EE1J2 - Slide 1 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,

EE1J2 - Slide 2 Tutorial arrangements 3 Tutorial groups: X, Y and Z Thursdays 3pm, starting 31 st 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

EE1J2 - Slide 3 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.

EE1J2 - Slide 4 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

EE1J2 - Slide 5 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))

EE1J2 - Slide 6 Formalised Statement (A  L  P  M)  ((  M   A)  P)  ((  L   P)  (  A  M))

EE1J2 - Slide 7 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

EE1J2 - Slide 8 Interpreting logical statements in NL NL interpretation of propositional connectives ConnectiveInterpretation  p 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

EE1J2 - Slide 9 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”

EE1J2 - Slide 10 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)

EE1J2 - Slide 11 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

EE1J2 - Slide 12 Form and Content Although the content is different, the forms are the same…

EE1J2 - Slide 13 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))

EE1J2 - Slide 14 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

EE1J2 - Slide 15 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))

EE1J2 - Slide 16 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?

EE1J2 - Slide 17 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)

EE1J2 - Slide 18 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’

EE1J2 - Slide 19 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’

EE1J2 - Slide 20 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

EE1J2 - Slide 21 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

EE1J2 - Slide 22 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.

EE1J2 - Slide 23 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

EE1J2 - Slide 24 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

EE1J2 - Slide 25 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

EE1J2 - Slide 26 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

EE1J2 - Slide 27 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