → ( α n → a )) ∈ [ ϕ j ( x , . . . , x n ))] } . Remark:
Two categorical statements “no P is Q ” and “some P are not Q ” aremissing in E . They are added in the next section in the context of negation. Thisis an expository choice. In fact they could have been added here separately, whichwould make E the full categorical language. Predications as defined above are “factual” interpretations, they produce a setof facts [ P ] · x · · · x n . Our interpretation of the language E resulted in a calculusof facts and as such cannot really be called a logical calculus. It lacks the logicalconnectives and judgements about the truth of a predication.Predications lend themselves to logical composition by the connectives ∧ , ∨ and ¬ .These constitute a language extension E of E . The interpretation is extended to E recursively on the structure of the logical composition: the evaluation of RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 9 [ ϕ ( x , . . . , x n )] x · · · x n ∧ [ ψ ( x , . . . , x n )] x · · · x n is[ ϕ ( x , . . . , x n )] x · · · x n ∩ [ ψ ( x , . . . , x n )] x · · · x n ,correspondingly with ∨ and ∪ , where we conformed the two predications to com-bined variables x , . . . , x n by comprehension.For negation we set [ ¬ ϕ ( x , . . . , x n )] x · · · x n = ext [ ϕ n ] ([ x n ]) − [ ϕ ( x , . . . , x n )] x · · · x n .This concludes the definition of the logical predications. In particular, we can nowexpress all the syllogistic statements “some P are Q ”, . . . , “some P are not Q ”,and our ad hoc formalism in section 1 is thereby legitimised. Observation:
The operation of negation could have been added separately inthe definition of the language E which would make it possible to add to it the twomissing categorical statements “no P is Q ” and “some P are not Q ”. The extendedlanguage E thus includes the full Aristotelian language of categorical statements.The intuition behind our Truth Definition for a unary predicate P , modelledby a set of expressions α i → a is that [ P ] x is true if x has all the attributes thatare required by P , that is [ P ] x = { a : α → a ∈ [ P ] } . Correspondingly the truthdefinition for arbitrary predications in E is[ ϕ ( x , . . . , x n ) x · · · x n is true , denoted by ⊤ ,if [ ϕ j ( x , . . . , x n ) x · · · x j − · x j +1 · · · x n · x j = ext [ ϕ j ] ([ x j ]) for each j .[ ϕ ( x , . . . , x n ) x · · · x n is false , denoted by ⊥ ,if it is a proper non-empty subset of ext [ ϕ j ] ([ x j ]) for some j .In all other cases it is indeterminate , denoted by △ . The “facts” produced by predications [ P ] · x i · · · x n in E do not reflect the actualrelations between the arguments x i . This can be accomplished by introducing relational predications [ P R ]: If [ P ] consists of elements( α → · · · → ( α n → a )) with a ∈ G ( A ) , α i ⊆ G ( A ) finite and nonempty, then[ P ] R · x · · · x n is the set of all h a , . . . , a n i : ∃ ( α → · · · → ( α n → a )) ∈ [ P ]such that a i ⊆ x i ⊆ α i , i = 1 , . . . , n . The subscript R is tacitly understood in thefollowing.Let E be extended to E Λ ( C , . . . , C m ) by introducing a valuation of the relationalpredicate constants C , . . . , C m of E .The factual interpretation of relational predications distinguishes items thatverify or falsify it. This distinction is based on a valuation Λ i for each one ofthe constant predicates [ C i ]. We denote the valuation of a tuple h a , . . . , a n i by h a , . . . , a n i Λ i = h a , . . . , a n i ⊤ if it is a verifying fact, by h a , . . . , a n i ⊥ if it is falsi-fying .For a n -ary predicate constants [ C i ] the facts that are valued by the valuation Λ i are the tuples h a , . . . , a n i ∈ D i , where D i = {h a , . . . , a n i : ( α → . . . ( α n − → a j )) ∈ [( C i ) j ] , j = 1 , . . . , n } We define the corresponding predication [ C i ] Λ i by the set of objects( α → . . . ( α n → h a , . . . , a n i ) Λ i where( α → ( α → · · · → ( α n → a n ))) ∈ [ C i ] , h a , . . . , a n i ∈ D i .As a result, the factual interpretation of the relational predication C i is the set function [ C i ] Λ i · x · · · x n which produces a set containing elements h a , . . . , a n i ⊤ , h a , . . . , a n i ⊥ and △ , (for the cases where the predication returns the empty set onthe given inputs x , . . . , x n ).The factual interpretation of the language E Λ ( C , . . . , C m ) is based on the val-uation Λ i for each C i , and then extended over logical composition, the ε - and α -operations and the quantifiers ∃ and ∀ as follows: The valuation Λ maps eachpredication into a set of the above objects. We represent each such set as a proposi-tional formula consisting of the valued tuples h a , . . . , a n i . The valuation Λ assignsto each of them the truth-value true or false as indicated by the superscripts. Toobtain the truth-value of a constant predication, the set produced by it is inter-preted as the conjunction of these elements as a formula in a propositional logic.The presence of △ in a propositional formula assigns to it the value indeterminate .This interpretation is then extended as follows to predications obtained by the log-ical operations:The conjunction and disjunction of predications [ ϕ ( x , . . . x n )] Λ · x · · · x n and[ ψ ( y , . . . y m )] Λ · y · · · y m are conjunctions respectively disjunctions of the corre-sponding propositional expressions. The interpretation of negation is obtained byinverting all ⊤ to ⊥ and all ⊥ to ⊤ in the valuations of the tuples h a , . . . , a n i ⊤ , h a , . . . , a n i ⊥ .The factual interpretation of the ε - and α operators on a predicate ϕ ( x , . . . , x n )is obtained as[ ε x j ( ϕ ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n = {h a , . . . , a j − , a j +1 , . . . , a n , a j i ⊤ : a j ∈ [ ε x j ( ϕ j ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n · x j } ,[ α x j ( ϕ ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n = {h a , . . . , a j − , a j +1 , . . . , a n , a j i ⊤ : a j ∈ [ ϕ j ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n · ext [ ϕ j ] ([ x j ]) }∪ {h a , . . . , a j − , a j +1 , . . . , a n , a j i ⊥ : a j ∈ [ ¬ ϕ j ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n · ext [ ϕ j ] ([ x j ]) } . Existential and universal quantifiers can now be introduced as follows:[ ∃ x j ϕ ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n = [ ε x j ( ϕ ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n ,[ ∀ x j ϕ j ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n = [ α x j ( ϕ ( x , . . . , x n )] Λ · x · · · x j − · x j +1 · · · x n .Altogether, we have now extended the factual interpretation of predications toall of the language E Λ ( C , . . . , C m ). Each predication produces a propositionalformula containing elements of the form h a , . . . , a n i ⊥ , h a , . . . , a n i ⊤ and △ . Thiscompletes the factual interpretation of E Λ ( C , . . . , C m ); it assigns a propositionalformula of verifying, falsifying tuples and △ -s.The truth-value of the predication is obtained by evaluating the tuples of the propo-sitional components as true, false and indeterminate as above. RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 11
Looking for an example to discuss our interpretation E and E Λ of the Organon ,recall Plato’s advice in the
Republic , (Chapter VII), that with geometry you caneducate the mind.We take “geometry” here as a first-order theory in mathematical logic, and forsimplicity restrict to projective geometry.The first-order models in mathematical logic are relational structures h A, R , . . . , R n , c , . . . , c n i with relations R i ⊆ A k i and individual constants c j ∈ A .In our modelling the relations R i correspond to combinatory constants that denote k i -ary predicates; the constants c j also correspond to combinatory constants. Theseobjects are then interpreted as subsets of G ( A ) according to the above definitions.As an example, consider a mathematical structure such as a projective plane, un-derstood as a set P of “points”, L as a set of “lines” with the binary relation of“incidence” Inc ⊆ P × L , where P ∩ L = ∅ . These are subject to some axioms suchas: “For any two points there is a unique line on which they are incident.”The combinatory model is based on this A for the construction of G ( A ). In E theincidence relation is interpreted as[ Inc ] Λ = { ( { p } → ( { p } → l )) → ( { p } → l ) : p ∈ P, l ∈ L, h p, l i ∈ Inc } .In E Λ ( Inc ) it would be[
Inc ] Λ = { ( { p } → ( { p } → l )) → ( { p } → h p, l i ) : p ∈ P, l ∈ L, h p, l i ∈ Inc } An equality predicate is needed here only for points and lines and can therefore beviewed as a binary predicate constant with the interpretation[ eq ] = { ( { x } → ( { y } → y )) → ( { y } → y ) : x = y, x, y ∈ P ∪ L } in E ,with the corresponding the relational predication[ eq ] Λ = { ( { x } → ( { y } → y )) → ( { x } → h x, y i ) : x = y, x, y ∈ P ∪ L } in E Λ .Using the quantifiers introduced earlier, the above axiom is interpreted as ∀ x ∀ x ([ eq ] Λ x x ∨∃ y ([ Inc ] Λ x y ∧ [ Inc ] Λ x y ∧∀ z ([ ¬ Inc ] Λ x z ∨ [ ¬ Inc ] Λ x z ∨ [ eq ] Λ yz ))).with the parameter [ Inc ]. – The other axioms would be represented in the samefashion and combined into a logical predicate denoted by π ( Inc ). It then turns intoan exercise of inventiveness in finding the required fixpoints, and of formal persis-tence to verify that it returns the value true on some given model of projectiveplane geometry, that is for a given binary relation of incidence, e.g. the
Fano plane of seven points and seven lines.Of course, projective geometry itself may be considered as a defined predicate,obtained by using a suitably defined recursion on a predication: Let the variable X be substituted for Inc in the axiom-expression π ( Inc ) . The recursion equation X = ε X π ( X )) defines the geometric concept of a projective plane. The modellingof this formula starts with fixing on a given set of points and lines, these may comefrom defniitions in terms of finite fields or Q , R or C . Here again the modellingconsists in the finding of a fixpoint, which is the crucial matter in obtaining anactual model. If the language has more than one predicate constant, e.g. one forbetweenness, one would of course use joint recursion for incidence and betweenness. Remark : Perhaps it is worth mentioning that this use of the predicate π ( X ) is anexample of introducing additional predicate constants with which one may extend the language to encompass additional concepts. This corresponds to the familiarway to define new mathematical concepts and structures. The restriction of “facts” to the the basic set A as the extension of the variablesis natural in simple contexts like the Aristotle family. We also used it in the contextof projective geometry treating of ”facts” about points and lines. But this is toorestrictive even in this case: Consider the notion of point-functions, for exampleprojectivities. The object f is a point-function if f pq = f pq for all q , q ∈ P .This can be expressed by a predication [ fun ] defined by[ fun ] f = [ eq ] · α q ( f pq ) · α q ( f pq ),using the predication [ eq ] from above. Observe that [ fun ] is an element of G ( A ).The ε -operator also creates “facts”: Consider the line connecting two point p , p ,expressed by ε l ([ eq ] · [ Inc ]( p l ) · ( p l )),which is a function on p , p , called a Skolem-function in logic. It is an element of G ( A ).A “Skolem-function” f is thus the result of a recursion, a concept that we tracedback to Aristotle’s notion of a mentally completed definition of induction. It istherefore legitimate to call this object a thought . – Anyway, I would have preferred “thoughts” over “ predicates ” and “ predications ”. The latter are naturals in theAristotelean context. But I see predicates as thoughts, as sets or patterns of smalland big notions: The predication [ P ] · x is perceived as applying a thought P to athought x , checking to what extent the thought [ P ] applies to x . This perceptionis the background of my modelling and is connected to my work on neural algebrawhich treats of thoughts as patterns of firing neurons. Let me not forget my own brain-child, algorithmic logic, [4], now approachingretirement age after a long career. It treats of algorithmic properties of structures.Its predicates are of the form π ( x , . . . , x n ) which denotes a program π with theinput variables x , . . . x n of elements of the structure. Since combinatory logic candeal with the notions of computation and termination, there is an important pointof contact here which merits elaboration.Programs π ( x , . . . , x n ) are composed of individual instructions, namely assign-ments of the form z := f ( x, y ), decisions such as x < y . These correspond, looselyspeaking, to our predicate constants. Program statements are composed by succes-sive execution ( π ( x , . . . , x n )); ( π ( x , . . . , x n )) and recursion. These essentiallycorrespond to composition and the ε -operation. Finally, the factual interpretation[ π ( x , . . . , x n )] Λ · x · · · x n of a program is the so-called denotational semantics ofthe program. The valuation Λ is understood as the valuation of tuples of elementsof the relations and functions in some relational structure. The program statement[ π ( x , . . . , x n )] Λ x · · · x n evaluates to the result of executing the program on theinput assignment.For the logic of programs, the “ algorithmic logic ”, we chose to evaluate a programas “true” in a relational structure if it halts on all inputs. RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 13
This discussion may be conducted in the interpretation of the language E Λ insection 3.4 above which captures, extends and completes the ad hoc formalism usedin our motivational section 1. The example fits nicely into this framework: The set A lists the names of the members of the family. The binary relations for mother-hood, fatherhood and marriage are lists of pairs of names, and singletons for beingmale or female. Thus, the interpretation [ f ather ] Λ for fatherhood is a case of bi-nary relations which are represented in the form {{ x } → ( { y } → y )) → ( { x } → h x, y i ).Therefore fatherhood in the Aristotle family would be represented by a set contain-ing the substitution instances x := Niarchus1 , y := Aristotle and x := Niarchus1 , y := Arimnestus as well as x := Aristotle , y := Niarchus2 .A more adequate modelling of fatherhood would need more attributes from of thevital statistics of the family members, e.g. profession, place and date of birth, etc.to exclude false claims of fatherhood. Above, we have artificially distinguished thetwo people called “Niarchus” by adding the distinction to the names.In this setting it becomes clear that Aristotle had the conceptual means to con-ceive of, and formally treat, relations. But the primary goal of the
Organon didnot require this.This concludes our search for Aristotle’s relations. The missing relations ofEuclid, a blemish on his axioms, were not noticed at the time because geometrywas understood by Aristotle as describing geometric properties of lines and otherobjects unquestioned as continuous. As pointed out by de Risi in [2 ] this was onlyput into question in the 16th century and not fully received into the understandingof space till the 19th century. What have we learned from our combinatory experiment on the
Organon ?I: The objects of syllogisms, the categorical statements, were interpreted as ele-ments of the language E which expands the set of terms of combinatory logic. Thiscombinatory interpretation thus becomes a model of syllogistics comparable to themodels of Lucasiewicz and others.II: The categorical statements themselves are statements about properties of facts,traditionally about individual facts. This aspect is captured by our language E .III: The logical interpretation of the language E Λ ( C , . . . , C m ) is based on factsabout relations as n -ary predications. This is probably the most adequate ren-dering of my understanding of an extension of the Organon if it were to includerelations.IV: Logical interpretations of predications turns them into
Judgements . Our dis-tinction between factual and logical interpretation recalls the famous distinctionbetween judgements de re and judgements de dicto going back to 12th century The de Risi references were pointed pout to me by Prof.M.Beeson.
Scholastics, when Peter Abelard derived de dicto from de re , as we do. V: Our modelling addressed the semantics of E and E Λ and not its deductions . Butthis is another chapter. Apologia and Dedication
Who knows how Aristotle would react to my experiment. I picture him and Eu-clid as little Raffael-angels looking down on our travails with mischievous interest.My interpretation is based on my individual reading of his
Organon . Individualdoes not mean indivisible; I may have had two minds about a number of things.Also, while I avoided the pitfalls of turning the logical connectives into predicatesand moreover turn syllogisms themselves into statements, I do not propose to justifythis here.Finally, I claim the privilege of a very old man and refrain from the labours ofperforming all the verifications and of following up my own suggestions. Let thefriendly reader smile and forgive. Here, thanks are due to Michael Beeson whosedetailed comments on the manuscript were very helpful.I dedicate this little essay to all my friends and students who troubled themselvesto remember me and my birthday at the symposium in Zurich, early 2020. Andto all those that could not be present because of the distances that destiny putbetween us, in particular to my late friend and colleague Ernst Specker whom thissymposium was meant to honour too. Special mention goes to the three organisersGerhard Jaeger, Reinhard Kahle and Giovanni Sommaruga, the sponsors that theyfound, and to my alma mater , the ETH, for its hospitality.
References [1] John Corcoran. Aristotle’s Prior Analytics and Boole’s Laws of Thoughts.
His-tory and Philosophy of Logic , 24: 261-288,2003[2] Vincenzo de Risi. Francesco Patrizi and the New Geometry of Space. In: KoenVermeir and Jonathan Regier, editors.
Boundaries, Extents and Circulations: Spaceand Spaciality in Early Modern Natural Philosophy , number 41 in Studies in His-tory and Philosophy of Science, chapter 3, pages 55 - 100. Springer InternationalPublishing, Switzerland, 2016.[3]) Vincenzo de Risi. The Development of Euclidean Axiomatics.
Archiv for theHistory of the Exact Sciences , 70: 561-676, (2016)..[4]) Erwin Engeler. Algorithmic Properties of Structures.
Mathematical SystemsTheory , 1: 183-193, 1967.[5]) Erwin Engeler. Algebras and Combinators.
Algebra Universalis , 13: 389-392,1981. This was strongly contested at the time by Bernard de Clairvaux, who maintained that thedogmata of the Church (to which Abelard addressed himself) were de dicto statements and notto be made dependent on judgements de re . Bernard was sainted, Abelard not.
RISTOTLE’S RELATIONS: AN INTERPRETATION IN COMBINATORY LOGIC 15 [6]) Jan Lukasiewicz.
Aristotle’s Syllogistics from the Standpoint of Modern Logic .Clarendon Press, Oxford, second edition,1957.[7]) Octavius Freire Owen.
The Organon, or Logical Treatisies of Aristotle with anIntroduction to Porphyry , (two volumes). Henry G. Bohn, London, 1853.[8]) Bertrand Russell.
A Critical Exposition of the Philosophy of Leibniz . Cam-bridge Univ.Press, Cambridge, 1900.[6]) John C. Shepherdson. On the Interpretation of Aristotelean Syllogistics.
Jour-nal of Symbolic Logic , 21: 137-147, 1956[10]) Robin Smith.
Prior Analytics . Hackett Publ.Co., Indianapolis Cambridge,1989.. Hackett Publ.Co., Indianapolis Cambridge,1989.