A Platonic Paradox
#John #Colarusso
I came into linguistics with two degrees in philosophy, in which discipline I had studied a fair bit of logic. Because of this background the theory of the #language #acquisition #device (LAD) as conceived by Chomsky at the time seemed flawed to me.
#Chomsky argued more or less that the #LAD was a sort of super grammar, one that took a language input (from parent to child) and formulated the rules (grammar) for that language on the basis of (super) grammatical rules.
While this seemed a reasonable position I knew that it could not work; Chomsky had stumbled into the #halting #problem.
Chomsky’s LAD was a device that matched grammatical rules to an input, in effect positing an “algorithm,” a coherent way of establishing a set of rules for a language “input.” One might frame it as a question,
(1):
QLAD: Does G, a grammar, applied to an utterance, U, have a derivation, Δ?
This may be put into a close association (an isomorphism, one for one match) with the “halting” question,
(2):
QW: Does M applied to a ‘M’ eventually stop on ‡?
Here M is a “Turing machine,” a formal mechanism than can calculate (derive) anything that can be derived by an algorithm, and ‘M’ is some string, some formal object that must be derived by some procedure (read “algorithm”). The diamond, ‡, is the “halting symbol,” signifying that M has in fact reached an end, that is, that it has in fact derived.
As Crossley et al. show this is an unsolvable problem
https://inference-review.com/letter/a-platonic-paradox