Semantics and Inference Part II Johan Bos

1. Semantics and Inference Part II Johan Bos

2. Summary of last lecture • Inferences on the sentence level • Entailment • Paraphrase • Contradiction • Using logic to understand semantics • Introduction to propositional logic • Syntax • Semantics

3. Propositions • What is a proposition? • Something that is expressed by a declarative sentence making a statement • Something that has a truth-value • Propositions can be true or false • There are only two possible truth-values • True, T or 1 • False, F or 0

4. Ingredients of propositional logic • Propositional variables • Usually: p, q, r, … • Connectives • The symbols: , ,, ,  • Often called logical constants • Punctuation symbols • The round brackets ( )

5. Syntax of propositional logic • All propositional variables are propositional formulas • If  is a propositional formula, then so is  • If  and  are propositional formulas, then so are (), (), ()and () • Nothing else is a propositional formula

6. Negation • Symbol:  • Pronounced as: “not” •  is called the negation of  • Truth-table:

7. Conjunction • Symbol:  • Pronounced as: “and” • () is called the conjunction of the conjuncts  and  • Truth table:

8. Disjunction • Symbol:  • Pronounced as: “or” • () is called the disjunction of the disjuncts  and  • Truth table:

9. (Material) Implication • Symbol:  • Pronounced as: “implies” or “arrow” • Truth table:

10. Equivalence (biconditional) • Symbol:  • Pronounced as: “if and only if” • Truth table:

11. Summary

12. This lecture • We will look at the role of tautologies in propositional logic • Explain the method of truth tables to detect tautologies • Apply this formal method to textual entailment • Look further at the notion of truth

13. Tautologies • A formula that is true in all situations is called a tautology or a semantic theorem • Examples of tautologies: (pp)(qq)(pp)(p(qp))(p(qp))((pq)((pq)(q p))

14. Checking for tautologies • How can we systematically check whether some formula is a tautology? • This is the business of theorem proving • This is what mathematicians do, and • therefore is not our main concern here • There are many methods • Using semantic tableaux (intuitive) • Using resolution (advanced) • Using truth-tables (nice for simple cases)

15. Using a truth-table Example: (pp) 1) Make a column for all propositional variables with possible truth-values

16. Using a truth-table Example: (pp) 2) Add columns for all sub-formulas

17. Using a truth-table Example: (pp) 3) Put the formula itself in the last column

18. Using a truth-table Example: (pp) 4) Fill in the truth values for the columns using the tables of the connectives

19. Using a truth-table Example: (pp) 4) Fill in the truth values for the columns using the tables of the connectives

20. Using a truth-table Example: (pp) 4) Fill in the truth values for the columns using the tables of the connectives

21. Using a truth-table Example: (pp) 4) Fill in the truth values for the columns using the tables of the connectives

22. Using a truth-table Example: (pp) 4) Fill in the truth values for the columns using the tables of the connectives

23. Using a truth-table Example: (pp) 5) Check the values in the last column All true in this column, hence tautology

24. Another example Example: (p q) 1) Make a column for all propositional variables with possible truth-values

25. Another example Example: (p q) 2) Add columns for sub-formulas

26. Another example Example: (p q) 3) Add formula itself in last column

27. Another example Example: (p q) 4) Fill in the truth values

28. Another example Example: (p q) 5) Check values in last column Not all true in this column, hence no tautology

29. Which of the following are tautologies? • (p(pq)) • (p(qr)) • ((pq)p) • ((pq)(pq))

30. Tautologies and inference • We are now ready to formalise the notions of • Entailment • Paraphrase • Contradiction • Some notational convention • If S is a sentence, then we will write S' meaning the logical translation of S.

31. Entailment • Let S be a sentence and S' the logical translation of S. Then: If (S1'S2') is a tautology, then S1entailsS2

32. Paraphrase • Let S be a sentence and S' the logical translation of S. Then: • Note: If S1 entails S2, and S2 entails S1, then S1 and S2 are paraphrases If (S1'S2') is a tautology, then S1 and S2 are paraphrases

33. Contradiction • Let S be a sentence and S' the logical translation of S. Then: • Note: If a set of a sentences is not contradictory, they are called consistent If (S1' S2') is a tautology, then S1 and S2 are contradictory or inconsistent

34. Entailment, example 1 • Translate into propositional logic and check if entailment holds: Diabolik found the treasure. Eva will be happy if Diabolik found the treasure. ----------------------------------------------------- Eva will be happy.

35. Entailment, example 2 • Translate into propositional logic and check if entailment holds: Diabolik found the treasure. Eva will be disappointed if Diabolik didn’t find the treasure. ----------------------------------------------------- Eva won’t be disappointed.

36. More about truth • This is what logicians claim • In any situation, a declarative sentence is true or false • In other words: it has one truth-value • But does this make sense? • Life seems to be full of half-truths, grey areas, and borderline cases • Logic divides the world into two parts: the True and the False

37. Is logic an illusion? • Maybe there are grades of truth? • Can something be more true than something else? • We will explore this question by looking at scaling and non-scaling adjectives • Scaling: small, big, fat, happy • Non-scaling: straight, silent, perfect

38. Scaling adjective: small • If you pick a really small doll, it is still possible to pick an even smaller doll • There can be two small dolls, one smaller than the other

39. Non-scaling adjective: straight • Line (a) is straighter then line (b) • This means line (b) is not really straight • Line (a) could be straight, but not necessarily so (a) (b)

40. Is “true” a scalar adjective? • It is not: if your statement is “truer” than mine, then mine is not wholly true • “more true” can only mean “nearer the truth” • There are no degrees of truth • Truth is absolute

41. Borderline cases • There is no precise cut-off point between small and not small • Both scaling and non-scaling adjectives can have borderline cases

42. What about “truth”? • Although “truth” doesn’t have degrees, it does have borderline cases • Look at the dolls and ask whether it is true this doll is small.

43. What about “truth”? • Although “truth” doesn’t have degrees, it does have borderline cases • Look at the dolls and ask whether it is true this doll is small.

44. What about “truth”? • Although “truth” doesn’t have degrees, it does have borderline cases • Look at the dolls and ask whether it is true this doll is small.

45. The truth about “truth” • Logicians are forced to admit that: where borderline cases may arise, logic is not an exact science • Logicians therefore stick to matter-of-fact notions, and leave the vague matters to philosophers

46. Misleading statements • Some common English words like and, some and all can give rise to misleading statements • Often the choice is to go for a weak or strong reading • Logicians normally opt for a weak reading, but there are good arguments to opt for strong readings too

47. Misleading statement 1 • A witness in a case:The poiliceman hit Mr Unlucky three times with the stick, and Mr Unlucky fell to the floor • What the witness actually saw was that Mr Unlucky fell to the floor just before the policeman came into the room, and the policeman hit him three times with the stick before he could get up

48. Misleading statement 2 • After a dinner party, Diabolik admits to Eva:I did kiss some of the girls… • In fact, Diabolik kissed all nineteen girls that were are the party.

49. Misleading statement 3 • Groucho boasts to Dylan Dog:All the girls at the party kissed me! • In fact, there were no girls at this particular party

50. Misleading statements • All of these three statements are misleading • But are they true or false? • There are two views here: • All of these statements were false (the strong readings of the sentences) • The statements expressed the truth but not the whole truth (weak readings)