EE1J2 Discrete Maths Lecture 2

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EE1J2 Discrete Maths Lecture 2

• Tutorials
• Revision of formalisation
• Interpreting logical statements in NL
• Form and Content of an Argument
• Formalisation of NL grammatical analysis,
  production rules, parsing, parse trees
• Propositional logic as a formal language
  symbols and formulae
• Parsing and parse trees in Propositional Logic,

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Tutorial arrangements

• 3 Tutorial groups X, Y and Z
• Thursdays 3pm, starting 31st January
• X Room 220/221, A Teye
• Y Room 523, K Hussein
• Z Room 521/522, G Philips
• Hand in work Tuesday before tutorial
• Drawers marked X, Y, Z downstairs

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Revision - formalisation

• Either Arsenal, Leeds, Liverpool, or ManU will
  win the league. If neither ManU nor Arsenal win
  it, then Liverpool will win. If Leeds or
  Liverpool fail to win, then Arsenal will not win
  and ManU will win it.

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Elementary propositions

• A Arsenal will win the league
• L Leeds will win the league
• P Liverpool will win the league
• M ManU will win the league

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Formalised statement

• Either Arsenal, Leeds, Liverpool, or ManU will
  win the league
• (A ? L ? P ? M)
• If neither ManU nor Arsenal win it, then
  Liverpool will win
• ((?M ? ?A) ? P)
• If Leeds or Liverpool fail to win, then Arsenal
  will not win and ManU will win it.
• ((?L ? ?P) ?(?A ? M))

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Formalised Statement

• (A ? L ? P ? M) ? ((?M ? ?A) ? P) ? ((?L ? ?P)
  ?(?A ? M))

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Formalisation (continued)

• Statement
• If Polonius is not behind that curtain then
  Polonius is well
• Atomic propositions
• C Polonius is behind that curtain
• W Polonius is well
• Formalisation in Propositional Logic
• (?C) ? W

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Interpreting logical statements in NL
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Example

• Consider the statement p ? q ? ?r ? ?s where
• p the thief is young
• q the thief is hanged
• r the thief will grow old
• s the thief will steal
• In NL, this equates to if the thief is young
  and the thief is hanged, then the thief will
  neither grow old nor steal

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Exclusive and inclusive OR

• The English word or can be ambiguous. The two
  possible meanings are denoted by inclusive or and
  exclusive or
• Inclusive or is represented by the propositional
  connective ?
• Exclusive or is represented by
• (p? q) ? ?(p? q)

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Separating Form and Content

• If I play cricket or go to work, but not both,
  then I will not be going shopping. Therefore, if
  I go shopping then neither would I play cricket
  nor would I go to work
• An object remaining stationary or moving at a
  constant velocity means that there is no external
  force acting upon it. Therefore, if there is a
  force acting upon the object, it is not
  stationary and it is not moving at a constant
  velocity

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Form and Content

• Although the content is different, the forms are
  the same

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Argument 1

• If I play cricket or go to work, but not both,
  then I will not be going shopping. Therefore, if
  I go shopping then neither would I play cricket
  nor would I go to work.
• Atomic Propositions
• P I play cricket
• Q I go to work
• R I go shopping
• Formal Argument
• ((P? Q) ? ?(P? Q) ? ?R)?(R?(?P)?(?Q))

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Argument 2

• An object remaining stationary or moving at a
  constant velocity means that there is no external
  force acting upon it. Therefore, if there is a
  force acting upon the object, it is not
  stationary and it is not moving at a constant
  velocity
• Atomic propositions
• S the object is stationary
• M the object is moving at a constant velocity
• F there is an external force acting upon the
  object

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Argument 2 (cont.)

• Atomic propositions
• S the object is stationary
• M the object is moving at a constant velocity
• F there is an external force acting upon the
  object
• Formal Argument
• ((S? M) ? ?(S? M) ? ?F)?(F?(?S)? (?M))

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Re-cap

• Propositional logic motivated by analogies with
  natural language
• Formalisation of statements in NL
• Naturalisation of formulae in PL
• Separation of form and meaning
• Now move on to study propositional logic as a
  formal language
• What is a formal language?

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Formalisation of Natural Language

• Remember grammar lessons in primary school?
• The purpose is to expose the underlying
  grammatical or syntactic structure of the
  sentence
• Or, to decide whether the given sentence is
  grammatical (i.e. in the language)

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Grammatical analysis in NL

• Consider S The cat devoured the tiny mouse
• S is made up of of
• the noun phrase NP The cat, and
• the verb phrase VP devoured the tiny mouse

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Grammatical Analysis

• NP comprises the determiner The and the noun
  cat
• VP comprises the verb devoured and the noun
  phrase the tiny mouse
• The noun phrase the tiny mouse comprises the
  determiner the, the adjective tiny, and the
  noun mouse

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Production Rules

• Formally, this analysis of the sentence is with
  respect to a set of production rules
• Production rules determine how non-terminal
  elements in a language can be expanded into
  sequences of non-terminal elements and terminal
  elements.
• The non-terminals are structures like sentence,
  noun-phrase, verb-phrase, adjective, etc
• The terminals are actual words

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Production Rules

• The first production rule which we used was
• S ? NP VP
• Then we applied more production rules, formally
  denoted as
• NP ? DET N
• VP ? V NP
• NP ? DET ADJ N

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Parsing

• This process is called parsing
• The sequence of production rules which transforms
  S into the sequence of words in the sentence is a
  parse of the sentence.

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Grammatical sentences

• In a formal language, a sequence of words is
• a sentence in the language
• or is grammatical
• if and only if
• a parse of the word sequence exists

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Parse Trees

• The parse of the sentence The cat devoured the
  tiny mouse given by the above set of production
  rules can be represented simply, intuitively and
  usefully as a tree structure
• This tree structure is called a parse tree

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Parse Tree for the cat devoured the tiny mouse
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Parsing in NL

• The bases of the branches of the tree correspond
  to non-terminal units of the language.
• The leaves of the tree correspond to the
  terminal unit.
• Local structure of the tree at a non-terminal
  unit corresponds to the production rule employed
  in the parse

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Summary of Lecture 2

• Revision of formalisation
• Interpreting logical statements in NL
• Form and Content of an Argument
• Formalisation of NL grammatical analysis,
  production rules, parsing, parse trees
• Propositional logic as a formal language
  symbols and formulae
• Parsing a formula in Propositional Logic