The Common Logic Standard Christopher Menzel Texas A&M University

Traditional First-order Languages Lexicon Basic vocabulary Grammar Recursive rules for forming complex expressions Semantics (Model Theory) Mathematical theory of meaning Formal definitions of validity, entailment, satisfiability Proof Theory Metatheory Soundness, Completeness, Compactness

Lexicon Terms Individual constants E.g.: a, b, c, a, b, c, … Individual variables E.g.: x, y, z, x, y, z, … Predicate constants E.g.: P, Q, R, Boy, Girl, Kissed Boolean operators Typically: , , , ,  Quantifiers Typically: , 

Grammar If P is a predicate and t 1, …, t n are terms, then P(t 1,…,t n ) is an (atomic) formula. If A and B are formulas, so are  A, (A  B), (A  B), (A  B), and (A  B). If A is a formula and x any variable, then  x A and  x A are formulas. Nothing else is a formula.

Semantics: Interpretations An interpretation I =  D,R,ext,V  consists of nonempty sets D and R, a function ext such that ext(r)   n <  D n and a valuation function V such that V(t)  D, for all terms t, V(P)  R, for all predicates P. Let I [x/e] =  D,R,ext,V , where V is just like V except that V (x) = e. I [x/e] is known as an x- variant of I.

Semantics: Truth Let I =  D,W,ext,V  be an interpretation. P(t 1,…,t n ) is true in I iff  V(t 1 ),…, V(t n )   V(P) Boolean cases ~A is true in I iff A is not true in I. (A  B) is true in I iff both A and B are. …  xA is true in I iff A is true in all x-variants of I  xA is true in I iff A is true in some x-variant of I

KIF: An Alternative Syntax The development of high-speed computers and associated knowledge engineering technologies led to the development of alternative first-order (or stronger) languages. The Knowledge Interchange Format (KIF) is perhaps the best known and most widely used.

A Standard First-Order KIF Lexicon Terms Individual constants Strings of unicode characters Individual variables Strings of unicode chars prefixed by “ ? ” Predicate constants Strings of unicode characters Boolean operators not, and, or, =>, Quantifiers forall, exists

A Standard First-order KIF Grammar If P is a predicate and t 1, …, t n are terms, then ( P t 1 … t n ) is an (atomic) formula. If A and B are formulas, so are (not A ), (and A B ), ( or A B ), (=> A B ), and ( A B ). If A is a formula and x any variable, then (forall ( x ) A ) and (exists ( x ) A ). Nothing else is a formula.

Common Structures and a Standard Logic Standard first-order languages and KIF languages are structurally identical. It would be untenable in a standard for logic to force users to adopt one particlar syntax over another. Hence, the Common Logic Standard specifies languages by means of an abstract syntax that can have many concrete instances.

A Simple Abstract Syntax for FOL A lexicon consists of 1) A denumerable set Var of individual variables 2) A countable set Con of individual constants 3) A nonempty set Pred of predicate constants Let App be a 1-1 function on Pred   { Con n : n <  }. Range ( App ) is the set AT of atomic formulas.

Formula Classes A formula class F (for a given lexicon) is a smallest class that includes AT and is closed under a set Op of operations Id, Neg, Disj, Cond, Bicond, ExQuant, and UnivQuant that satisfy the following conditions: Each operation in Op is 1-1 The ranges of the operation are pairwise disjoint and disjoint from AT. Id : TRM  TRM  F Neg : F  F Disj ( Conj, Cond, Bicond ): F  F  F ExQuant ( UnivQuant ) : Var  F  F Our two grammars above are both simply concrete instances of this abstract grammar.

“Every boy kissed a girl.” (forall (?x) (=> (Boy ?x) (exists (?y) (and (Girl ?y) (Kissed ?x ?y)))))  x(Boy(x)   y(Girl(y)  Kissed(x,y))) Abstractly: UnivQuant(v 1,Cond(App(Boy,v 1 ),ExQuant(v 2,Conj(App(Girl,v 2 ),App(Kissed,v 1,v 2 ))))) Our standard first-order semantics can be applied to our abstract syntax. The semantics is then simply inherited by any instance of the syntax.

Extensions of Standard FOL Full KIF and other logic-based KR languages include features that extend standard FOL. A logic standard should be flexible enough to incorporate such extensions as options.

Type Freedom The Central Intuition All entities – individuals, propositions, properties, and relations alike – are first-class logical citizens that jointly constitute a single domain of quantification. Hence, such entities can themselves have properties, stand in relations, and serve as potential objects of reference.

Nominalization The verb phrase ‘is famous’ is used to predicate fame of Quentin. “Quentin is famous.” Its nominalization ‘being famous’ is used to denote fame. “Being famous is all Quentin thinks about. Since the verb phrase and its nominalization have the semantic value, the same constant can be used to symbolize both sentences. (Famous quentin) (forall (?x) ( (ThinksAbout quentin ?x) (= ?x Famous)))

Self-exemplification Classes in most KR frameworks can be thought of as properties. Some classes are members of themselves The “root” class Entity or Thing in standard inheritance hierarchies. The classes Class and Property. Type-freedom enables one to express self- membership (as self-exemplification) (Entity Entity) (Class Class) (Property Property)

Paradoxes: Not to Worry Self-membership or self-exemplification is often cited as the culprit behind the paradoxes E.g., the Russell class {x | x  x} Type-freedom appears to open the door to such undesirable fellows. ( (Russell ?X) (not (?X ?X)) The problem lies no more with type-freedom than with negation. The real villain is the idea that we can postulate classes/properties/functions satisfying any given condition. This is known as the naïve comprehension principle in set theory.

Predicates as function symbols It is often convenient to use predicates as function symbols: (gardener (father-of cain)) (exists (?x) (and (father-of cain ?x) (gardener ?x))) But often the reverse is true: (father-of cain adam) (= adam (father-of cain)) CL syntax allows both usages, and its semantics provides the right interpretations in each context.

Sequence Variables Sequence variables go hand in glove with variable polyadicity. Range over all finite sequences over the domain of quantification. Increase expressive power to that of a sublanguage of the infinitary language L  1 . The model theory requires a proof theory with an infinitary rule of inference. Completeness likely but not proved. Clearly not compact (hence not first-order). { (not (exists ?x 1 (S ?x 1 )), …, (not (exists (?x 1 … ?x n (S ?x 1 … ?x n )), … (exists }

Translating into FOL Full CL languages without sequence variables can be thought of as notational variants of first- order theories. Introduce Pred n for each n P(t 1, …, t n ) goes to (Pred n P t1 … tn) (forall (?Op) (?Op ?x)) (forall (?Op) (R ?Op))

Translating Full CL into Infinitary Languages Full CL languages with sequence variables can be translated into (a sublanguage of) the infinitary language L  1  ((Nom Op) ?x) Rel(Nom(Op))