Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
sentence: every student sings
Q A B
derivation: sings every student
B Q A
14.1 Monotonicity inferences for subject-predicate
And this sentence is true iff 〈A, B〉 ∈Q.
It is now easy to represent sound patterns of inference for different kinds of quantifiers.
B(Q(A)) C(every(B))
[Q ↑] (for any Q ↑: all, most, the, at least N, infinitely many,...)
C(Q(A))
B(Q(A)) B(every(C))
C(Q(A))
[Q↓] (for any Q ↓: no, few, fewer than N, at most N,...)
B(Q(A)) C(every(A))
B(Q(C))
[↑Q] (for any ↑Q: some, at least N, ...)
B(Q(A)) A(every(C))
[↓Q] (for any ↓Q: no, every, all, at most N, at most finitely many,...)
B(Q(C))
Example: Aristotle noticed that “Darii syllogisms” like the following are sound:
Some birds are swans All swans are white
Therefore, some birds are white
We can recognize this now as one instance of the Q↑ rule:
birds(some(swans)) white(every(swan))
white(some(birds))
The second premise says that the step from the property [bird] to [white] is an “increase,” and since we know
some is increasing in its second argument, the step from the first premise to the conclusion always preserves
truth.
Example: We can also understand the simplest traditional example of a sound inference:
Socrates is a man
All men are mortal
Therefore, Socrates is mortal
Remember we are interpreting Socrates as denoting a quantifier! It is the quantifier that maps a property to
true just in case Socrates has that property. Let’s call this quantifier socrates. Then, since socrates↑ we
have just another instance of the Q↑ rule:
socrates(man) mortal(every(man))
socrates(mortal)
Example: Aristotle noticed that “Barbara syllogisms” like the following are sound:
[Q↑]
All birds are egg-layers
All seagulls are birds
Therefore, all seagulls are egg-layers
Since the second premise tells us that the step from [birds] to [seagulls] is a decrease and ↓all, we can recognize
this now as an instance of the ↓Q rule:
egg−layer(all(bird)) bird(every(seagull))
egg−layer(all(seagull))
238
[Q↑]
[↓Q]