The document discusses the evolution of first-order logic (FOL) to accommodate knowledge representation on the open web. It describes four adaptations: 1) Removing fixed arities of functions and predicates, 2) Allowing functions to also be predicates, 3) Allowing terms to be predicates and functions, establishing identity between denotations and extensions, and 4) Treating functions and relations as intensional objects through type-free quantification over all objects. These changes abandon the rigid syntactic and semantic categories of FOL for a more anarchic framework capable of representing knowledge on the open web.
1. 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
2. Background Evolution Metatheory Beyond FOL
Where We Are
1 Background
In Praise of āTraditionalā 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 Reļ¬ections
Common Logic: An Evolutionary Tale Christopher Menzel
3. Background Evolution Metatheory Beyond FOL
Open Networks, Expressiveness, and Monotonicity
ā¢ Publishers need the intended meaning of their content to be
properly interpreted and retained by consumers
ā¢ Hence, just as they have adopted the HTML presentation
standard, publishers must agree on a KR standard
ā¢ Requirements:
ā¢ Clearly deļ¬ned syntax and rigorous semantics
ā¢ No constraints on (ļ¬rst-order) expressiveness
ā¢ Meaning must be stable across contexts, i.e., monotonic
ā¢ Logical consequence should be axiomatizable to support
automated reasoning (as far as possible)
ā¢ Points to the need for some sort of standardized version of
ļ¬rst-order logic
Common Logic: An Evolutionary Tale Christopher Menzel
4. Background Evolution Metatheory Beyond FOL
In Praise of āTraditionalā FOL: Representation
ā¢ āTraditionalā FOL ā TFOL ā is wonderfully expressive
ā¢ As a rule if you canāt say it in TFOL, you canāt say it!
ā¢ The simplest reasons for this:
ā¢ There are names for denoting things
ā¢ āPatHayesā, āNGC1976ā, āĻā
ā¢ There are predicates for describing the properties of, and
relations among, things
ā¢ Curmudgeon(PatHayes), Nebula(NGC1976), Ļ < Ļ + 17
ā¢ There are quantiļ¬ers for expressing generality
ā¢ Nebulas exist ā (āx)Nebula(x)
ā¢ If anyone is a curmudgeon, Hayes is ā
(āx)(Curmudgeon ā Curmudgeon(PatHayes))
Common Logic: An Evolutionary Tale Christopher Menzel
5. Background Evolution Metatheory Beyond FOL
In Praise of TFOL: Theory
ā¢ A simple, rigorous syntax
ā¢ A clear, well-understood, monotonic semantics
ā¢ A.k.a., āTarskianā model theory
ā¢ Semantically complete proof theory
ā¢ Albeit only semi-decidable
ā¢ For these reasons, TFOL has become a virtually universal
framework for formal representation and a standard (though
obviously not unique) platform for automated reasoning
ā¢ Notably, OWL is basically a class theory expressed in a
fragment of FOL
ā¢ Otter, Prover9, Tau, E-SETHEO, Vampire, Waldmeister, etc
are all ļ¬rst-order theorem provers
Common Logic: An Evolutionary Tale Christopher Menzel
6. Background Evolution Metatheory Beyond FOL
TFOLās Fregean Heritage
ā¢ TFOL is typically traced back to Frege
ā¢ Yes, and Peirce and others...
ā¢ Fregeās semantical and metaphysical views in many ways out
of favor
ā¢ Notably, the inviolable divide between concept and object
ā¢ A.k.a., between the meanings of predicates and names
ā¢ TFOL generalizes these divisions
ā¢ Segregates objects from functions from n-place relations
ā¢ Segregates functions and relations internally according to arity
ā¢ Reļ¬ects these divisions in its syntax
ā¢ These divisions represent a signiļ¬cant ā and questionable ā
metaphysical viewpoint
ā¢ And, in the context of the Web, an untenable syntactic rigidity
Common Logic: An Evolutionary Tale Christopher Menzel
7. Background Evolution Metatheory Beyond FOL
Features of TFOL: Syntax
ā¢ A tripartite lexicon
ā¢ A set Con of individual constants
ā¢ A set Fn of function symbols, for n ā N
ā¢ A set Pr of predicates, for n ā N
ā¢ Fixed signatures
ā¢ Every Ī± ā Fn has a ļ¬xed adicity n, i.e., Ī± can only be applied
to exactly n arguments
ā¢ Every n-place Ļ ā Pr has a ļ¬xed adicity n, i.e., Ļ can only be
predicated of n arguments
ā¢ Strict syntactic typing
ā¢ No self-application Ī±(Ī±, Ī²) or self-predication Ļ (Ļ )
ā¢ Individual constants cannot be applied or predicated
ā¢ No function symbol or predicate quantiļ¬ers
Common Logic: An Evolutionary Tale Christopher Menzel
8. Background Evolution Metatheory Beyond FOL
Features of TFOL: Semantics
ā¢ A tripartite ontology
ā¢ A set D of individuals serving as the denotations of individual
constants (den(Īŗ ) ā D, for Īŗ ā Cn)
ā¢ A set F of n-place functions over D serving as the denotation
of n-place function symbols (fext(Ī±) ā F, for Ī± ā Fn)
ā¢ A set R of relations over D (rext(Ļ ) ā R, for Ļ ā Pr)
ā¢ Fixed arities
ā¢ Every f ā F and r ā R has a ļ¬xed arity n, i.e., f ās extension is
a set of n + 1-tuples, rās a set of n-tuples
ā¢ The adicity of a lexical item Ī± ā Fn, Ļ ā Pr must match the
arity of its semantic value fext(Ī±), rext(Ļ )
ā¢ Strict semantic typing
ā¢ No function or relation a constituent of its own extension
ā¢ Individuals cannot be functionally applied or exempliļ¬ed
ā¢ Functions and relations not in the range of any quantiļ¬ers
Common Logic: An Evolutionary Tale Christopher Menzel
9. Background Evolution Metatheory Beyond FOL
Features of TFOL: Additional Semantic Features
ā¢ Extensionality
ā¢ Functions and relations understood extensionally
ā¢ Functions identical if they map the same input to the same
output
ā¢ Relations identical if they are true of the same (n-tuples of)
objects
ā¢ Typically assured by deļ¬ning them as sets
ā¢ Variable assignments
ā¢ Variables are assigned individuals relative to a ļ¬xed
interpretation for the lexicon
ā¢ Truth is deļ¬ned in terms of variable assignments.
Common Logic: An Evolutionary Tale Christopher Menzel
10. Background Evolution Metatheory Beyond FOL
Features of TFOL: Semantics
Common Logic: An Evolutionary Tale Christopher Menzel
11. Background Evolution Metatheory Beyond FOL
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
ļ¬exibly as possible have led to a logic ā Common Logic
ā in which all of these syntactic and semantic features
ultimately disappear.
Common Logic: An Evolutionary Tale Christopher Menzel
12. Background Evolution Metatheory Beyond FOL
Entailment and Open Networks
ā¢ To illustrate
ā¢ Entailment should commute with communication...
Common Logic: An Evolutionary Tale Christopher Menzel
13. Background Evolution Metatheory Beyond FOL
ā¢ ...but the open milieu of the Web raises challenges that a
language in the ātraditionalā mold (e.g., KIF) may not be able
to deal with:
ā
Common Logic: An Evolutionary Tale Christopher Menzel
14. Background Evolution Metatheory Beyond FOL
Where We Are
1 Background
In Praise of āTraditionalā 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 Reļ¬ections
Common Logic: An Evolutionary Tale Christopher Menzel
15. Background Evolution Metatheory Beyond FOL
I: Variable Polyadicity
ā¢ The data: The number of arguments a predicate or function
symbol can take can vary from context to context.
ā¢ (Teacher Plato)
ā¢ (Teacher Plato Aristotle)
ā¢ (Teacher Plato Aristotle 364-360BCE)
ā¢ Syntactic change:
ā¢ Eliminate ļ¬xed adicity constraint on Fn and Pr
ā¢ Semantic change:
ā¢ Eliminate ļ¬xed arity constraint on F and R
ā¢ For function symbols Ī±, fext(Ī±) ā {f : f : Dā āā D}1
ā¢ For predicates Ļ, rext(Ļ ) ā ā(Dā )
1 Dā = Dn , where D0 = { }, D1 = D, and Dn+1 = D Ć Dn , for n ā„ 1.
nāN
Common Logic: An Evolutionary Tale Christopher Menzel
17. Background Evolution Metatheory Beyond FOL
III: Complete Cross-categoricity: āObjectiļ¬edā Relations
ā¢ The breakdown of inviolable lexical boundaries of TFOL
extends to terms
ā¢ Relations often treated both as predicables and as logical
āļ¬rst-class citizensā in KR contexts (e.g., in DLs)
ā¢ (TeacherOf Aristotle Plato)
ā¢ (ConverseOf TeacherOf StudentOf)
ā¢ Second-order treatment leads to ramiļ¬cation
ā¢ (Binary TeacherOf),(Binary ConverseOf)
ā¢ Syntactic change:
ā¢ Remove all disjointness conditions on Con, Fn, and Pr
ā¢ Semantic consequence:
ā¢ Constants Ī³ that are also function symbols or predicates given
a denotation in D as well as a function and/or relation
Common Logic: An Evolutionary Tale Christopher Menzel
18. Background Evolution Metatheory Beyond FOL
III: Complete Cross-categoricity: Identity
ā¢ Nominalization also motivates complete cross-categoricity
ā¢ āWhenever Bo is running, he hates it (i.e., running).ā
ā¢ (āt (if (time t) ((running Bo t) (hates Bo running t)))
ā¢ āBeing married is the same as being hitched.ā
ā¢ 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) ā“ (hitched Jo Bo)
ā¢ Invalid under our current revisions
ā¢ For constants Ī² that are predicates, there is no coordination
between denotation den( Ī²) and relational extension rext( Ī²)
ā¢ Hence: no guarantee that den(married) = den(hitched)
implies rext(married) = rext(hitched)
Common Logic: An Evolutionary Tale Christopher Menzel
19. Background Evolution Metatheory Beyond FOL
III: Complete Cross-categoricity: Denotation and Extension
ā¢ Semantic Change:
ā¢ For constants Ī² that are preds, require den( Ī²) = rext( Ī²)
ā¢ Likewise for constants that are function symbols
ā¢ This puts extensional relations ā sets of objects ā among the
objects in the domain
ā¢ A radical change!
ā¢ Requires non-well-founded set theory:
ā¢ If a constant Ī² is also a predicate, (Ī² Ī²) is well-formed
ā¢ (Ī² Ī²) is true iļ¬ den( Ī²) ā rext( Ī²)
ā¢ But den( Ī²) = rext( Ī²); hence, (Ī² Ī²) is true iļ¬
rext( Ī²) ā rext( Ī²).
ā¢ Raises the specter of paradox...
ā¢ By Cantorās Theorem, D is smaller than ā(D)
ā¢ So D canāt accommodate all possible extensional relations
over D
Common Logic: An Evolutionary Tale Christopher Menzel
20. Background Evolution Metatheory Beyond FOL
IV: Type-free Intensionality: Objects
ā¢ A better solution: Take functions and relations to be
intensional objects
ā¢ That is, they are not themselves extensions, rather they are
objects in D that have extensions
ā¢ Semantic change:
ā¢ F and R are now subsets of D
ā¢ fext : F āā {f | f : Dā āā D}
ā¢ rext : R āā ā(Dā )
ā¢ den : Cn āŖ Fn āŖ Pr āā D such that
ā¢ den(Ī±) ā F, for Ī± ā Fn
ā¢ den Pr(Ļ ) ā R, for Ļ ā Pr
ā¢ (r (f a) b) is true iļ¬ fext(f)(den(a)), den(b) ā rext(den(r))ā
Common Logic: An Evolutionary Tale Christopher Menzel
21. Background Evolution Metatheory Beyond FOL
IV: Type-free Intensionality: Quantiļ¬cation
ā¢ From
(āt (if (time t) (if (running Bo t) (hates Bo running t))))
ā¢ we can infer only
(āx (āt (if (time t) (if (running Bo t) (hates Bo x t)))))
āThere is something that Bo hates whenever he is running.ā
ā¢ But clearly, that is not all that follows. We also get
āThere is something that Bo hates whenever he is doing it.ā
ā¢ Syntactic change:
ā¢ Variables can occur in function and predicate position
(āR (āt (if (time t) (if (R Bo t) (hates Bo R t)))))
Common Logic: An Evolutionary Tale Christopher Menzel
22. Background Evolution Metatheory Beyond FOL
Taking Stock
ā¢ The web is anarchic
ā¢ One does not ļ¬nd, nor can one expect, authors of logical KBs,
and even logical KR languages, to comply with traditional
lexical boundaries
ā¢ Recognizing this has led us to loosen the boundaries between
traditional syntactic and semantic categories
ā¢ Yet we retain them ā leaving us with the complications in
question
ā¢ These boundaries are vestiges of our Fregean ontological
heritage!
ā¢ We have loosed our Fregean shackles ā it is time we freed
ourselves from them altogether!
Common Logic: An Evolutionary Tale Christopher Menzel
23. Background Evolution Metatheory Beyond FOL
An Anarchic Ontology: Things
Three Principles
ā¢ There are things.
ā¢ Some things can be (truly) predicated of other
things.
ā¢ All things can have some things (truly)
predicated of them.
Common Logic: An Evolutionary Tale Christopher Menzel
24. Background Evolution Metatheory Beyond FOL
An Anarchic Syntax: Names
One (Non-logical) Lexical
Category
ā¢ Names
Common Logic: An Evolutionary Tale Christopher Menzel
25. Background Evolution Metatheory Beyond FOL
An Anarchic Syntax: Grammar
One (Basic) Grammatical
Rule
ā¢ Every name can be predicated of any number of
names
Common Logic: An Evolutionary Tale Christopher Menzel
26. Background Evolution Metatheory Beyond FOL
An Anarchic Semantics
Two (Basic) Semantic
Principles
ā¢ Names name things
ā¢ Names can be true of things
Common Logic: An Evolutionary Tale Christopher Menzel
27. Background Evolution Metatheory Beyond FOL
Syntax: Lexicon of a CLIF Language
A CLIF language consists of the following lexical items:
ā¢ Logical operators: if, not, forall
ā¢ Identity: =
ā¢ Names: A denumerable set NL of nonempty strings of unicode
text characters (i.e., no whitespace) other than the logical
operators
ā¢ The unicode SPACE character (U+0200)
ā¢ Parentheses: (, )
Deļ¬nition
A CLIF language L is inclusive if it includes the identity symbol ā=
among its names. L is conventional if it does not.
Common Logic: An Evolutionary Tale Christopher Menzel
28. Background Evolution Metatheory Beyond FOL
Syntax: Grammar
Let L be an arbitrary CLIF language.
1 Every name of L is a term of L.
2 If Ī±, Ī² 1 , ..., Ī² n are terms of L (n ā„ 0), then the expression
(Ī± Ī² 1 ... Ī² n ) is both a term and a sentence of L.
ā If L is conventional and Ī² is a term of L, then the expression
(= Ī± Ī²) is a sentence of L.
3 If Ļ is a sentence of L, so is (not Ļ).
4 If Ļ and Ļ are sentence of L, so is (if Ļ Ļ).
5 If Ļ is a sentence of L and Ī½ ā NL , then (forall (Ī½) Ļ) is
a sentence of L ((āĪ½Ļ), for short).
6 Nothing else is a term or sentence of L.
Common Logic: An Evolutionary Tale Christopher Menzel
29. Background Evolution Metatheory Beyond FOL
Features of the Syntax
ā¢ Type freedom
ā¢ There are only logical operators and names in the lexicon
ā¢ Traditional lexical categories ā Cn, Fn, Pr ā are simply
contextual roles that any name can play
ā¢ Self-predication and self-application are legit
ā¢ (Abstract Abstract), (P (f f) a), etc.
ā¢ Signature freedom
ā¢ There is no speciļ¬cation of adicity
ā¢ Same name be predicated of any ļ¬nite number of arguments
ā¢ Including 0: (P) is a 0-place atomic formula
ā¢ (P), (P P), (P (P P) P), (P (P P) (P P (P P) P), ...
ā¢ āHigher-orderā quantiļ¬cation permitted
ā¢ (āR (āc (iff (R c) (not (c c)))))
Common Logic: An Evolutionary Tale Christopher Menzel
30. Background Evolution Metatheory Beyond FOL
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 āā {f | f : Dā āā D}, erel : D āā ā(Dā ), V : N āā D,
and if L is inclusive, erel (V (=)) = { a, a : a ā D}.
Denotation and Truth
ā¢ For names Ī½ of L, dV (Ī½) = V (Ī½).
ā¢ dV ((Ī± Ī² 1 ... Ī² n )) = efn (dV (Ī±))(dV ( Ī² 1 ), ..., dV ( Ī² n )).
ā¢ (Ī± Ī² 1 ... Ī² n ) is true in I iļ¬ dV ( Ī² 1 ), ..., dV ( Ī² n ) ā erel (dV (Ī±)).
ā¢ If L is conventional, (= Ī± Ī²) is true in I iļ¬ dV (Ī±) = dV ( Ī²).
ā¢ (not Ļ) is true in I iļ¬ Ļ is not true in I .
ā¢ (if Ļ Ļ) is true in I iļ¬ either Ļ is not true in I or Ļ is true in I .
ā¢ (āĪ½ Ļ) is true in I iļ¬, for all a ā D, Ļ is true in I a .
Ī½
ā¢ Satiļ¬ability, validity, logical consequence (|=L ) deļ¬ned as usual
Common Logic: An Evolutionary Tale Christopher Menzel
31. Background Evolution Metatheory Beyond FOL
Recall: Semantics of TFOL
Common Logic: An Evolutionary Tale Christopher Menzel
32. Background Evolution Metatheory Beyond FOL
Semantics: CL Model Theory
Common Logic: An Evolutionary Tale Christopher Menzel
33. Background Evolution Metatheory Beyond FOL
Abstract Syntax: Web Sensitive Features
ā¢ A text is either a set or list or bag of phrases.
ā¢ A piece of text may be identiļ¬ed by a name.
ā¢ A phrase is either a comment, a module, a sentence, or an
importation.
ā¢ A comment is a piece of data.
ā¢ No particular restrictions are placed on comments.
ā¢ Comments can be attached to other comments.
ā¢ A module consists of a name and a text called the body text.
ā¢ The module name indicates the local domain of discourse in
which the text is to be understood
ā¢ An importation contains a name. (More below)
Common Logic: An Evolutionary Tale Christopher Menzel
34. Background Evolution Metatheory Beyond FOL
Abstract Syntax: Representational Features
ā¢ A sentence is either an atom, a boolean sentence, or a
quantiļ¬ed sentence.
ā¢ A sentence may have an attached comment.
ā¢ A boolean sentence has a type, called a connective, and a
number of sentences, called the components of the sentence.
ā¢ The number depends on the type.
ā¢ 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.
ā¢ A quantiļ¬ed sentence has (i) a type, called a quantiļ¬er, (ii) a
ļ¬nite, nonrepeating sequence of names called the binding
sequence, each element of which is called a binding of the
quantiļ¬ed sentence, and (iii) a sentence called the body of the
quantiļ¬ed sentence.
Common Logic: An Evolutionary Tale Christopher Menzel
35. Background Evolution Metatheory Beyond FOL
ā¢ An atom is either an equation containing two arguments,
which are terms, or an atomic sentence.
ā¢ An atomic sentence consists of a term, called the predicate,
and a term sequence called the argument sequence.
ā¢ Each term in the term sequence of an atomic sentence is called
an argument of the sentence.
ā¢ Any name can be the predicate in an atomic sentence.
ā¢ A term is either a name or a functional term.
ā¢ Terms may have attached comments.
ā¢ A functional term consists of a term, called the operator and a
term sequence called the argument sequence.
ā¢ Parallel qualiļ¬cations to atomic sentences.
ā¢ A term sequence is a (possibly null) ļ¬nite sequence of terms or
sequence markers.
Common Logic: An Evolutionary Tale Christopher Menzel
36. Background Evolution Metatheory Beyond FOL
Features of the Abstract Syntax
ā¢ Abstraction!
ā¢ No speciļ¬cation of any concrete syntactic forms
ā¢ Speciļ¬c form left to the KR designers.
ā¢ A given KR language neednāt use all the features of CL
ā¢ E.g., Description Logics lacking negation
ā¢ Conformance deļ¬ned ļ¬exibly enough to allow a side range of
CL dialects, including ātraditionalā ļ¬rst-order languages
ā¢ āEvery cloud has a silver liningā in PM-ese, CGs, and KIF
ā¢ āx(Cloud(x) ā āy(Lining(y) ā§ Silver(y) ā§ Has(x, y)))
ā¢ [@every*x] [If: (Cloud ?x) [Then: [*y] (Lining ?y) (Silver ?y) (Has ?x ?y)]]
ā¢ (forall (?x ?y)
(if (Cloud ?x)
(exists (?y)
(and (Lining ?y) (Silver ?y) (Has ?x ?y)))))
Common Logic: An Evolutionary Tale Christopher Menzel
37. Background Evolution Metatheory Beyond FOL
Where We Are
1 Background
In Praise of āTraditionalā 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 Reļ¬ections
Common Logic: An Evolutionary Tale Christopher Menzel
38. Background Evolution Metatheory Beyond FOL
Proof Theory: The System CL
Any generalization of any of the following is an axiom of CL :
1 Propositional tautologies
2 (if (āĪ½ Ļ) ĻĪ½ ), where Ī± is free for Ī½ in Ļ
Ī±
3 (if (āĪ½ (if Ļ Ļ)) (if (āĪ½ Ļ) (āĪ½ Ļ)))
4 (if Ļ (āĪ½ Ļ)), where Ī½ does not occur free in Ļ
5 (= Ī½ Ī½), for any name Ī½ of L
6 (if (= Ī½ Āµ) (if Ļ ĻĪ½ )), where Āµ is free for Ī½ in Ļ
Āµ
The system CL has one rule of inference:
ā¢ Modus Ponens (MP): From Ļ and (if Ļ Ļ), infer Ļ.
Common Logic: An Evolutionary Tale Christopher Menzel
39. Background Evolution Metatheory Beyond FOL
+
Soundness of CL and CL
ā¢ Deļ¬ne the notion of an interpretation+ by adding semantic
conditions M and C
ā¢ Truth in an interpretation+ deļ¬ned as above
+
ā¢ All derivative notions (satisļ¬ability+ , model+ , validity+ , |=L ,
etc) deļ¬ned accordingly
+
ā¢ Let CL be the resulting of adding schemas 7 and 8 to CL
+
Theorem (Soundness of CL and CL )
If Ī CL Ļ, then Ī |=L Ļ; and if Ī +
CL Ļ, then Ī |=L Ļ.
+
Common Logic: An Evolutionary Tale Christopher Menzel
40. Background Evolution Metatheory Beyond FOL
+
Completeness of CL and CL
+
Theorem (Completeness of CL and CL )
If Ī |=L Ļ, then Ī CL Ļ; and if Ī |=L Ļ, then Ī
+
+
CL Ļ.
Corollary (Lƶwenheim-Skolem)
If a set Ī of sentences of L has an L-model (L-model+ ), it has a
countable L-model (L-model+ ).
Corollary (Compactness)
If every ļ¬nite subset of a set Ī of sentences of L has an L-model
(L-model+ ), then Ī has a model (model+ ).
Common Logic: An Evolutionary Tale Christopher Menzel
41. Background Evolution Metatheory Beyond FOL
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 ā) as
well as the following:
ā¢ The set NL of names of L, which are known as the individual constants of L*.
ā¢ For every n ā N, an n + 1-place predicate Holdsn
ā¢ For every n ā N, an n + 1-place function symbol Appn .
ā¢ 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
ā¢ Individual constants and variables of L* together with those expressions of L* of
the form (Appn Ī± Ī² 1 ... Ī² n ), for terms Ī±, Ī² 1 , ..., Ī² n of L*.
Formulas
ā¢ Those expressions of the form (Holdsn Ī± Ī²1 ... Ī²n ) for terms Ī±, Ī²1 , ..., Ī²n of L*
ā¢ For formulas Ļ, Ļ of L*, those expressions of the form (not Ļ), (if Ļ Ļ), and
(forall (Ļ) Ļ) ((āĻ Ļ)), for variables Ļ of L*.
Common Logic: An Evolutionary Tale Christopher Menzel
42. Background Evolution Metatheory Beyond FOL
Standard Translations
Let L* be a traditional counterpart of L. Let x be a ļ¬xed one-to-one
correspondence from the set NL of names of L onto VarL* .
ā¢ For names Ī½ ā NL , Ī½ = Ī½
ā¢ For terms Ī±, Ī² 1 , ..., Ī² n of L,
ā¢ (= Ī² 1 Ī² 2 )ā = (= Ī² 1 Ī² 2 )
ā¢ (Ī± Ī² 1 ... Ī² n ) = (Appn Ī± Ī² 1 ... Ī² n )
ā¢ (Ī± Ī² 1 ... Ī² n )ā = (Holdsn Ī± Ī² 1 ... Ī² n )
ā¢ For sentences Ļ,Ļ of L and Ī½ ā NL ,
ā¢ (not Ļ)ā = (not Ļā )
ā¢ (if Ļ Ļ)ā = (if Ļā Ļā )
ā¢ (āĪ½ Ļ)ā = (āxĪ½ Ļā xĪ½ )
Ī½
Call the pair , ā of functions a standard translation of L into L*.
Common Logic: An Evolutionary Tale Christopher Menzel
43. Background Evolution Metatheory Beyond FOL
Standard Translations: Examples
ā¢ (Married Bill Hillary) = (Holds2 Married Bill Hillary)
ā¢ (not (F (f a b)))) = (not (Holds1 F (App2 f a b)))
ā¢ (if (F a b) (not (G a))) =
(if (Holds2 F a b) (not (Holds1 G a))))
ā¢ (āx (if (F (f x a)) (G x))) =
(āx (if (Holds2 F (App2 f x a)) (Holds1 G x)))
Common Logic: An Evolutionary Tale Christopher Menzel
45. Background Evolution Metatheory Beyond FOL
Standard Translations are Meaning Preserving
Theorem. For sentences Ļ and interpretations I = D, erel , efn , V
of L, Ļ is true in I iļ¬ Ļā is true in I *= D, V āŖ WI .
Corollary 1. For sentences Ļ of L, Ī |=L Ļ if and only if
Īā |=L* Ļā .
Common Logic: An Evolutionary Tale Christopher Menzel
46. Background Evolution Metatheory Beyond FOL
Completeness via TFOL
Fact. For any sentence Ļ of L* and any set Ī£ of sentences of L*,
if Ī£ CL* Ļ, then there is a proof of Ļ from Ī£ consisting entirely of
sentences of L* (i.e., formulas of L* in which no variables occur
free).
Lemma. If Ļ1 , ..., Ļn is a proof in CL* of Ļā from Īā , then there
ā ā
are sentences Ļ1 , , ..., Ļn of L such that Ļ1 , , ..., Ļn is a proof of
Ļ ā from Īā in C ā .
L
Lemma. If Ļ1 , ..., Ļn is a proof from Īā in CL* , then Ļ1 , ..., Ļn is a
ā ā
proof from Ī in CL .
Corollary 2. If Īā CL* Ļā , then Ī CL Ļ.
Common Logic: An Evolutionary Tale Christopher Menzel
47. Background Evolution Metatheory Beyond FOL
Completeness via TFOL
Theorem (Completeness of CL via TFOL)
If Ī |=L Ļ, then Ī CL Ļ.
Proof. If Ī |=L Ļ, then by Corollary 1, Īā |=L* Ļā . Hence, by the
completeness of CL* , we have Īā CL* Ļā and thus, by Corollary 2,
Ī CL Ļ.
Common Logic: An Evolutionary Tale Christopher Menzel
48. Background Evolution Metatheory Beyond FOL
Where We Are
1 Background
In Praise of āTraditionalā 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 Reļ¬ections
Common Logic: An Evolutionary Tale Christopher Menzel
49. Background Evolution Metatheory Beyond FOL
Beyond First-order: Sequence Markers
ā¢ Sequence markers are a natural mechanism vis-Ć -vis
signature-freedom
ā¢ But: They push CL beyond FOL in expressiveness
ā¢ Chaining
ā¢ (forall (F x) ((Chain F) x))
(forall (F x y)
(iff ((Chain F) ... x y)
(and (F x y) ((Chain F) ... x)))))
ā¢ (= AscendingOrder (Chain LessThan))
ā¢ (AscendingOrder 2 5 17 25)
ā¢ Axioms for Relations
ā¢ (iff (Unary F)
(and (not (F))
(not (exists (... x y) (F ... x y)))))
Common Logic: An Evolutionary Tale Christopher Menzel
50. Background Evolution Metatheory Beyond FOL
Sequence Markers: Chained Identity and Diļ¬erence
ā¢ Chained Identity
(AllEq x)
(iff (AllEq x y ...)
(and (= x y) (AllEq y ...)))
ā¢ Chained Diļ¬erence
(iff (AllDiff x)) (Comment "a.k.a. āNoRepeatsā")
(iff (AllDiff x y ...)
(and (not (= x y))
(AllDiff x ...)
(AllDiff y ...)))
Common Logic: An Evolutionary Tale Christopher Menzel
51. Background Evolution Metatheory Beyond FOL
Sequence Markers: Finitude
ā¢ SeqOf
((seqOf F)) (Comment "Holds only of seqs of Fs")
(iff ((seqOf F) x ...) (and ((seqOf F) ...) (F x))
ā¢ Finitude of properties
(iff (Finite F)
(and (Unary F)
(exists (...)
(and ((seqOf F) ...)
(AllDiff ...)
(forall (x)
(if (F x) (not (AllDiff x ...))))))))
Common Logic: An Evolutionary Tale Christopher Menzel
52. Background Evolution Metatheory Beyond FOL
Final Reļ¬ections
ā¢ Given the Holds/App translation, why not just use TFOL?
ā¢ The Holds/App translation is ontologically artiļ¬cial
ā¢ Schizophrenic regarding relations
ā¢ Automated reasoning tools built for TFOL
ā¢ But can still use them via translators
ā¢ Horrocks sentences ā deep or superļ¬cial?
ā¢ The following is a logical truth of CLIF
(if (x (iff (F x) (not (G x)))) (āxāy (not (= x y))))
ā¢ This form is not a logical truth of TFOL
ā¢ Theoretically innocuous but user-unfriendly?
Common Logic: An Evolutionary Tale Christopher Menzel