There can be gramatically correct sentences which don’t constrain anticipations at all, and not only the self-referential cases. All mathematical statements somehow fall in this category, just imagine, what observations one anticipates because believing “the empty set is an empty set”.
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about. (More easily, you may find that mathematicians agree with correct statements and disagree with incorrect ones, and you can predict agreement/disagreement from knowledge about correctness.)
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about.
That’s why I have said “[t]he thing is a little complicated with mathematical statements because, at least for the more complicated theorems, believing in them causes the anticipation of being able to derive them using valid inference rules”.
What’s the distinction between the two? (Useful for deriving propositions about smth vs. referring.)
The latter are sentences which directly mention the object (“the planet moves along an elliptic trajectory”) while the former are statements that don’t (“an ellipse is a closed curve”). Perhaps a better distinction would be based on the amount of processing between the statement and sensory inputs; on the lowest level we’ll find sentences which directly speak about concrete anticipations (“if I push the switch, I will see light”), the higher level statements would contain abstract words defined in terms of more primitive notions. Such statements could be unpacked to gain a lower-level description by writing out the definition explicitly (“the crystal has O_h symmetry” into “if I turn the crystal 90 degress, it will look the same and if I turn it 180 degrees...”). If a statement can be unpacked in finite number of recursions down to the lowest level containing no abstractions, I would say it refers to the external world.
“[t]he thing is a little complicated with mathematical statements because, at least for the more complicated theorems, believing in them causes the anticipation of being able to derive them using valid inference rules”.
This doesn’t look to me like a special condition to be excused, but as a clear demonstration that mathematical truths can and do constrain anticipation.
“Directly mentioning” passes the buck of “referring”, you can’t mention a planet directly, the planet itself is not part of the sentence. I don’t see how to make sense of a statement being “unpacked in finite number of recursions down to the lowest level containing no abstractions” (what’s “no abstractions”, what’s “unpacking”, “recursions”?).
(I understand the distinction between how the phrases are commonly used, but there doesn’t appear to be any fundamental or qualitative distinction.)
There has to be a definition of base terms standing for primitive actions, observations and grammatical words (perhaps by a list, to determine what to put on the list would ideally need some experimental research of human cognition). An “abstraction” is then a word not belonging to the base language defined to be identical to some phrase (possibly infinitely long) and used as an abbreviation thereof. By “unpacking” I mean replacing all abstractions by their definitions.
If you build an inference system that outputs statements it proves, or lights up a green (red) light when it proves (disproves) some statement, then your anticipations about what happens should be controlled by the mathematical facts that the inference system reasons about. (More easily, you may find that mathematicians agree with correct statements and disagree with incorrect ones, and you can predict agreement/disagreement from knowledge about correctness.)
That’s why I have said “[t]he thing is a little complicated with mathematical statements because, at least for the more complicated theorems, believing in them causes the anticipation of being able to derive them using valid inference rules”.
The latter are sentences which directly mention the object (“the planet moves along an elliptic trajectory”) while the former are statements that don’t (“an ellipse is a closed curve”). Perhaps a better distinction would be based on the amount of processing between the statement and sensory inputs; on the lowest level we’ll find sentences which directly speak about concrete anticipations (“if I push the switch, I will see light”), the higher level statements would contain abstract words defined in terms of more primitive notions. Such statements could be unpacked to gain a lower-level description by writing out the definition explicitly (“the crystal has O_h symmetry” into “if I turn the crystal 90 degress, it will look the same and if I turn it 180 degrees...”). If a statement can be unpacked in finite number of recursions down to the lowest level containing no abstractions, I would say it refers to the external world.
This doesn’t look to me like a special condition to be excused, but as a clear demonstration that mathematical truths can and do constrain anticipation.
“Directly mentioning” passes the buck of “referring”, you can’t mention a planet directly, the planet itself is not part of the sentence. I don’t see how to make sense of a statement being “unpacked in finite number of recursions down to the lowest level containing no abstractions” (what’s “no abstractions”, what’s “unpacking”, “recursions”?).
(I understand the distinction between how the phrases are commonly used, but there doesn’t appear to be any fundamental or qualitative distinction.)
There has to be a definition of base terms standing for primitive actions, observations and grammatical words (perhaps by a list, to determine what to put on the list would ideally need some experimental research of human cognition). An “abstraction” is then a word not belonging to the base language defined to be identical to some phrase (possibly infinitely long) and used as an abbreviation thereof. By “unpacking” I mean replacing all abstractions by their definitions.