November 21, 2003

Section notes

Let’s establish the coherence of the following discourse, and determine the pronoun reference while we’re at it.

The discourse:

(1) John ate Bill’s peaches.

(2) He went ballistic.

The general coherence axioms:

If one utterance results from another, there’s a coherence relation:

(3) "e_i, e_j Result(e_i, e_j) Þ CoherenceRel(e_i, e_j)

If one utterance causes another, that second one results from the first:

(4) "e_i, e_j cause(e_i, e_j) Þ Result(e_i, e_j)

Causation is transitive:

(5) "e_i, e_j, e_k cause(e_i, e_j) Ù cause(e_j, e_k) Þ cause(e_i, e_k)

Real-world axioms:

If somebody has peaches, they want to eat them:

(6) "e_i, x, y Have(e_i, x, y) Ù Peaches(y) Þ $e_j, e_k Want(e_j, x, e_k) Ù Eat(e_k, x, y)

If somebody eats something, that causes it to be gone:

(7) "e_i, x, y Eat(e_i, x, y) Þ $e_j Gone(e_j, y) Ù cause(e_i, e_j)

If something is gone, that makes it impossible to eat it:

(8) "e_i, e_j, x, y Gone(e_i, x) Ù Eat(e_j, y, x) Þ $e_k Impossible(e_k, e_j) Ù cause(e_i, e_k)

If somebody wants something that’s impossible, they go ballistic

(9) "e_i, e_j, e_m, x Want(e_i, x, e_j) Ù Impossible(e_m, e_j) Þ $e_k GoBallistic(e_k, x) Ù cause(e_m, e_k)

The content of the utterances themselves:

From (1):

(10) Eat(e₁, John, p) Ù Peaches(p) Ù Have(e₂, Bill, p)

From (2):

(11) GoBallistic(e₃, he)

Begin hypothesizing (abduction):

Assume we have a Coherence relation:

(12) Coherence(e₁, e₃)

From (12) & (3), we can hypothesize a Result relation:

(13) Result(e₁, e₃)

From (13) & (4), hypothesize that (1) caused (2)

(14) cause(e₁, e₃)

Start making deductions:

From (10) & (6), deduce that Bill wants to eat the peaches:

(15) Want(e₄, Bill, e₅) Ù Eat(e₅, Bill, p)

From (7) & (10), deduce that the peaches are gone:

(16) Gone(e₆, p) Ù cause(e₁, e₆)

From (8) and (16), deduce that it is impossible for Bill to eat the peaches:

(17) Impossible(e₇, e₅) Ù cause(e₆, e₇)

From (5), (16) & (17), deduce that John’s eating the peaches caused it to be impossible for Bill to eat them:

(18) cause(e₁, e₇)

From (9), (15) & (17), deduce that Bill goes ballistic:

(19) GoBallistic(e₈, Bill) Ù cause(e₇, e₈)

From (5), (18) & (19), deduce that John’s eating the peaches caused Bill to go Ballistic:

(20) cause(e₁, e₈)

If we replace he in (11) with Bill, we can assert that e₃ = e₈, and recast (19):

(21) GoBallistic(e₃, Bill) Ù cause(e₁, e₃)

Which is proof of what we hypothesized, establishing a coherence relation given a particular resolution for the pronoun he.