Logical Representations. LING 7800/CSCI 7000 Martha Palmer 9/30/2014
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1 Logical Representations LING 7800/CSCI 7000 Martha Palmer 9/30/2014 1
2 The Semantic Wall Physical Symbol System World +BLOCKA+ +BLOCKB+ +BLOCKC+ P 1 :(IS_ON +BLOCKA+ +BLOCKB+) P 2 :((IS_RED +BLOCKA+)
3 Truth depends on Interpretation Representation 1 World A B ON(A,B) T ON(A,B) F ON(A,B) F A ON(A,B) T B
4 Logical Interpretations Representation: Sentences entails Sentence Refers to (semantics) follows World: Facts Fact
5 Logic Truth. A sentence is true if the state of affairs it describes is actually the case in the world. Valid. A sentence is valid if it s true no matter what the world is actually like or what the semantics is. tautologies or analytic truths.
6 Two KR Languages Representation Language Inference Procedure Propositional Logic Unit Resolution First Order Logic General Resolution (Basis for Prolog)
7 Propositional Logic Syntax. Sentences T True, F False P, Q, etc. (Sentence), Sentence Connective Sentence, Sentence Connectives negation, conjunction, disjunction, implication, also (, ) equivalence (biconditional implication)
8 Truth Tables P: The sun is shining Q: The day is warm R: The sun is shining and the day is warm. P Q P /\ Q F F T T F T F T 8
9 Propositional Logic Semantics. Specifies what facts in the world sentences refer to. P. Richard is a pacifist, Q. Richard is a Quaker. Truth Table gives an interpretation P Q P P /\ Q P \/ Q P Q F F T F F T F T T F T T T F F F T F T T F T T T
10 Unit Clauses P Q is equivalent to P \/ Q P Q P P /\ Q P \/ Q P Q P\/ Q F F T F F T F T T F T T T F F F T F T T F T T T
11 Flourishes Exclusive OR You will pay the fine or you will go to jail Bi-conditionals If and only if P Q /\ Q P; P Q /\ P Q A triangle is isosceles if and only if the triangle has two congruent (equal) sides. We ll leave if and only if we re forced to. 11
12 Flourishes, cont. Inclusive \/; exclusive \/ We spend the afternoons swimming or sunbathing. They can resuscitate him or allow him to die. If the site is in a particularly sensitive area, or there are safety considerations, we can refuse planning permission. 12
13 Linguistic intuitions about logic? If Patricia goes to the party then Emmet will go too. If wishes were money then we d all be rich. If I were an ostrich, then I would be a bird. If I were an ostrich, then I would not be a bird. 13
14 Linguistic intuitions about and P /\ Q equivalent to Q /\ P? He woke up and saw on TV that he had won the lottery. Combine the egg yolks with water in a bowl and whisk the mixture until foamy. He made two false starts and was disqualified from the race. Move and I ll shoot! 14
15 Necessary/analytic truth of propositions (sentences) Logically true by virtue of the meaning of the terms Either the Buffs will beat Florida State or the Buffs will not beat Florida State. P \/ P If the Buffs beat Florida State then Florida State loses to the Buffs. Reversal All teams who win are teams. Quantifiers The train will either arrive or it won t arrive. 15
16 Entailment 4.6 P entails Q when the truth of P guarantees the truth of Q, and the falsity of Q guarantees the falsity of P. Composite Truth Table for entailment: P T F F Q T T or F F Tor F T 16
17 Entailment P entails Q when the truth of P guarantees the truth of Q, and the falsity of Q guarantees the falsity of P. Composite Truth Table for entailment: P T F F Q T T or F F Tor F T 17 P Q P Q F F T F T T T F F T T T
18 Entailment VS. Bi-conditional P Q P s truth is sufficient for Q s truth Q s truth is necessary for P s truth P Q P s truth is necessary AND sufficient for Q s truth, and vice versa 18
19 Examples Q is necessary for P, P is sufficient for Q Human Mammal Jump Leave_ground Prime (but not 2) Odd Senator 30_years_old P is necessary and sufficient for Q, IFF A person is a bachelor iff that person is a marriageable man who has never married. 19
20 Entailment Examples The anarchist assassinated the emperor. P The emperor died. Q P Q If P is true is Q true? If Q is False, is P false? If P is false, then what about Q? 20
21 Sources for Entailments Hyponyms I bought a dress today. P I bought clothes today. Q Syntax Caesar won the war. P The war was won by Caesar. Q Paraphrases mutually entail each other 21
22 And now also. Necessarily true tautologies Venus is Venus Necessarily false contradictions Here is not here. Presuppositions The King of France is bald. presupposes There is a King of France. 22
23 Presuppositions Her husband is a drunk. She has a husband I don t regret leaving Philadelphia I left Philadelphia The Prime Minister of Pakistan is in negotiations with the US. Pakistan has a Prime Minister. 23
24 Presuppositions as a Truth Relation If P is true then Q (the presupposition) is true. If P is false then Q is still true. If Q is true, P could be true or false Composite Truth Table for presupposition: P T F Q T T T or F T 24
25 Entailment P entails Q when the truth of P guarantees the truth of Q, and the falsity of Q guarantees the falsity of P. Composite Truth Table for entailment: P T F F Q T T or F F Tor F T 25 P Q P Q F F T F T T T F F T T T
26 Entailments compared to Presuppositions as Truth Relations Negate the entailing sentence, the entailment fails I saw my father today/i saw someone today. I didn t see my father today/i didn t see someone today. Negate a presupposing sentence, the presupposition still holds The mayor of Denver is/is not in Boulder. Denver has a mayor 26
27 Unit Clauses P Q is equivalent to P \/ Q P Q P P /\ Q P \/ Q P Q P\/ Q F F T F F T F T T F T T T F F F T F T T F T T T
28 And Elimination Inference Rules α 1 α 1 α 2,... α n And Introduction α 1,..., α n α 1 α 2,... α n
29 Inference Rules (cont d) Or Introduction α i α 1 α 2,... α i α n Double Negation Elimination α α
30 Inference Rules (cont d) Modus Ponens (Implication Elimination) α β,α β
31 De Morgan s laws (P /\ Q) ( P) \/ ( Q) "not (A and B)" is the same as "(not A) or (not B) The negation of a conjunction is the disjunction of the negations (P \/ Q) ( P) /\ ( Q) "not (A or B)" is the same as "(not A) and (not B) The negation of a disjunction is the conjunction of the negations. 31
32 Example Search A: NOT (cats OR dogs) Search B: (NOT cats) AND (NOT dogs) Document 1: Contains only the word cat(s). Document 2: Contains only dog(s). Document 3: Contains both cat(s) and dog(s). Document 4: Contains neither cat(s) nor dog(s). 32
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