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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)

StatementIf Polonius is not behind that curtain then Polonius is well

Atomic propositions:C – Polonius is behind that curtainW – Polonius is well

Formalisation in Propositional Logic:(C) W

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Interpreting logical statements in NL

NL interpretation of propositional connectives

Connective Interpretation

p not p, p does not hold, p is false

p q p and q, p but q, not only p but q, p while q, p despite q, p yet q, p although q

p q p or q, p or q or both, p and/or q, p unless q

p q p implies q, if p then q, q if p, p only if q, q when p, p is sufficient for q, p materially implies q

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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 1If 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 + NVP V + NPNP 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”

The cat devoured the tiny mouse

DET ADJ NOUN

DET NOUN VERB NP

NP VP

S

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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