Mathematical Logic and Linguistics
Glyn Morrill & Oriol Valent´ın
Department of Computer Science Universitat Politecnica` de Catalunya [email protected] & [email protected]
BGSMath Course Class 10 ”The Montague Test” is the challenge of providing a computational cover grammar of the Montague fragment.
Syntactic and Semantic Analyses: the PTQ Fragment Syntactic and Semantic Analyses: the PTQ Fragment
”The Montague Test” is the challenge of providing a computational cover grammar of the Montague fragment. The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality I coordination
Montague Grammar I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment: I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics The fragment includes:
I quantification I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) I quantification I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes: I anaphora I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I intensionality I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I coordination
Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality Montague Grammar
The Montague fragment:
I overturned the view that semantics cannot be formalised I generated a paradigm of logical semantics I gave rise to a field of formal semantics (L&P, A’dam Colloquium, and for forth) The fragment includes:
I quantification I anaphora I intensionality I coordination CatLog2 is a type logical parser/theorem prover. It:
I comprises 6000 lines of prolog I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − ×
CatLog2 I comprises 6000 lines of prolog I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − ×
CatLog2
CatLog2 is a type logical parser/theorem prover. It: I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − ×
CatLog2
CatLog2 is a type logical parser/theorem prover. It:
I comprises 6000 lines of prolog I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − ×
CatLog2
CatLog2 is a type logical parser/theorem prover. It:
I comprises 6000 lines of prolog I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − ×
CatLog2
CatLog2 is a type logical parser/theorem prover. It:
I comprises 6000 lines of prolog I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × CatLog2
CatLog2 is a type logical parser/theorem prover. It:
I comprises 6000 lines of prolog I has 20 primitive categorial connectives, 29 defined connectives, and 1 metalogical connective: a total of 50 I has typically 2 rules for each connective: a rule of use and a rule of proof: roughly 50 2 = 100 rules × I uses backward chaining focused sequent proof search so that for a binary connective for half of the rules there are 4 cases: +/+, +/ , /+, / : 50 4 + 50 = a total of about 250 focused− rules− − − × Concretely, we define the Montague test as the task of:
semantically parsing the mini-corpus comprising the example sentences of Chapter Seven of Dowty et al. (1981)
as a baseline criterion for grammar formalisms.
I.e. we propose as a sine qua non being able to simulate computationally the Montague syntax-semantics interface of quantification, anaphora, intensionality, coordination, . . .
The Montague Test semantically parsing the mini-corpus comprising the example sentences of Chapter Seven of Dowty et al. (1981)
as a baseline criterion for grammar formalisms.
I.e. we propose as a sine qua non being able to simulate computationally the Montague syntax-semantics interface of quantification, anaphora, intensionality, coordination, . . .
The Montague Test
Concretely, we define the Montague test as the task of: as a baseline criterion for grammar formalisms.
I.e. we propose as a sine qua non being able to simulate computationally the Montague syntax-semantics interface of quantification, anaphora, intensionality, coordination, . . .
The Montague Test
Concretely, we define the Montague test as the task of:
semantically parsing the mini-corpus comprising the example sentences of Chapter Seven of Dowty et al. (1981) I.e. we propose as a sine qua non being able to simulate computationally the Montague syntax-semantics interface of quantification, anaphora, intensionality, coordination, . . .
The Montague Test
Concretely, we define the Montague test as the task of:
semantically parsing the mini-corpus comprising the example sentences of Chapter Seven of Dowty et al. (1981)
as a baseline criterion for grammar formalisms. The Montague Test
Concretely, we define the Montague test as the task of:
semantically parsing the mini-corpus comprising the example sentences of Chapter Seven of Dowty et al. (1981)
as a baseline criterion for grammar formalisms.
I.e. we propose as a sine qua non being able to simulate computationally the Montague syntax-semantics interface of quantification, anaphora, intensionality, coordination, . . . Taking on the Montague Test
We invoke CatLog2 on this mini-corpus . . . str(dwp(’(7-7)’), [b([john]), walks], s(f)). str(dwp(’(7-16)’), [b([every, man]), talks], s(f)). str(dwp(’(7-19)’), [b([the, fish]), walks], s(f)). str(dwp(’(7-32)’), [b([every, man]), b([b([walks, or, talks])])], s(f)). str(dwp(’(7-34)’), [b([b([b([every, man]), walks, or, b([every, man]), talks])])], s(f)). str(dwp(’(7-39)’), [b([b([b([a, woman]), walks, and, b([she]), talks])])], s(f)). str(dwp(’(7-43, 45)’), [b([john]), believes, that, b([a, fish]), walks], s(f)). str(dwp(’(7-48, 49, 52)’), [b([every, man]), believes, that, b([a, fish]), walks], s(f)). str(dwp(’(7-57)’), [b([every, fish, such, that, b([it]), walks]), talks], s(f)). str(dwp(’(7-60, 62)’), [b([john]), seeks, a, unicorn], s(f)). str(dwp(’(7-73)’), [b([john]), is, bill], s(f)). str(dwp(’(7-76)’), [b([john]), is, a, man], s(f)). str(dwp(’(7-83)’), [necessarily, b([john]), walks], s(f)). str(dwp(’(7-86)’), [b([john]), walks, slowly], s(f)). str(dwp(’(7-91)’), [b([john]), tries, to, walk], s(f)). str(dwp(’(7-94)’), [b([john]), tries, to, b([b([catch, a, fish, and, eat, it])])], s(f)). str(dwp(’(7-98)’), [b([john]), finds, a, unicorn], s(f)). str(dwp(’(7-105)’), [b([every, man, such, that, b([he]), loves, a, woman]), loses, her], s(f)). str(dwp(’(7-110)’), [b([john]), walks, in, a, park], s(f)). str(dwp(’(7-116, 118)’), [b([every, man]), doesnt, walk], s(f)). Categorial connectives
limited cont. disc. norm. brack. contr. mult. mult. add. qu. mod. mod. exp. & weak. V / & [] 1 ! \ ↑ ↓ − | primary 2 •I J W ? W ⊕ hi 3 (
sem. ( ( inactive u ∀ G# H# variants t ∃ G# H#
1 1 det. /− .− ˇ diff. synth. /. ˆ
nondet. ÷ ⇑ ⇓ synth. } − × a: g( f((Sf Nt(s(g))) Sf)/CNs(g)) : λAλB C[(AC) (BC)] ∀ ∀ ↑ 1 1 ↓ n+ ∃ ∧ and: f((?Sf []− []− Sf)/Sf) : (Φ 0 and) ∀ \ 1 1 n+ and: a f((?( Na Sf) []− []− ( Na Sf))/( Na Sf)) : (Φ (s 0) and) believes∀ :∀(( gNthi (s\(g))\Sf)/(CPthathi \Sf)) : ˆhiλAλ\B(Pres ((ˇbelieve A) B)) bill: Nt(s(mhi∃)) : b \ t catch: (( aNa Sb)/ aNa): ˆλAλB((ˇcatch A) B) doesnt: hi∃g a((Sg\ (( ∃Na Sf)/( Na Sb))) Sg): λA (A λBλC(BC)) eat: (( ∀aNa∀ Sb)/↑ aNahi ):\ ˆλAλhiB((ˇ\eat A)↓B) ¬ every: hi∃g( f((\Sf Nt∃(s(g))) Sf)/CNs(g)) : λAλB C[(AC) (BC)] finds: ((∀ ∀gNt(s↑(g)) Sf)/ ↓aNa): ˆλAλB(Pres ((∀ˇfind A) B))→ fish: CNshi∃(n): fish \ ∃ 1 he: []− g((Sg Nt(s(m)))/( Nt(s(m)) Sg)) : λAA her: g ∀a((( Na| Sg) Nt(s(fhi))) (( Na\ Sg) Nt(s(f)))) : λAA in: ( ∀a ∀f(( Nahi Sf\ ) ( ↑Na Sf))/ ↓aNa):hi ˆλ\AλB|λC((ˇin A)(BC)) is: ((∀ ∀gNthi(s(g\)) Sf\ )hi/( aNa\ ( ∃g((CNg/CNg) (CNg CNg)) I))) : λAλB(Pres (A C.[B = C]; D.((hi∃D λE[E = B\]) B)))∃ ⊕ ∃ t \ − → john: Nt(s(m)) : j loses: (( gNt(s(g)) Sf)/ aNa): ˆλAλB(Pres ((ˇlose A) B)) loves: ((hi∃gNt(s(g))\Sf)/∃aNa): ˆλAλB(Pres ((ˇlove A) B)) man: CNshi∃(m): man \ ∃ necessarily: (SA/SA): Nec 1 1 n+ or: f((?(Sf/( gNt(s(g)) Sf)) []− []− (Sf/( gNt(s(g)) Sf)))/(Sf/( gNt(s(g)) Sf))) : (Φ (s 0) or) park:∀CNs(n): hi∃park \ \ hi∃ \ hi∃ \ seeks: (( gNt(s(g)) Sf)/ a f(((Na Sf)/ bNb) (Na Sf))) : ˆλAλB((ˇtries ˆ((ˇA ˇfind) B)) B) slowly: hi∃a f(( Na Sf\ ) ( ∀Na∀ Sf)) :\ˆλAλ∃B(ˇslowly\ ˆ\(ˇA ˇB)) such+that∀: ∀ n((hiCNn\CNn\)hi/(Sf \Nt(n))) : λAλBλC[(BC) (AC)] talks: ( gNt∀ (s(g)) \Sf): ˆλA(Pres| (ˇtalk A)) ∧ that: (CPthathi∃ /Sf):\λAA the: n(Nt(n)/CNn): ι to: ((∀PPto/ aNa) n(( Nn Si)/( Nn Sb))) : λAA unicorn: CNs∃ (n):u∀unicornhi \ hi \ walks: ( gNt(s(g)) Sf): ˆλA(Pres (ˇwalk A)) woman: hi∃CNs(f): woman\