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Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.

Where does this leave us? First, Williamson has not shown that an assertion theoretic account of meaning is impossible because of a commitment to luminosity. For as the PAT view demonstrates by its existence, it is possible to hold to a view that the meaning of a proposition is determined by its assertion conditions without thereby committing oneself to the luminosity of those assertion conditions. What Dummett insists on, and what he claims a realist, truth-conditional theory of meaning cannot obviously explain, is that a speaker should know the meanings of the sentences understood. So what is required is knowledge of the assertion conditions of these sentences. That, as we’ve seen, can be formulated in the manner employed by Martin-Löf and Tait, under which such knowledge doesn’t require luminosity, i.e., one in which it is possible that a speaker be mistaken in all the expected ways about whether those conditions obtain in a particular case.

However, as we have also seen, there are serious deficiencies with PAT if it is offered as an account of mathematical meaning. In particular, the supposition that all propositions have their meanings given by what counts as a canonical proof seems unlikely to be able to give us a story about the meaning of everything which ought to count as a meaningful mathematical statement. While this version of the assertion theoretic account of the meaning of mathematical sentences doesn’t fall prey to Williamson’s objections, I think the facts reviewed above suggest that it’s a much more seriously constrained account of mathematical meaning than is sometimes recognized.