Introduction to DynamicSemantics

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Introduction to Dynamic SemanticsNorihiro OgataOsaka UniversityWhat is a logical language? Term(項, 名辞)denotes individuals, numbers, and other elements.can have complex structures:a+b, (a+b)-c, f(a), g(f(a)), …can be or contain variables(変数,変項) x, (a+x)-y, f(x), g(f(x)), …x,y∈Varcan be an individual constant (定数,定項) a, b, c, d, …∈ConTermt∈What is a logical language?Formula (式, 文)denotes a proposition (命題)can have complex structures: can be a logical constant (論理定数)Fma ϕ∈,⊥⊤12(),x x ϕϕϕ∧∃What is a logical language? More Formally, Terms are defined by a BNF grammar:1,,()::||(,...,)|(.)n x Var c Con f Fun t x c f t t x ιϕ∈∈∈=What is a logical language? More Formally, Formulas are defined by a BNF grammar:R(t1,…,tn) and t1=t2 are called atomicformulas 12112121212()::()|(,...,)|||||.|.|||n R Rel t t R t t x x ϕϕϕϕϕϕϕϕϕϕϕϕ∈==∧∨→¬∀∃↔⊥⊤What is a logical language? Example:.(()).((,)(,))()(,)(,)x f x c x R a x R b x P x R a b R c b ∃¬=∀∧→∨language denote?Terms denotes individuals in the model . Formulas denotes Truth Values {1,0} Dis a set of individuals.,,Model D =M ,,:()({1,0})g Term D Fma →∪→ Mlanguage denote?Variables are assigned their values of individuals in the model by variable assigments. 01010102,,......()(,,...)(/),,...,[/],...,,,......i i ii i g d d D D g x d d d g e x d d e d g d d D D π∞∞⊥⊥=∈××====⊥∈××order languageThe static semantics of First-Order Language is defined by recursion on the complexity of the expressions:,,,,,,11,()(,...,)(,...,)(...)g g g g g g n n n times g x g x c Df t t f t t f D D D −=∈=∈××→M M M M M M Morder languageThe static semantics of First-Order Language is defined by recursion on the complexity of the expressions:,,,1212,,,,11,1(,...,)1(,...,)...=g g g g g g g n n n times g t t t t R t t t t R R D D −==⇔=⇔∈⊆××M M M M M M M Morder languageThe static semantics of First-Order Languageis defined by recursion on the complexity of the expressions:,,,1212,,,1212,,,1212,,,12121111111==1 or =0 or = g g ggggg g ggggϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕ∧=⇔=∨=⇔=→=⇔=↔=⇔M M M M M M M M M M M Morder languageThe static semantics of First-Order Languageis defined by recursion on the complexity of the expressions:,,,,11=0ggggϕϕ¬=⇔=⊥=M M M M ⊤order languageThe static semantics of First-Order Languageis defined by recursion on the complexity of the expressions:,,(/),,(/).1.1=1 for some =1 for all gg d x gg d x x d D x d D ϕϕϕϕ∃=⇔∈∀=⇔∈M M M Morder languageExample,,,,,,,..1..11,, (1)g ggggx y x y x y y x x y z x z z y =∀∃<=∀∃=+==∀∀∃<∧<=ℕℝM M M M M M MDonkey sentencesEvery farmer who owns a donkey beats it.If a farmer owns a donkey, he beats it.Donkey sentencesE-type reading: Every farmer who owns a donkey beats the donkey he owns.Universal reading: For every farmer and every donkey, if he owns it, he beats it.Weak reading: Every farmer who owns a donkey beats a donkey he owns.Strong reading: Every farmer who owns a donkey beats every donkey he owns.Donkey sentencesDirect translation:E-type reading :Universal reading :Weak reading :Strong reading :.()(.()(,))(,)x F x y D y O x y B x y ∀∧∃∧→.()(.()(,))(,(.()(,)))x F x y D y O x y B x z D y O x y ι∀∧∃∧→∧..()()(,)(,)x y F x D y O x y B x y ∀∀∧∧→.()(.()(,))(.()(,)(,))x F x y D y O x y y D y O x y B x y ∀∧∃∧→∃∧∧.()(.()(,))(.()(,)(,))x F x y D y O x y y D y O x y B x y ∀∧∃∧→∀∧→Donkey sentencesThe Problem of Direct translation:y is unbound (free)Desired translation ::.()(.()(,))(,)x F x y D y O x y B x y ∀∧∃∧→..()()(,)(,)x y F x D y O x y B x y ∀∀∧∧→Order Language,() if if gAsg Asg Fma D Termαααα⊆×∈∈∈MMorder language12,,121,,,1''&(,...,)''&(,...,)=ggn gggn g t t g g g t t g R t t g g g t t R =⇔=⇔=∈MM M M M M Morder language12,121212''''&'''''''&'''''' for some or for some g g g g g g g g g g g g g g g g g ϕϕϕϕϕϕϕϕ∧⇔∨⇔=M M M MM Morder language1212121221''&'':'',''','''''''&'':'',''','''''&'':'',''','''''for all for som e for all for som e for all for som e g g g gg g g g g g g g g gg g g g g g g g g g g g ϕϕϕϕϕϕϕϕϕϕ→⇔=↔⇔= MM M M M M M Morder language''&''..''','','there is no For all There is no g g g gg s t g g g g g g g g ϕϕ¬⇔=⊥ MM M M ⊤order language.'(/)'.''&,'',(/)'' for som e for all for som e g x g g d x g d D g x g g gd D g g d x g ϕϕϕϕ∃⇔∈∀⇔=∈ M MM Morder language(.())()''',.()''&''()''',,(/)()''&''()',(/)()(/)&(/)()'.()()'for som e for som e for som e P R g x P x R x g g g x P x g g R x g g d D g d x P x g g R x g d D g d x P x g d x g d x R x g g x P x R x g ∩≠∅∃∧⇔∃⇔∈⇔∈⇔∃∧ M MM M M M M M M Example 1telescoping ∃order language()(.())''',()''&''.()''',()''&,''(/)()'()&,(/)()(/),(/)()(/)&(for som e for som e for som e for som e for som e g P x xR x g g g P x g g x P x g g g P x g d D g d x R x g g P x g d D g d x R x g d x d D g d x P x g d x g d ∧∃⇔∃⇔∈⇔∈≠∈ MM M M M M M M /)()(/)',(.())()' for som e x R x g d x g g x P x R x g ⇔∃∧ M M Example 2 (non-commutativity ∧)order language(.())()':.()','':'()''',:(/)()','':'()'',',(/)()','':'()for all for som e for all for som e for som e for all for all for som e g x P x R x gg g x P x g g g R x g g d D g d x P x g g g R x g d D g g d x P x g g g R x ∃→⇔∃⇔∈⇔∈ MM M M M M M ',(/)()()(/).()()for all g d D g d x P x R x g d x g x P x R x g ⇔∈→⇔∀→ M M Example 3 (first-order invalid equivalence)Dynamic Semantics of DonkeysentencesThe Direct translation:y is boundequals to the desired one.()(.()(,))(,) x F x y D y O x y B x y ∀∧∃∧→..()()(,)(,)x y F x D y O x y B x y∀∀∧∧→Exceptions to this dynamic semanticsJohn doesn’t own a car, and he drives it on Sunday.Either there is no bathroom in this house or it’s in a funny place.If a theory is inconsistent, it isn’t necessarily trivial. If it is classical, it will be.The Basic BibliographyJ. Barwise (1984) “Noun Phrases, Generalized Quantifiers and Anaphora”, in P.G”ardenfors ed., Generalized Quantifiers. Reidel: Dordrecht, pp. 1-29.J. Groenendijk & M. Stokhof (1991) “Dynamic Predicate Logic”, Linguistics and Philosophy14:39-100.。