Supply exceeds demand in research related to donkey sentences, and posts like this only drive the price further down. So, I will say this: I have no answers of my own to the intractable donkey sentence problem, but I also don’t share the dilemmas which others readily admit. Instead, I have my own donkey-sentence-related dilemmas.
In a nutshell, donkey sentences are like (1):
(1) If a farmer owns a donkey, he beats it.
The pronouns ‘he’ and ‘it’ are anaphoric on ‘farmer’ and ‘donkey’, and the latter pair especially has polarized philosophers, logicians and linguists. Is the sentence true of, say, ten farmers, each of whom has one donkey which he beats, or is it true of ten farmers, each of whom has an unspecified number of donkeys of which he beats every single donkey that he owns.
By convention, first-order logic follows the second approach. It is convenient because the use of a universal quantifier allows us to bind the variable ‘y’ in the predicate expression ‘Bxy’.
(x){Fx ⊃ (y)[(Dy • Oxy) ⊃Bxy]}
It suits me. The indefinite article ‘a’ in English is called so for a reason. It is intended to capture both ‘one’ and ‘every’ of the items mentioned. It could be argued though that this strategy distorts the meaning (truth) of some sentences, such as:
(2) If a gentleman fancies a lady, he marries her.
I can live with it. It is not the only place where logic and language come apart. I tend to agree with Quine: the logic is clear, it is the language that is ambiguous. Where the problem becomes interesting for me is the preservation or loss of equivalence upon conversion of conditional sentences into sentences containing relative clauses as we go from indefinite to definite descriptions, as can be seen from the following pairs of sentences:
(3a) If a gentleman compliments a lady, he fancies her.
(3b) A gentleman who compliments a lady fancies her.
(4a) If a politician criticizes the queen, he envies her.
(4b) A politician who criticizes the queen envies her.
(5a) If the lady fancies a gentleman, she humours him.
(5b) The lady who fancies a gentleman humours him. *
(5a) If the dog sees the postman, it chases him.
(5b) The dog that sees the postman chases him. *
The sentences sound plausible at first, but less so as we move towards the last pair. (I have deliberately kept the Simple Present tense throughout to enable a same-basis comparison.) Sentences (3a) and (3b) are equivalent, with ‘a gentleman’ taking wide scope. Sentences (5a) and (5b), on the other hand, present the problem of scope. If, as in the symbolization of (1), we quantify universally over ‘the postman’ to ensure that all the variables are bound, we will shoot ourselves in the foot over ‘the postman’ being a definite description.
(5a) does not convert elegantly into (5b) in English, and I wonder if it is the question of logic or the question of semantics. If we substitute the verb ‘hate’ for the verb ‘see’, the difficulty works in the opposite direction.
(6a) If the dog hates the postman, it chases him. *
(6b) The dog that hates the postman chases him.
(6b) can be paraphrased as:
(6c) The dog chases the postman it hates.
This can be captured straightforwardly as:
(∃x){Px • (y)(Py ⊃ y = x) • (∃z){Dz • Hzx • (w)[(Dw • Hwx) ⊃w = z] • Czx}}
Plug in proper names, and it is (5a) in turn that becomes easiest to handle (‘If Razor sees Jasper, it chases him,’ is Srj ⊃Crj). Double anaphora coupled with definite descriptions is a mind-bender, and I wonder if there is a systematic treatment that ties all the ends.
In a nutshell, donkey sentences are like (1):
(1) If a farmer owns a donkey, he beats it.
The pronouns ‘he’ and ‘it’ are anaphoric on ‘farmer’ and ‘donkey’, and the latter pair especially has polarized philosophers, logicians and linguists. Is the sentence true of, say, ten farmers, each of whom has one donkey which he beats, or is it true of ten farmers, each of whom has an unspecified number of donkeys of which he beats every single donkey that he owns.
By convention, first-order logic follows the second approach. It is convenient because the use of a universal quantifier allows us to bind the variable ‘y’ in the predicate expression ‘Bxy’.
(x){Fx ⊃ (y)[(Dy • Oxy) ⊃Bxy]}
It suits me. The indefinite article ‘a’ in English is called so for a reason. It is intended to capture both ‘one’ and ‘every’ of the items mentioned. It could be argued though that this strategy distorts the meaning (truth) of some sentences, such as:
(2) If a gentleman fancies a lady, he marries her.
I can live with it. It is not the only place where logic and language come apart. I tend to agree with Quine: the logic is clear, it is the language that is ambiguous. Where the problem becomes interesting for me is the preservation or loss of equivalence upon conversion of conditional sentences into sentences containing relative clauses as we go from indefinite to definite descriptions, as can be seen from the following pairs of sentences:
(3a) If a gentleman compliments a lady, he fancies her.
(3b) A gentleman who compliments a lady fancies her.
(4a) If a politician criticizes the queen, he envies her.
(4b) A politician who criticizes the queen envies her.
(5a) If the lady fancies a gentleman, she humours him.
(5b) The lady who fancies a gentleman humours him. *
(5a) If the dog sees the postman, it chases him.
(5b) The dog that sees the postman chases him. *
The sentences sound plausible at first, but less so as we move towards the last pair. (I have deliberately kept the Simple Present tense throughout to enable a same-basis comparison.) Sentences (3a) and (3b) are equivalent, with ‘a gentleman’ taking wide scope. Sentences (5a) and (5b), on the other hand, present the problem of scope. If, as in the symbolization of (1), we quantify universally over ‘the postman’ to ensure that all the variables are bound, we will shoot ourselves in the foot over ‘the postman’ being a definite description.
(5a) does not convert elegantly into (5b) in English, and I wonder if it is the question of logic or the question of semantics. If we substitute the verb ‘hate’ for the verb ‘see’, the difficulty works in the opposite direction.
(6a) If the dog hates the postman, it chases him. *
(6b) The dog that hates the postman chases him.
(6b) can be paraphrased as:
(6c) The dog chases the postman it hates.
This can be captured straightforwardly as:
(∃x){Px • (y)(Py ⊃ y = x) • (∃z){Dz • Hzx • (w)[(Dw • Hwx) ⊃w = z] • Czx}}
Plug in proper names, and it is (5a) in turn that becomes easiest to handle (‘If Razor sees Jasper, it chases him,’ is Srj ⊃Crj). Double anaphora coupled with definite descriptions is a mind-bender, and I wonder if there is a systematic treatment that ties all the ends.
No comments:
Post a Comment