Propositional Logic   Johan Bos

Propositional Logic Johan Bos PowerPoint PPT Presentation


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Overview of this lecture. Inferences on the sentence levelEntailmentParaphraseContradictionUsing logic to understand semanticsIntroduction to propositional logicSyntax and SemanticsDifferent kinds of logics. Making inferences. Meaning relations between expressions in a languageEntailmentsParaphrasesContradictions.

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Propositional Logic Johan Bos

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1. Propositional Logic Johan Bos

2. Overview of this lecture Inferences on the sentence level Entailment Paraphrase Contradiction Using logic to understand semantics Introduction to propositional logic Syntax and Semantics Different kinds of logics

3. Making inferences Meaning relations between expressions in a language Entailments Paraphrases Contradictions

4. Entailment, definition A sentence expressing a proposition X entails a sentence expressing a proposition Y if it is not possible to think of a situation where X is true and Y is false Or, put differently: the truth of Y follows necessarily from the truth of X

5. Entailments, examples the sentences 2, 3, and 4 are entailed by sentence 1: Dylan stroked the cat and hugged the dog. Dylan hugged the dog. Someone stroked the cat. Dylan hugged an animal.

6. Paraphrases, definition Two sentences are paraphrases of each other if they entail each other Put differently: whenever one is true, the other must also be true

7. Paraphrases, examples The following sentences are paraphrases of each other: Dylan stroked the cat and hugged the dog. Dylan hugged the dog and stroked the cat. The cat was stroked by Dylan and the dog was hugged by Dylan.

8. Contradiction, definition Two sentences are contradictory if it is impossible to think of a situation where both sentences can be true

9. Contradiction, examples Sentence 1 and 2 are contradictions, and so are sentence 1 and 3: Dylan stroked the cat and hugged the dog. The dog wasn’t hugged. Nobody stroked anything.

10. Side remark Usually references and contexts are kept constant in natural language inferences! Example: Dylan likes Groucho. Dylan hates Groucho. Contradiction or not?

11. Side remark Usually references and contexts are kept constant in natural language inferences! Example: Dylan likes Groucho. Dylan hates Groucho. Contradiction or not? “Of course the sentences are contradictory. You can’t hate and like someone at the same time.” “The sentences are not contradictory. I meant Dylan Dog in the first sentence, and Bob Dylan in the second…”

12. Formal Semantics Study of meaning with the help of (mathematical) logic Has been controversial for some time, but now widely accepted "Aren’t human languages imperfect and illogical anyway?" "Human languages have their own internal logic!"

13. Human vs. logical languages Languages such as Italian, English and Dutch are human languages (natural or ordinary languages) Logical systems are also referred to as languages by logicians; these are of course artificial languages; to avoid confusion they are sometimes called formal languages or calculi

14. Basic idea of formal semantics Provide a mapping from ordinary language to logic

15. Logical languages propositional logic modal logic description logic first-order logic (predicate logic) second-order logic higher-order logic

16. This lecture In this lecture we will try to map English to Propositional Logic Propositional logic is suitable to model the basics of sentence semantics As we will see it is not very useful for modelling sub-sentential semantics, for which usually more expressive logics are needed

17. Why is this a useful exercise? Description of some aspects of meaning in language Detect ambiguities or imprecisions in natural language Most of the literature in formal semantics presuppose familiarity with propositional and first-order logic

18. 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 (or T, or 1) False (or F, or 0)

19. Propositional logic Propositional logic is a language So we will look at its ingredients We will define the syntax, or in other words, the grammar We will define the semantics

20. Ingredients of propositional logic Propositional variables Usually: p, q, r, … Connectives The symbols: ?, ?,?, ?, ? Often called logical constants Punctuation symbols The round brackets ( )

21. Propositional variables Variables are used to stand for propositions Usually, the letters p, q, r are used for propositional variables Example p ? “It is raining outside." q ? “Eva Kant is reading a newspaper.“ Note: the internal structure of propositions is not of our concern in this lecture

22. 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

23. Which of these are propositional formulas? (p?p) p ???q ((p?q)?(?q)) ((p?q)?q?r) (p?(p?(p?p))) (?r?q) (?((p?q)??q)) ?(p?p))

24. Which of these are propositional formulas? (p?p) p ???q ((p?q)?(?q)) ((p?q)?q?r) (p?(p?(p?p))) (?r?q) (?((p?q)??q)) ?(p?p)) Yes Yes Yes No No Yes No No No

25. Logicians are only human Even though logicians and mathematicians are usually very precise in their formulations, they sometimes drop punctuation symbols if this does not give rise to confusion Often outermost brackets are dropped; also other brackets if no confusion arises

26. Logicians are only human Even though logicians and mathematicians are usually very precise in their formulations, they sometimes drop punctuation symbols if this does not give rise to confusion Often outermost brackets are dropped; also other brackets as long as no confusion arises Examples: p ? q instead of (p ? q) p ? (q ? r) instead of (p ? (q ? r)) (p ? q ? r) instead of (p ? (q ? r))

27. Negation Symbol: ? Pronounced as: “not” ?? is called the negation of ? Truth-table:

28. Negation: examples p ? “It is raining." ?p ? “It is not raining.” p ? “Eva said something." ?p ? “Eva didn’t say anything.” p ? “Diabolik sometimes lies." ?p ? “Diabolik never lies.”

29. Conjunction Symbol: ? Pronounced as: “and” (???) is called the conjunction of the conjuncts ? and ? Truth table:

30. Conjunction: examples p ? “The plan is simple.” q ? “The plan is effective." (p?q) ? “The plan is simple and effective." p ? “Diabolik was in trouble.” q ? “Diabolik managed to escape." (p?q) ? “Although he was in trouble, Diabolik managed to escape."

31. Disjunction Symbol: ? Pronounced as: “or” (???) is called the disjunction of the disjuncts ? and ? Truth table:

32. Disjunction: examples p ? “Eva has a gun." q ? “Eva has a knife." (p?q) ? “Eva has a gun or a knife (or both)." p ? “He is a fool." q ? “He is a liar." (p?q) ? “He is a fool or a liar (or both)."

33. (Material) Implication Symbol: ? Pronounced as: “implies” or “arrow” Truth table:

34. Implication: examples p ? “Ginko shot Diabolik." q ? “Diabolik is wounded." (p?q) ? “If Ginko shot Diabolik then Diabolik is wounded." p ? “The paper will turn red." q ? “The solution is acid." (p?q) ? “If the paper turns red, then the solution is acid."

35. Equivalence (biconditional) Symbol: ? Pronounced as: “if and only if” Truth table:

36. Equivalence: examples p ? “The number is even." q ? “The number is divisible by two." (p?q) ? “The number is even precisely if it is divisible by two." p ? “The company has to be registered." q ? “The annual turnover of the company is above Euro 5,000." (p?q) ? “The company has to be registered just if its annual turnover is above Euro 5,000."

37. Problematic cases “Eva wants a black and white cat.” “Eva wants a black cat” ? “Eva wants a white cat” “Geller can read your mind or he can bend spoons” “Geller can read your mind” v “Geller can bend spoons” “If you want to wash your hands, the bathroom is first on the left” “You want to wash your hands” ? “the bathroom is first on the left”

38. Problematic cases “Eva wants a black and white cat.” “Eva wants a black cat” ? “Eva wants a white cat” “Geller can read your mind or he can bend spoons” “Geller can read your mind” v “Geller can bend spoons” “If you want to wash your hands, the bathroom is first on the left” “You want to wash your hands” ? “the bathroom is first on the left”

39. Problematic cases “Eva wants a black and white cat.” “Eva wants a black cat” ? “Eva wants a white cat” “Geller can read your mind or he can bend spoons” “Geller can read your mind” v “Geller can bend spoons” “If you want to wash your hands, the bathroom is first on the left” “You want to wash your hands” ? “the bathroom is first on the left”

40. Problematic cases “Eva wants a black and white cat.” “Eva wants a black cat” ? “Eva wants a white cat” “Geller can read your mind or he can bend spoons” “Geller can read your mind” v “Geller can bend spoons” “If you want to wash your hands, the bathroom is first on the left” “You want to wash your hands” ? “the bathroom is first on the left”

41. Translate these in logic Neither I nor my wife speak Russian. If I am not Italian then I am not allowed to play for the Italian football team. You will get a room provided you have no pets. Diabolik will not fail to find the diamonds.

42. Semantic relations

43. Semantic relations

44. Different Logics propositional logic modal logic description logic first-order logic (predicate logic) second-order logic higher-order logic

45. Logics and how they relate

46. Logics and how they relate

47. Logics and how they relate

48. Logics and how they relate

49. Why different logics? Why don’t we take the most expressive logic and use that to analyse semantics? Answer: different logics have different computational properties There exists an algorithm to decide whether a formula is a validity (a theorem) for propositional and modal logic But there is no such algorithm for first-order logic (or higher-order logic) [In logical terms: first-order logic “is undecidable”]

50. A note on notation… Negation: ? or ? Conjunction: ? or & Implication: ? or ? Equivalence: ? or ? Brackets: (…) or […]

51. Summary

52. Truth Tables 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

53. 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))

54. 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)

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

56. Using a truth-table Example: (p??p) 2) Add columns for all sub-formulas

57. Using a truth-table Example: (p??p) 3) Put the formula itself in the last column

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

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

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

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

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

63. Using a truth-table Example: (p??p) 5) Check the values in the last column

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

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

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

67. Another example Example: (p ? q) 4) Fill in the truth values

68. Another example Example: (p ? q) 5) Check values in last column

69. Which of the following are tautologies? (p?(p?q)) (p?(q??r)) ((p?q)?p) ((?p?q)?(p?q))

70. 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.

71. Entailment Let S be a sentence and S' the logical translation of S. Then:

72. 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

73. 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

74. Consistency If someone supports Inter today, and Milan tomorrow, that is clearly inconsistent, but not the logically kind of consistency that we are interested in A set of formulas is consistent if they are true in at least one situation

75. 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.

76. 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.

77. Entailment or not? Milan is a more expensive city than Rome. Rome is an expensive city.

78. Entailment or not? All churches in Rome are beautiful. All old churches in Rome are beautiful.

79. Further reading Cann (1993): Formal Semantics; An introduction, Chapter 7 Hodges (1977): Logic. An introduction to elementary logic. Hurford & Heasley (1983): Semantics. A coursebook, Unit 10 Lyons (1977): Semantics, Volume 1, Chapter 6

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