semantics and inference part ii johan bos. summary of last lecture inferences on the sentence level...
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Semantics and Inference
Part II
Johan Bos
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Summary of last lecture
• Inferences on the sentence level– Entailment– Paraphrase– Contradiction
• Using logic to understand semantics– Introduction to propositional logic– Syntax– Semantics
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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
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Ingredients of propositional logic
• Propositional variables– Usually: p, q, r, …
• Connectives– The symbols: , ,, , – Often called logical constants
• Punctuation symbols– The round brackets ( )
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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
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Negation
• Symbol: • Pronounced as: “not” is called the negation of
• Truth-table:
True False
False True
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Conjunction
• Symbol: • Pronounced as: “and”
• () is called the conjunction of the conjuncts and
• Truth table: ()
True True True
True False False
False True False
False False False
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Disjunction
• Symbol: • Pronounced as: “or”
• () is called the disjunction of the disjuncts and
• Truth table:
()
True True True
True False True
False True True
False False False
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(Material) Implication
• Symbol: • Pronounced as: “implies” or “arrow”
• Truth table: ()
True True True
True False False
False True True
False False True
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Equivalence (biconditional)
• Symbol:
• Pronounced as: “if and only if”
• Truth table: ()
True True True
True False False
False True False
False False True
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Summary
()
True True True
True False False
False True False
False False True
()
True True True
True False False
False True True
False False True
()
True True True
True False True
False True True
False False False
()
True True True
True False False
False True False
False False False
True False
False True
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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
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Tautologies
• A formula that is true in all situations is called a tautology or a semantic theorem
• Examples of tautologies: (pp)(qq)(pp)(p(qp))(p(qp))((pq)((pq)(q p))
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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)
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Using a truth-table
Example: (pp)
1) Make a column for all propositional variables with possible truth-values
p
True
False
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Using a truth-table
Example: (pp)
2) Add columns for all sub-formulas
p p
True
False
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Using a truth-table
Example: (pp)
3) Put the formula itself in the last column
p p (pp)
True
False
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Using a truth-table
Example: (pp)
4) Fill in the truth values for the columns using the tables of the connectives
p p (pp)
True ?
False
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Using a truth-table
Example: (pp)
4) Fill in the truth values for the columns using the tables of the connectives
p p (pp)
True False
False ?
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Using a truth-table
Example: (pp)
4) Fill in the truth values for the columns using the tables of the connectives
p p (pp)
True False ?
False True
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Using a truth-table
Example: (pp)
4) Fill in the truth values for the columns using the tables of the connectives
p p (pp)
True False True
False True ?
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Using a truth-table
Example: (pp)
4) Fill in the truth values for the columns using the tables of the connectives
p p (pp)
True False True
False True True
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Using a truth-table
Example: (pp)
5) Check the values in the last column
p p (pp)
True False True
False True True
All true in this column, hence tautology
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Another example
Example: (p q)
1) Make a column for all propositional variables with possible truth-values
p q
True True
True False
False True
False False
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Another example
Example: (p q)
2) Add columns for sub-formulas
p q
True True
True False
False True
False False
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Another example
Example: (p q)
3) Add formula itself in last column
p q (p q)
True True
True False
False True
False False
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Another example
Example: (p q)
4) Fill in the truth values
p q (p q)
True True True
True False False
False True True
False False True
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Another example
Example: (p q)
5) Check values in last column
p q (p q)
True True True
True False False
False True True
False False True
Not all true in this column, hence no tautology
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Which of the following are tautologies?
1) (p(pq))
2) (p(qr))
3) ((pq)p)
4) ((pq)(pq))
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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.
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Entailment
• Let S be a sentence and S' the logical translation of S. Then:
If (S1'S2') is a tautology, then S1 entails S2
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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
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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
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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.
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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.
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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
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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
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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
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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)
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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
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Borderline cases
• There is no precise cut-off point between small and not small
• Both scaling and non-scaling adjectives can have borderline cases
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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.
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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.
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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.
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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
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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
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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
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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.
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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
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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)
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Strong readings
• and implies and then in narratives
• some implies not all
• all implies at least one
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Weak readings
• Omitting to mention something which any honest person would have mentioned, but not by saying something untrue
• Imagine Diabolik decided to be completely honest to Eva:
I did kiss some of the girls, in fact I kissed all nineteen of them
• Does Diabolik contradicts himself or not?
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Entailment or not?
• Milan is more expensive city than Rome.• Rome is an expensive city.
• Pisa is a small town with a leaning tower.• Pisa is a town with a leaning tower.
• Bologna is the cultural capital of Italy.• Bologna is the capital of Italy.
• Milan is a smaller city than Rome.• Milan is a small city.
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Entailment or not?
• All churches in Rome are beautiful.• All old churches in Rome are beautiful.
• Some churches in Rome are beautiful.• Some old churches in Rome are beautiful.
• Turin has the highest tower of Italy.• Turin has the highest leaning tower of Italy.
• Pisa has the highest leaning tower of Italy.• Pisa has the highest tower of Italy.
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Entailment Test Suites
• Fracas Deliverable D16• Pascal Recognising Textual Entailment
– RTE-1– RTE-2– RTE-3
• Microsoft Research Paraphrase Corpus
• See:http://homepages.inf.ed.ac.uk/jbos/rte/
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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