Formal Conceptual Analysis

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Formal Conceptual Analysis

• Octavian Popescu
• 11/15/2009

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Conceptual Structure

• A pervasive feeling of hidden structure
  whenever it comes about the language a tacit
  knowledge representation of it.
• Conceptual structures in knowledge representation
  are models of a perceived reality
• Practical view knowledge as a collection of
  facts, rules and procedures.
• It is not sufficient

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Truth functionality price

• paradoxes of material implication
• p ? (q p).
• p ? (p? q).
• (p? q) ? (q ? r).
• (p p) ? q.
• p ? (q? q).
• p? (q q).
• Whenever the antecedent is false, the whole
  conditional is true
• Whenever the consequent is true, the conditional
  is true

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Human Behavior

• A bus is driven with 60km/h. If the bus do not
  have a flat tire it will reach its destination
  point in time. The bus didnt arrived in time.
  Has the bus had a flat tire?
• Linda majored in philosophy in college. As a
  student, she was deeply concerned social issues.
• (1) Linda is active in the feminist movement.
• (2) Linda is a bank teller.
• (3) Linda is a bank teller and active in the
  feminist movement.

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Conceptual Structure

• Intersubjective community of communication and
  argumentation
• Relevance
• Truth preserving vs. relevance
• Relevant logic
• I argue that this is just a matter of
  conceptualizing
• Boole Scroeder vs. Russell Whitehead
• Logical structures respond productively to the
  application of algebraic techniques

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Conceptual Structure

• Universal properties for natural languages
• Phonological change
• Grammaticality
• Context coherence
• Classical formal logics has little to say about
• Probably inadequate treatment of meaning
• Alternative approaches ?

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Formal Concepts

• Concept
• Extension / Intension
• The unity of Objects / Attributes
• Let X be a set of objects and consider X the
  set of common attributes of these objects
• Let Y be a set of attributes, Y the set of
  object that have those attributes
• Concept
• (A,B) is a formal concept iff A B and A B

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Formal Concepts
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Formal Concept
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Formal Concepts

• a needs water
• b lives in water
• c lives on land
• d chlorophyll
• e two leaf
• f one leave
• g can move
• h has limbs
• i mammal

• X leech, bream 1,2
• Y lives in water, has limbs b,
  h
• ?
• X needs water, lives in water, can move
  a, g, h
• Y bream, frog 2 , 3

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Formal Concepts

• Example of formal concepts
• B1 (2,3 , a, b, g, h)
• B2 (1,2,3, a, b, g)
• We say that B2 is a super-concept of B1
• (A1,B1) (A2, B2) iff A1 A2 iff B2 B1
• Smallest concept for an object / attribute
• G g (g, g)
• M m (m, m)

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Formal Conceptual Analysis

• The study of formal concepts within the framework
  of lattice theory
• FCA has been developed at the Faculty of
  Mathematics at Darmstadt University of Technology
• By the end of eighties first major articles
• 1996 Formale Begriffsanalyse Mathematische
  Grudlagen, which gathers the whole theory in one
  place.
• 1999 its English translation

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Mathematical Support

• A set (V, ) on which a partial order relation
  has been defined is a lattice if for any x, y in
  V supx,y), infx,y exist.
• A lattice (V, ) is complete if sup and inf
  exist for any subset of V.
• Any finite lattice is complete
• Any complete lattice has greatest /lowest element
  - 1v / 0v respectively
• An v in V is sup-irreducible if
• v sup x in V x v ? v

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Mathematical Support

• Galois connection of (P, ) (Q, )
• Let fP?Q ?Q?P. (f,?) is a GC if
• f,? are monotonic
• p ?fp and q f?q
• For every GC (f, ?)
• f f?f and ? ?f?

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Formal Conceptual Analysis

• A formal context K (G, M, I) consists of two
  sets G, M and I a relation between G and M. The
  elements of G are called objects, the elements of
  M are called attributes, gIm it is read g has
  m.
• For A in G A m in M gIm for all g in A
• For B in M B g in G gIm for all m in B
• (A,B) is a formal concept A B and A B

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Formal Conceptual Analysis

• Properties
• 1) A1 in A2 then A2 in A1
• 2) A in A
• 3) A A
• 4) A in B iff B in A iff AxB in I
• 5) (U At) n At
• 6) The set of concepts is a complete lattice.

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Formal Conceptual Analysis

• Theorem
• Inf(At, Bt) (nAt , (UBt))
• Sup(At,Bt) ((UAt), (nBt))
• Clarified and reduced Contexts
• A (G, M, I) is clarified if for g, h in G such
  that g h then g h (m n then m n in M)
• A clarified context is row reduced if any object
  concept (g, g) is sup irreducible and column
  reduced if any attribute concept (m, m) is inf
  irreducible.

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Implication between Attributes

• A formal context leads to a concept lattice and
  vice versa
• A concept lattice can be viewed as a
• hierarchical conceptual clustering (extends)
• representation of all implication between
  attributes (intents)
• Every x with a, b, c has also l, k, i
• For M1,M2 subsets in M we write M1 ? M2

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Implications between Attributes

• Attribute logic the rules concerning the
  combination of attributes
• Implication holds in a context
• An implication M1 ? M2
• M1 ? M2 holds in (G, M,I) iff M2 in M1
• Determination of a set of implications as a basis
  of all implications in the given context

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Phonological Systems

• The PS of a language is the inventory of used
  phonemes in a certain age
• PS are not fixed. They endure various systematic
  changes.
• The PSs of more or less related languages do
  exhibited same changes
• Examples
• Grimm laws of Indo-European drift
• Lost of long vowels from Latin
• Great vowel shift in English and in Germanic
  languages

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Phonological Systems

• The most frequent phonological phenomenon
  assimilation
• Phonemes (allophones) are characterized by a set
  of features
• For consonants manner and place of articulation
• For vowels lip position, closure point

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Phonological Systems

• Features
• A oral cavity openness
• I palatality
• U labiality
• ? Occlusion
• h aperiodic energy
• N nasality
• R apicality/coronality
• H voiceless
• Harris Lindsey - 1993

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Phonological system
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Phonological System

• Associate to each PS a formal context
• Compute the concepts and the concept lattice
• Analyze the change in terms of lattice
  transformations
• See the evolution of a particular language in
  time or within its language family

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Metaphors

• Lakoff (Metaphors we live by)
• Metaphors the understating of a concept
  partially in terms of another concept
• The analysis of metaphors as formal concepts
  whose intents are direct attributes primary
  functions

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Referential language

• Definite description
• Russell, Strawson, Donnellan, Kripke, Sag, Reimer
• Where Russells predictions go wrong
• An inhabitant of London vs. the inhabitant of
  London
• Where Donnellans prediction go wrong
• To predict whether a use is referential or
  attributive

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Conclusions

• There are certain properties of natural languages
  that may be described in a different way that
  classical truth preserving logic propose.
• Conceptual representation may show a way to what
  generally is considered as truth.