Common Logic: An Evolutionary Tale

Background Evolution Metatheory Beyond FOL

Common Logic:
An Evolutionary Tale

Christopher Menzel

Texas A&M University
Munich Center for Mathematical Philosophy
cmenzel@tamu.edu

PhiloWeb 2012
WWW2012, Lyon
17 April 2012

Common Logic: An Evolutionary Tale Christopher Menzel


Where We Are
1 Background
In Praise of “Traditional” First-order Logic
Open Networks
2 Evolution
Four Evolutionary Adaptations
Common Logic: The Next Evolutionary Step
3 Metatheory
A Complete Proof Theory
CL and TFOL
4 Beyond FOL
Sequence Markers
Final Reﬂections



Open Networks, Expressiveness, and Monotonicity

• Publishers need the intended meaning of their content to be
properly interpreted and retained by consumers
• Hence, just as they have adopted the HTML presentation
standard, 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 of
first-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, and
relations 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 universal
framework for formal representation and a standard (though
obviously not unique) platform for automated reasoning
• Notably, OWL is basically a class theory expressed in a
fragment of FOL
• Otter, Prover9, Tau, E-SETHEO, Vampire, Waldmeister, etc
are 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 out
of 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 individual
constants (den(κ ) ∈ D, for κ ∈ Cn)
• A set F of n-place functions over D serving as the denotation
of 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 is
a set of n + 1-tuples, r’s a set of n-tuples
• The adicity of a lexical item α ∈ Fn, π ∈ Pr must match the
arity 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 same
output
• 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 of
logic with the growth of the Semantic Web and the
corresponding need to represent natural language as
ﬂexibly as possible have led to a logic — Common Logic
— in which all of these syntactic and semantic features
ultimately disappear.



Entailment and Open Networks

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



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

‘



I: Variable Polyadicity
• The data: The number of arguments a predicate or function
symbol 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∗ )

1 D∗ = Dn , where D0 = { }, D1 = D, and Dn+1 = D × Dn , for n ≥ 1.
n∈N



II: Cross Categoricity: Function Symbols and Predicates

• Inﬂuenced by “frame-based” KR languages, traditional role of
many binary predicates can be subsumed by function symbols
• (TeacherOf Aristotle Plato)
• (= (TeacherOf Aristotle) Plato)

• 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 TFOL
extends 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)

• 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 are
married. 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 the
objects 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 be
intensional objects
• That is, they are not themselves extensions, rather they are
objects 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.”

• 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 traditional
lexical boundaries
• Recognizing this has led us to loosen the boundaries between
traditional syntactic and semantic categories
• Yet we retain them — leaving us with the complications in
question
• These boundaries are vestiges of our Fregean ontological
heritage!
• We have loosed our Fregean shackles — it is time we freed
ourselves from them altogether!



An Anarchic Ontology: Things

Three Principles
• There are things.
• Some things can be (truly) predicated of other
things.
• All things can have some things (truly)
predicated of them.



An Anarchic Syntax: Names

One (Non-logical) Lexical
Category
• Names



An Anarchic Syntax: Grammar

One (Basic) Grammatical
Rule
• Every name can be predicated of any number of
names



An Anarchic Semantics

Two (Basic) Semantic
Principles
• 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 unicode
text characters (i.e., no whitespace) other than the logical
operators
• The unicode SPACE character (U+0200)
• Parentheses: (, )

Deﬁnition
A 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, β 1 , ..., β n are terms of L (n ≥ 0), then the expression
(α β 1 ... β n ) 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 nonempty
set, 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 β 1 ... β n )) = efn (dV (α))(dV ( β 1 ), ..., dV ( β n )).
• (α β 1 ... β n ) is true in I iff dV ( β 1 ), ..., dV ( β n ) ∈ 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 I a .
ν

• 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 an
importation.
• 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 a
quantified sentence.
• A sentence may have an attached comment.
• A boolean sentence has a type, called a connective, and a
number 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, and
biconditional with, respectively, one, any number, any number,
two and two components.
• A quantified sentence has (i) a type, called a quantifier, (ii) a
finite, nonrepeating sequence of names called the binding
sequence, each element of which is called a binding of the
quantified sentence, and (iii) a sentence called the body of the
quantified 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 called
an 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 a
term sequence called the argument sequence.
• Parallel qualiﬁcations to atomic sentences.

• A term sequence is a (possibly null) ﬁnite sequence of terms or
sequence 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 tautologies
2 (if (∀ν ϕ) ϕν ), where α is free for ν in ϕ
α
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 L
6 (if (= ν µ) (if ϕ ϕν )), where µ is free for ν in ϕ
µ

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



+
Soundness of CL and CL

• Define the notion of an interpretation+ by adding semantic
conditions M and C
• Truth in an interpretation+ defined as above
+
• All derivative notions (satisfiability+ , model+ , validity+ , |=L ,
etc) defined accordingly
+
• Let CL be the resulting of adding schemas 7 and 8 to CL

+
Theorem (Soundness of CL and CL )
If Γ CL ϕ, then Γ |=L ϕ; and if Γ +
CL ϕ, then Γ |=L ϕ.
+



+
Completeness of CL and CL

+
Theorem (Completeness of CL and CL )
If Γ |=L ϕ, then Γ CL ϕ; and if Γ |=L ϕ, then Γ
+
+
CL ϕ.

Corollary (Löwenheim-Skolem)
If a set Γ of sentences of L has an L-model (L-model+ ), it has a
countable 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 L
Let L be a conventional CLIF language. The lexicon of a traditional counterpart L* of
L consists of the same logical operators not, if, and forall (written again as ∀) as
well 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 the
variables of L*.
Terms
• Individual constants and variables of L* together with those expressions of L* of
the form (Appn α β 1 ... β n ), for terms α, β 1 , ..., β n 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-one
correspondence from the set NL of names of L onto VarL* .
• For names ν ∈ NL , ν = ν
• For terms α, β 1 , ..., β n of L,
• (= β 1 β 2 )† = (= β 1 β 2 )
• (α β 1 ... β n ) = (Appn α β 1 ... β n )
• (α β 1 ... β n )† = (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† xν )
ν

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 (App2 f 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 (App2 f x a)) (Holds1 G x)))



Standard Translations are Meaning Preserving

Every L-interpretation I = D, efn , erel , V determines a unique
L*-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 into
a function V on of L* and NL and another W on the function symbols
and 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 it
yields 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 if
Γ† |=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 of
sentences of L* (i.e., formulas of L* in which no variables occur
free).

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

Lemma. If ϕ1 , ..., ϕn is a proof from Γ† in CL* , then ϕ1 , ..., ϕn is a
† †

proof 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 the
completeness 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-vis
signature-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?


Common Logic: An Evolutionary Tale

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