Common Logic: An Evolutionary Tale

Background Evolution Metatheory Beyond FOL

Common Logic:An Evolutionary Tale

Christopher Menzel

Texas A&M UniversityMunich Center for Mathematical Philosophy

[email protected]

PhiloWeb 2012WWW2012, Lyon17 April 2012

Common Logic: An Evolutionary Tale Christopher Menzel


Where We Are1 Background

In Praise of “Traditional” First-order LogicOpen Networks

2 EvolutionFour Evolutionary AdaptationsCommon Logic: The Next Evolutionary Step

3 MetatheoryA Complete Proof TheoryCL and TFOL

4 Beyond FOLSequence MarkersFinal Reflections



Open Networks, Expressiveness, and Monotonicity

• Publishers need the intended meaning of their content to beproperly interpreted and retained by consumers

• Hence, just as they have adopted the HTML presentationstandard, publishers must agree on a KR standard

• Requirements:• Clearly defined syntax and rigorous semantics• No constraints on (first-order) expressiveness• Meaning must be stable across contexts, i.e., monotonic• Logical consequence should be axiomatizable to support

automated reasoning (as far as possible)

• Points to the need for some sort of standardized version offirst-order logic



In Praise of “Traditional” FOL: Representation

• “Traditional” FOL — TFOL — is wonderfully expressive• As a rule if you can’t say it in TFOL, you can’t say it!

• The simplest reasons for this:• There are names for denoting things

• ‘PatHayes’, ‘NGC1976’, ‘ω’

• There are predicates for describing the properties of, andrelations among, things

• Curmudgeon(PatHayes), Nebula(NGC1976), ω < ω + 17

• There are quantifiers for expressing generality

• Nebulas exist — (∃x)Nebula(x)• If anyone is a curmudgeon, Hayes is —

(∀x)(Curmudgeon→ Curmudgeon(PatHayes))



In Praise of TFOL: Theory

• A simple, rigorous syntax• A clear, well-understood, monotonic semantics

• A.k.a., “Tarskian” model theory

• Semantically complete proof theory• Albeit only semi-decidable

• For these reasons, TFOL has become a virtually universalframework for formal representation and a standard (thoughobviously not unique) platform for automated reasoning

• Notably, OWL is basically a class theory expressed in afragment of FOL

• Otter, Prover9, Tau, E-SETHEO, Vampire, Waldmeister, etcare all first-order theorem provers



TFOL’s Fregean Heritage• TFOL is typically traced back to Frege

• Yes, and Peirce and others...

• Frege’s semantical and metaphysical views in many ways outof favor

• Notably, the inviolable divide between concept and object• A.k.a., between the meanings of predicates and names

• TFOL generalizes these divisions• Segregates objects from functions from n-place relations• Segregates functions and relations internally according to arity• Reflects these divisions in its syntax

• These divisions represent a significant — and questionable —metaphysical viewpoint

• And, in the context of the Web, an untenable syntactic rigidity



Features of TFOL: Syntax

• A tripartite lexicon• A set Con of individual constants• A set Fn of function symbols, for n ∈N

• A set Pr of predicates, for n ∈N

• Fixed signatures• Every α ∈ Fn has a fixed adicity n, i.e., α can only be applied

to exactly n arguments• Every n-place π ∈ Pr has a fixed adicity n, i.e., π can only be

predicated of n arguments

• Strict syntactic typing• No self-application α(α, β) or self-predication π(π)• Individual constants cannot be applied or predicated

• No function symbol or predicate quantifiers



Features of TFOL: Semantics• A tripartite ontology

• A set D of individuals serving as the denotations of individualconstants (den(κ) ∈ D, for κ ∈ Cn)

• A set F of n-place functions over D serving as the denotationof n-place function symbols (fext(α) ∈ F, for α ∈ Fn)

• A set R of relations over D (rext(π) ∈ R, for π ∈ Pr)• Fixed arities

• Every f ∈ F and r ∈ R has a fixed arity n, i.e., f ’s extension isa set of n + 1-tuples, r’s a set of n-tuples

• The adicity of a lexical item α ∈ Fn, π ∈ Pr must match thearity of its semantic value fext(α), rext(π)

• Strict semantic typing• No function or relation a constituent of its own extension• Individuals cannot be functionally applied or exemplified

• Functions and relations not in the range of any quantifiers



Features of TFOL: Additional Semantic Features

• Extensionality• Functions and relations understood extensionally

• Functions identical if they map the same input to the sameoutput

• Relations identical if they are true of the same (n-tuples of)objects

• Typically assured by defining them as sets

• Variable assignments• Variables are assigned individuals relative to a fixed

interpretation for the lexicon• Truth is defined in terms of variable assignments.



Features of TFOL: Semantics



Features of TFOL: Fate

Evolutionary adaptations springing from the interaction oflogic with the growth of the Semantic Web and thecorresponding need to represent natural language asflexibly as possible have led to a logic — Common Logic— in which all of these syntactic and semantic featuresultimately disappear.



Entailment and Open Networks

• To illustrate• Entailment should commute with communication...



• ...but the open milieu of the Web raises challenges that alanguage in the “traditional” mold (e.g., KIF) may not be ableto deal with:

‘










I: Variable Polyadicity

• The data: The number of arguments a predicate or functionsymbol can take can vary from context to context.

• (Teacher Plato)• (Teacher Plato Aristotle)• (Teacher Plato Aristotle 364-360BCE)

• Syntactic change:• Eliminate fixed adicity constraint on Fn and Pr

• Semantic change:• Eliminate fixed arity constraint on F and R

• For function symbols α, fext(α) ∈ {f : f : D∗ −→ D}1

• For predicates π, rext(π) ∈ ℘(D∗)

1D∗ =⋃

n∈N

Dn, where D0 = {〈〉}, D1 = D, and Dn+1 = D×Dn, for n ≥ 1.



II: Cross Categoricity: Function Symbols and Predicates

• Influenced by “frame-based” KR languages, traditional role ofmany binary predicates can be subsumed by function symbols

• (TeacherOf Aristotle Plato)• (= (TeacherOf Aristotle) Plato)

• Syntactic change:• Remove disjointness condition on Fn and Pr

• Semantic consequence:• β ∈ Fn∩ Pr assigned both a function fext(β) and relation

rext(β)

• Semantic change (optional; can be enforced axiomatically)• For β ∈ Fn∩ Pr , require, e.g., fext(β) ⊆ rext(β)



III: Complete Cross-categoricity: “Objectified” Relations

• The breakdown of inviolable lexical boundaries of TFOLextends to terms

• Relations often treated both as predicables and as logical“first-class citizens” in KR contexts (e.g., in DLs)

• (TeacherOf Aristotle Plato)• (ConverseOf TeacherOf StudentOf)• Second-order treatment leads to ramification

• (Binary TeacherOf),(Binary ConverseOf)

• Syntactic change:• Remove all disjointness conditions on Con, Fn, and Pr

• Semantic consequence:• Constants γ that are also function symbols or predicates given

a denotation in D as well as a function and/or relation



III: Complete Cross-categoricity: Identity• Nominalization also motivates complete cross-categoricity

• “Whenever Bo is running, he hates it (i.e., running).”• (∀t (if (time t) ((running Bo t) (hates Bo running t)))

• “Being married is the same as being hitched.”

• PROBLEM: Consider the following intuitive argument:

Being married is the same as being hitched. Jo and Bo aremarried. Therefore, Jo and Bo are hitched.(= married hitched), (married Jo Bo) ∴ (hitched Jo Bo)

• Invalid under our current revisions• For constants β that are predicates, there is no coordination

between denotation den(β) and relational extension rext(β)• Hence: no guarantee that den(married) = den(hitched)

implies rext(married) = rext(hitched)



III: Complete Cross-categoricity: Denotation and Extension• Semantic Change:

• For constants β that are preds, require den(β) = rext(β)• Likewise for constants that are function symbols

• This puts extensional relations — sets of objects — among theobjects in the domain

• A radical change!• Requires non-well-founded set theory:

• If a constant β is also a predicate, (β β) is well-formed• (β β) is true iff den(β) ∈ rext(β)• But den(β) = rext(β); hence, (β β) is true iff

rext(β) ∈ rext(β).

• Raises the specter of paradox...• By Cantor’s Theorem, D is smaller than ℘(D)• So D can’t accommodate all possible extensional relations

over D



IV: Type-free Intensionality: Objects

• A better solution: Take functions and relations to beintensional objects

• That is, they are not themselves extensions, rather they areobjects in D that have extensions

• Semantic change:• F and R are now subsets of D• fext : F −→ {f | f : D∗ −→ D}• rext : R −→ ℘(D∗)• den : Cn∪ Fn∪ Pr −→ D such that

• den(α) ∈ F, for α ∈ Fn• den � Pr(π) ∈ R, for π ∈ Pr

• (r (f a) b) is true iff 〈fext(f)(den(a)), den(b)〉 ∈ rext(den(r))‘



IV: Type-free Intensionality: Quantification

• From

(∀t (if (time t) (if (running Bo t) (hates Bo running t))))

• we can infer only

(∃x (∀t (if (time t) (if (running Bo t) (hates Bo x t)))))

“There is something that Bo hates whenever he is running.”

• But clearly, that is not all that follows. We also get

“There is something that Bo hates whenever he is doing it.”

• Syntactic change:

• Variables can occur in function and predicate position

(∃R (∀t (if (time t) (if (R Bo t) (hates Bo R t)))))



Taking Stock

• The web is anarchic• One does not find, nor can one expect, authors of logical KBs,and even logical KR languages, to comply with traditionallexical boundaries

• Recognizing this has led us to loosen the boundaries betweentraditional syntactic and semantic categories

• Yet we retain them — leaving us with the complications inquestion

• These boundaries are vestiges of our Fregean ontologicalheritage!

• We have loosed our Fregean shackles — it is time we freedourselves from them altogether!



An Anarchic Ontology: Things

Three Principles

• There are things.• Some things can be (truly) predicated of otherthings.

• All things can have some things (truly)predicated of them.



An Anarchic Syntax: Names

One (Non-logical) LexicalCategory

• Names



An Anarchic Syntax: Grammar

One (Basic) GrammaticalRule

• Every name can be predicated of any number ofnames



An Anarchic Semantics

Two (Basic) SemanticPrinciples

• Names name things• Names can be true of things



Syntax: Lexicon of a CLIF Language

A CLIF language consists of the following lexical items:• Logical operators: if, not, forall• Identity: =• Names: A denumerable set NL of nonempty strings of unicodetext characters (i.e., no whitespace) other than the logicaloperators

• The unicode SPACE character (U+0200)• Parentheses: (, )

DefinitionA CLIF language L is inclusive if it includes the identity symbol ‘=′

among its names. L is conventional if it does not.



Syntax: Grammar

Let L be an arbitrary CLIF language.

1 Every name of L is a term of L.2 If α, β1, ..., βn are terms of L (n ≥ 0), then the expression(α β1 ... βn) is both a term and a sentence of L.

– If L is conventional and β is a term of L, then the expression(= α β) is a sentence of L.

3 If ϕ is a sentence of L, so is (not ϕ).4 If ϕ and ψ are sentence of L, so is (if ϕ ψ).5 If ϕ is a sentence of L and ν ∈ NL, then (forall (ν) ϕ) is

a sentence of L ((∀νϕ), for short).6 Nothing else is a term or sentence of L.



Features of the Syntax

• Type freedom• There are only logical operators and names in the lexicon• Traditional lexical categories — Cn, Fn, Pr — are simply

contextual roles that any name can play• Self-predication and self-application are legit

• (Abstract Abstract), (P (f f) a), etc.

• Signature freedom• There is no specification of adicity• Same name be predicated of any finite number of arguments

• Including 0: (P) is a 0-place atomic formula• (P), (P P), (P (P P) P), (P (P P) (P P (P P) P), ...

• “Higher-order” quantification permitted• (∃R (∀c (iff (R c) (not (c c)))))



Semantics: L-interpretations and Truth

An L-interpretation I is a 4-tuple 〈D, efn, erel, V〉, where D is a nonemptyset, efn : D −→ {f | f : D∗ −→ D}, erel : D −→ ℘(D∗), V : N −→ D,and if L is inclusive, erel(V(=)) = {〈a, a〉 : a ∈ D}.

Denotation and Truth• For names ν of L, dV(ν) = V(ν).• dV((α β1 ... βn)) = efn(dV(α))(dV(β1), ..., dV(βn)).• (α β1 ... βn) is true in I iff 〈dV(β1), ..., dV(βn)〉 ∈ erel(dV(α)).

• If L is conventional, (= α β) is true in I iff dV(α) = dV(β).

• (not ϕ) is true in I iff ϕ is not true in I .• (if ϕ ψ) is true in I iff either ϕ is not true in I or ψ is true in I .• (∀ν ϕ) is true in I iff, for all a ∈ D, ϕ is true in Iν

a .

• Satifiability, validity, logical consequence (|=L) defined as usual



Recall: Semantics of TFOL



Semantics: CL Model Theory



Abstract Syntax: Web Sensitive Features

• A text is either a set or list or bag of phrases.• A piece of text may be identified by a name.

• A phrase is either a comment, a module, a sentence, or animportation.

• A comment is a piece of data.• No particular restrictions are placed on comments.• Comments can be attached to other comments.

• A module consists of a name and a text called the body text.• The module name indicates the local domain of discourse in

which the text is to be understood

• An importation contains a name. (More below)



Abstract Syntax: Representational Features• A sentence is either an atom, a boolean sentence, or aquantified sentence.

• A sentence may have an attached comment.

• A boolean sentence has a type, called a connective, and anumber of sentences, called the components of the sentence.

• The number depends on the type.• Every CL dialect must distinguish the following types:

negation, conjunction, disjunction, conditional, andbiconditional with, respectively, one, any number, any number,two and two components.

• A quantified sentence has (i) a type, called a quantifier, (ii) afinite, nonrepeating sequence of names called the bindingsequence, each element of which is called a binding of thequantified sentence, and (iii) a sentence called the body of thequantified sentence.



• An atom is either an equation containing two arguments,which are terms, or an atomic sentence.

• An atomic sentence consists of a term, called the predicate,and a term sequence called the argument sequence.

• Each term in the term sequence of an atomic sentence is calledan argument of the sentence.

• Any name can be the predicate in an atomic sentence.

• A term is either a name or a functional term.• Terms may have attached comments.

• A functional term consists of a term, called the operator and aterm sequence called the argument sequence.

• Parallel qualifications to atomic sentences.

• A term sequence is a (possibly null) finite sequence of terms orsequence markers.



Features of the Abstract Syntax

• Abstraction!• No specification of any concrete syntactic forms• Specific form left to the KR designers.

• A given KR language needn’t use all the features of CL

• E.g., Description Logics lacking negation• Conformance defined flexibly enough to allow a side range of

CL dialects, including “traditional” first-order languages

• “Every cloud has a silver lining” in PM-ese, CGs, and KIF• ∀x(Cloud(x)→ ∃y(Lining(y) ∧ Silver(y) ∧Has(x, y)))• [@every*x] [If: (Cloud ?x) [Then: [*y] (Lining ?y) (Silver ?y) (Has ?x ?y)]]• (forall (?x ?y)

(if (Cloud ?x)(exists (?y)

(and (Lining ?y) (Silver ?y) (Has ?x ?y)))))










Proof Theory: The System CL

Any generalization of any of the following is an axiom of CL:1 Propositional tautologies2 (if (∀ν ϕ) ϕν

α), where α is free for ν in ϕ

3 (if (∀ν (if ϕ ψ)) (if (∀ν ϕ) (∀ν ψ)))

4 (if ϕ (∀ν ϕ)), where ν does not occur free in ϕ

5 (= ν ν), for any name ν of L6 (if (= ν µ) (if ϕ ϕν

µ)), where µ is free for ν in ϕ

The system CL has one rule of inference:• Modus Ponens (MP): From ϕ and (if ϕ ψ), infer ψ.



Soundness of CL and C+L

• Define the notion of an interpretation+ by adding semanticconditions M and C

• Truth in an interpretation+ defined as above• All derivative notions (satisfiability+, model+, validity+, |=+

L ,etc) defined accordingly

• Let C+L be the resulting of adding schemas 7 and 8 to CL

Theorem (Soundness of CL and C+L )

If Γ `CL ϕ, then Γ |=L ϕ; and if Γ `C+L

ϕ, then Γ |=+L ϕ.



Completeness of CL and C+L

Theorem (Completeness of CL and C+L )

If Γ |=L ϕ, then Γ `CL ϕ; and if Γ |=+L ϕ, then Γ `C+

Lϕ.

Corollary (Löwenheim-Skolem)

If a set Γ of sentences of L has an L-model (L-model+), it has acountable L-model (L-model+).

Corollary (Compactness)

If every finite subset of a set Γ of sentences of L has an L-model(L-model+), then Γ has a model (model+).



The Traditional Counterpart of LLet L be a conventional CLIF language. The lexicon of a traditional counterpart L* ofL consists of the same logical operators not, if, and forall (written again as ∀) aswell as the following:

• The set NL of names of L, which are known as the individual constants of L*.• For every n ∈N, an n + 1-place predicate Holdsn

• For every n ∈N, an n + 1-place function symbol Appn.• A denumerable set VarL* of names (in the sense above) disjoint from NL and

not containing the predicates and function symbols above. These are thevariables of L*.

Terms

• Individual constants and variables of L* together with those expressions of L* ofthe form (Appn α β1 ... βn), for terms α, β1, ..., βn of L*.

Formulas

• Those expressions of the form (Holdsn α β1 ... βn) for terms α, β1, ..., βn of L*• For formulas ϕ, ψ of L*, those expressions of the form (not ϕ), (if ϕ ψ), and

(forall (χ) ϕ) ((∀χ ϕ)), for variables χ of L*.



Standard Translations

Let L* be a traditional counterpart of L. Let x be a fixed one-to-onecorrespondence from the set NL of names of L onto VarL*.• For names ν ∈ NL, ν′ = ν

• For terms α, β1, ..., βn of L,

• (= β1 β2)† = (= β′1 β′2)

• (α β1 ... βn)′ = (Appn α′ β′1 ... β′n)• (α β1 ... βn)† = (Holdsn α′ β′1 ... β′n)

• For sentences ϕ,ψ of L and ν ∈ NL,

• (not ϕ)† = (not ϕ†)• (if ϕ ψ)† = (if ϕ† ψ†)• (∀ν ϕ)† = (∀xν ϕ† ν

xν)

Call the pair 〈′, †〉 of functions a standard translation of L into L*.



Standard Translations: Examples

• (Married Bill Hillary)′ = (Holds2 Married Bill Hillary)

• (not (F (f a b))))′ = (not (Holds1 F (App2f a b)))

• (if (F a b) (not (G a)))′ =(if (Holds2 F a b) (not (Holds1 G a))))

• (∀x (if (F (f x a)) (G x)))′ =(∀x (if (Holds2 F (App2f x a)) (Holds1G x)))



Standard Translations are Meaning Preserving

Every L-interpretation I = 〈D, efn, erel, V〉 determines a uniqueL*-interpretation I* = 〈D, V ∪WI 〉 where:• WI (Appn) =

⋃ {{a} × (efn(a) � Dn) : a ∈ D

}• WI (Holdsn) =

⋃{{a} × (erel(a) ∩Dn) : a ∈ D}.

Every L*-interpretation is so determined by some (unique)L-interpretation. For if L* interpretation J = 〈D, U〉, U can be split intoa function V on of L* and NL and another W on the function symbolsand predicates of L*. Then let:

• efn =⋃{W(Appn) : n ∈N}

• erel =⋃{W(Holdsn : n ∈N}.

It is easy to check that 〈D, efn, erel, V〉 is an L-interpretation and that ityields J under the above mapping.



Standard Translations are Meaning Preserving

Theorem. For sentences ϕ and interpretations I = 〈D, erel, efn, V〉of L, ϕ is true in I iff ϕ† is true in I*=〈D, V ∪WI 〉.

Corollary 1. For sentences ϕ of L, Γ |=L ϕ if and only ifΓ† |=L* ϕ†.



Completeness via TFOL

Fact. For any sentence ψ of L* and any set Σ of sentences of L*,if Σ `CL* ψ, then there is a proof of ψ from Σ consisting entirely ofsentences of L* (i.e., formulas of L* in which no variables occurfree).

Lemma. If ψ1, ..., ψn is a proof in CL* of ϕ† from Γ†, then thereare sentences ϕ1, , ..., ϕn of L such that ϕ†

1, , ..., ϕ†n is a proof of

ϕ† from Γ† in CL∗ .

Lemma. If ϕ†1, ..., ϕ†

n is a proof from Γ† in CL*, then ϕ1, ..., ϕn is aproof from Γ in CL.

Corollary 2. If Γ† `CL* ϕ†, then Γ `CL ϕ.



Completeness via TFOL

Theorem (Completeness of CL via TFOL)

If Γ |=L ϕ, then Γ `CL ϕ.

Proof. If Γ |=L ϕ, then by Corollary 1, Γ† |=L* ϕ†. Hence, by thecompleteness of CL*, we have Γ† `CL* ϕ† and thus, by Corollary 2,Γ `CL ϕ.










Beyond First-order: Sequence Markers

• Sequence markers are a natural mechanism vis-à-vissignature-freedom

• But: They push CL beyond FOL in expressiveness• Chaining

• (forall (F x) ((Chain F) x))(forall (F x y)

(iff ((Chain F) ... x y)(and (F x y) ((Chain F) ... x)))))

• (= AscendingOrder (Chain LessThan))• (AscendingOrder 2 5 17 25)

• Axioms for Relations• (iff (Unary F)

(and (not (F))(not (exists (... x y) (F ... x y)))))



Sequence Markers: Chained Identity and Difference

• Chained Identity(AllEq x)

(iff (AllEq x y ...)(and (= x y) (AllEq y ...)))

• Chained Difference(iff (AllDiff x)) (Comment "a.k.a. ‘NoRepeats’")

(iff (AllDiff x y ...)(and (not (= x y))

(AllDiff x ...)(AllDiff y ...)))



Sequence Markers: Finitude

• SeqOf((seqOf F)) (Comment "Holds only of seqs of Fs")

(iff ((seqOf F) x ...) (and ((seqOf F) ...) (F x))

• Finitude of properties(iff (Finite F)

(and (Unary F)(exists (...)

(and ((seqOf F) ...)(AllDiff ...)(forall (x)

(if (F x) (not (AllDiff x ...))))))))



Final Reflections

• Given the Holds/App translation, why not just use TFOL?• The Holds/App translation is ontologically artificial

• Schizophrenic regarding relations

• Automated reasoning tools built for TFOL

• But can still use them via translators

• Horrocks sentences – deep or superficial?• The following is a logical truth of CLIF

(if (x (iff (F x) (not (G x)))) (∃x∃y (not (= x y))))

• This form is not a logical truth of TFOL• Theoretically innocuous but user-unfriendly?


Technology

Common Logic: An Evolutionary Tale