Common Logic: An Evolutionary Tale

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Christopher Menzel's presentation at the "Philosophy of the Web" seminar in Sorbonne, April 14 2012.

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Common Logic: An Evolutionary Tale

  1. 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 2012Common Logic: An Evolutionary Tale Christopher Menzel
  2. 2. Background Evolution Metatheory Beyond FOLWhere 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 ReflectionsCommon Logic: An Evolutionary Tale Christopher Menzel
  3. 3. Background Evolution Metatheory Beyond FOLOpen 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 logicCommon Logic: An Evolutionary Tale Christopher Menzel
  4. 4. Background Evolution Metatheory Beyond FOLIn 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))Common Logic: An Evolutionary Tale Christopher Menzel
  5. 5. Background Evolution Metatheory Beyond FOLIn 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 proversCommon Logic: An Evolutionary Tale Christopher Menzel
  6. 6. Background Evolution Metatheory Beyond FOLTFOL’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 rigidityCommon Logic: An Evolutionary Tale Christopher Menzel
  7. 7. Background Evolution Metatheory Beyond FOLFeatures 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 quantifiersCommon Logic: An Evolutionary Tale Christopher Menzel
  8. 8. Background Evolution Metatheory Beyond FOLFeatures 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 quantifiersCommon Logic: An Evolutionary Tale Christopher Menzel
  9. 9. Background Evolution Metatheory Beyond FOLFeatures 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.Common Logic: An Evolutionary Tale Christopher Menzel
  10. 10. Background Evolution Metatheory Beyond FOLFeatures of TFOL: SemanticsCommon Logic: An Evolutionary Tale Christopher Menzel
  11. 11. Background Evolution Metatheory Beyond FOLFeatures 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.Common Logic: An Evolutionary Tale Christopher Menzel
  12. 12. Background Evolution Metatheory Beyond FOLEntailment and Open Networks • To illustrate • Entailment should commute with communication...Common Logic: An Evolutionary Tale Christopher Menzel
  13. 13. Background Evolution Metatheory Beyond FOL • ...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: ‘Common Logic: An Evolutionary Tale Christopher Menzel
  14. 14. Background Evolution Metatheory Beyond FOLWhere 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 ReflectionsCommon Logic: An Evolutionary Tale Christopher Menzel
  15. 15. Background Evolution Metatheory Beyond FOLI: 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∈NCommon Logic: An Evolutionary Tale Christopher Menzel
  16. 16. Background Evolution Metatheory Beyond FOLII: 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) • 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)Common Logic: An Evolutionary Tale Christopher Menzel
  17. 17. Background Evolution Metatheory Beyond FOLIII: 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) • 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 relationCommon Logic: An Evolutionary Tale Christopher Menzel
  18. 18. Background Evolution Metatheory Beyond FOLIII: 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)Common Logic: An Evolutionary Tale Christopher Menzel
  19. 19. Background Evolution Metatheory Beyond FOLIII: 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 DCommon Logic: An Evolutionary Tale Christopher Menzel
  20. 20. Background Evolution Metatheory Beyond FOLIV: 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))‘Common Logic: An Evolutionary Tale Christopher Menzel
  21. 21. Background Evolution Metatheory Beyond FOLIV: 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)))))Common Logic: An Evolutionary Tale Christopher Menzel
  22. 22. Background Evolution Metatheory Beyond FOLTaking 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!Common Logic: An Evolutionary Tale Christopher Menzel
  23. 23. Background Evolution Metatheory Beyond FOLAn 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.Common Logic: An Evolutionary Tale Christopher Menzel
  24. 24. Background Evolution Metatheory Beyond FOLAn Anarchic Syntax: Names One (Non-logical) Lexical Category • NamesCommon Logic: An Evolutionary Tale Christopher Menzel
  25. 25. Background Evolution Metatheory Beyond FOLAn Anarchic Syntax: Grammar One (Basic) Grammatical Rule • Every name can be predicated of any number of namesCommon Logic: An Evolutionary Tale Christopher Menzel
  26. 26. Background Evolution Metatheory Beyond FOLAn Anarchic Semantics Two (Basic) Semantic Principles • Names name things • Names can be true of thingsCommon Logic: An Evolutionary Tale Christopher Menzel
  27. 27. Background Evolution Metatheory Beyond FOLSyntax: 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.Common Logic: An Evolutionary Tale Christopher Menzel
  28. 28. Background Evolution Metatheory Beyond FOLSyntax: 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.Common Logic: An Evolutionary Tale Christopher Menzel
  29. 29. Background Evolution Metatheory Beyond FOLFeatures 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)))))Common Logic: An Evolutionary Tale Christopher Menzel
  30. 30. Background Evolution Metatheory Beyond FOLSemantics: 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 usualCommon Logic: An Evolutionary Tale Christopher Menzel
  31. 31. Background Evolution Metatheory Beyond FOLRecall: Semantics of TFOLCommon Logic: An Evolutionary Tale Christopher Menzel
  32. 32. Background Evolution Metatheory Beyond FOLSemantics: CL Model TheoryCommon Logic: An Evolutionary Tale Christopher Menzel
  33. 33. Background Evolution Metatheory Beyond FOLAbstract 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)Common Logic: An Evolutionary Tale Christopher Menzel
  34. 34. Background Evolution Metatheory Beyond FOLAbstract 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.Common Logic: An Evolutionary Tale Christopher Menzel
  35. 35. Background Evolution Metatheory Beyond FOL • 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.Common Logic: An Evolutionary Tale Christopher Menzel
  36. 36. Background Evolution Metatheory Beyond FOLFeatures 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)))))Common Logic: An Evolutionary Tale Christopher Menzel
  37. 37. Background Evolution Metatheory Beyond FOLWhere 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 ReflectionsCommon Logic: An Evolutionary Tale Christopher Menzel
  38. 38. Background Evolution Metatheory Beyond FOLProof 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.Common Logic: An Evolutionary Tale Christopher Menzel
  39. 39. Background Evolution Metatheory Beyond FOL +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. +Common Logic: An Evolutionary Tale Christopher Menzel
  40. 40. Background Evolution Metatheory Beyond FOL +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+ ).Common Logic: An Evolutionary Tale Christopher Menzel
  41. 41. Background Evolution Metatheory Beyond FOLThe 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*.Common Logic: An Evolutionary Tale Christopher Menzel
  42. 42. Background Evolution Metatheory Beyond FOLStandard 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*.Common Logic: An Evolutionary Tale Christopher Menzel
  43. 43. Background Evolution Metatheory Beyond FOLStandard 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)))Common Logic: An Evolutionary Tale Christopher Menzel
  44. 44. Background Evolution Metatheory Beyond FOLStandard 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.Common Logic: An Evolutionary Tale Christopher Menzel
  45. 45. Background Evolution Metatheory Beyond FOLStandard 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† .Common Logic: An Evolutionary Tale Christopher Menzel
  46. 46. Background Evolution Metatheory Beyond FOLCompleteness 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.Common Logic: An Evolutionary Tale Christopher Menzel
  47. 47. Background Evolution Metatheory Beyond FOLCompleteness 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.Common Logic: An Evolutionary Tale Christopher Menzel
  48. 48. Background Evolution Metatheory Beyond FOLWhere 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 ReflectionsCommon Logic: An Evolutionary Tale Christopher Menzel
  49. 49. Background Evolution Metatheory Beyond FOLBeyond 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)))))Common Logic: An Evolutionary Tale Christopher Menzel
  50. 50. Background Evolution Metatheory Beyond FOLSequence 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 ...)))Common Logic: An Evolutionary Tale Christopher Menzel
  51. 51. Background Evolution Metatheory Beyond FOLSequence 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 ...))))))))Common Logic: An Evolutionary Tale Christopher Menzel
  52. 52. Background Evolution Metatheory Beyond FOLFinal 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 Christopher Menzel

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