Consider for a moment the the following proposition (or propositional schema, to be more precise):
(1) If [Insert your name here] believes (1) is true, [Insert your name here] believes every proposition is true.
To make things easier, I'll consider the paradox in my own case. Thus, I consider the following proposition:
(1') If Jason Zarri believes (1') is true, Jason Zarri believes every proposition is true.
So, I ask myself, do I believe (1')? Well, its truth-conditions are that, if I believe it is true, I believe every proposition is true. So if I believe (1') is true, I believe every proposition is true. But I'm pretty confident that I don't believe every proposition is true. I don't believe that the universe is less than five minutes old, for example. So I conclude (safely, so it seems) that I don't believe (1'). But wait a moment: In asserting what I just did, in the statement beginning "So if I believe (1')..." I asserted my belief in the truth of (1'), because the fact that if I believe (1') then I believe everything is precisely what (1') asserts! So it seems I do believe (1') after all. But, so I continue, if I believe (1'), and if I believe the antecdent of (1') -- both of which I have established-- then I am committed to believing the consequent of (1'), namely, that I believe every proposition is true! Thus it appears that, reasoning from fairly innocuous premises and rules of inference, I've reasoned myself into being a trivialist. And yet I remain certain--perhaps obstinately so--that I am not a trivialist: I don't believe everything is true, and if some demonstration purports to show that I do, then something is seriously wrong with it. The question is, then, what?
Wednesday, November 29, 2006
Tuesday, November 21, 2006
Non-truth-functional Connectives
Lately I’ve been considering a linguistic phenomenon which leads me to believe that, in addition to the familiar concept of truth, there is a more general concept which we can call satisfaction, truth being but one kind of satisfaction. Consider for a moment (1)
(1) Wash your clothes and put your dishes in the sink.
Normally, philosophers and logicians think of “and” as being a truth functional connective that conjoins propositions or declarative sentences, the conjunction being true if both conjuncts are true, and false if either is false. Yet the above, which seems (to me, at least) to be an obvious case of conjunction, has no truth conditions at all, for the statements being conjoined are commands, or imperatives, the term I will use. Though “and” cannot be a truth function as it occurs in (1), we can think of it as something highly similar: A satisfaction function. We can say that a proposition (or declarative sentence) is satisfied if and only if it is true, and an imperative is satisfied if and only if it is obeyed. (Extending the analogy, we might say that a yes or no question is satisfied if and only if its answer is yes, and that a rule is satisfied if and only if it is followed correctly). (1), though not true, is satisfied if and only if both conjuncts are satisfied, which seems intuitively correct: It has been obeyed if and only if its intended recipient has both washed their clothes and put their dishes in the sink. The similarity between truth and obediance can be confirmed by the fact that an analogue of the liar paradox can be constructed for imperatives. Thus, (2)
(2) Do not obey this imperative.
seems to be obeyed if and only if it is not. That such a similar paradox can be generated for imperatives counts as strong evidence of the affinity of truth and obedience as two different kinds of satisfaction. Statements of mixed kind are also possible:
(3) If John calls, tell him I’m at the video store.
(3) is satisfied if the antecedent is true an the consequent is obeyed. As a whole, the conditional is neither true nor obeyed. It is, in my terminology, purely satisfied, where a statement S is purely satisfied if and only if it is satisfied without being satisfied in some more specific way, that is, if it is satisfied without being true, or obeyed, or correctly answered… . In this it differs from (1), which is obeyed as a whole: if either conjunct is disobeyed, the conjunction is disobeyed. If the antecedent of (3) is true and the consequent is disobeyed, then it is unsatisfied. If the antecedent is false, the conditional is satisfied[1], and the consequent is not so much disobeyed as void. Being disobeyed, then, is more like negative satisfaction than a mere lack of satisfaction.
[1] If we held it to be unsatisfied if the antecedent is false, then the conditional would be satisfied if and only if the antecedent is true and the consequent is obeyed, making it a mixed conjunction—and a conditional “If A , then B” which turns out to be equivalent to “A and B” is a very poor conditional in my eyes—properly speaking, it isn’t a conditional at all.
(1) Wash your clothes and put your dishes in the sink.
Normally, philosophers and logicians think of “and” as being a truth functional connective that conjoins propositions or declarative sentences, the conjunction being true if both conjuncts are true, and false if either is false. Yet the above, which seems (to me, at least) to be an obvious case of conjunction, has no truth conditions at all, for the statements being conjoined are commands, or imperatives, the term I will use. Though “and” cannot be a truth function as it occurs in (1), we can think of it as something highly similar: A satisfaction function. We can say that a proposition (or declarative sentence) is satisfied if and only if it is true, and an imperative is satisfied if and only if it is obeyed. (Extending the analogy, we might say that a yes or no question is satisfied if and only if its answer is yes, and that a rule is satisfied if and only if it is followed correctly). (1), though not true, is satisfied if and only if both conjuncts are satisfied, which seems intuitively correct: It has been obeyed if and only if its intended recipient has both washed their clothes and put their dishes in the sink. The similarity between truth and obediance can be confirmed by the fact that an analogue of the liar paradox can be constructed for imperatives. Thus, (2)
(2) Do not obey this imperative.
seems to be obeyed if and only if it is not. That such a similar paradox can be generated for imperatives counts as strong evidence of the affinity of truth and obedience as two different kinds of satisfaction. Statements of mixed kind are also possible:
(3) If John calls, tell him I’m at the video store.
(3) is satisfied if the antecedent is true an the consequent is obeyed. As a whole, the conditional is neither true nor obeyed. It is, in my terminology, purely satisfied, where a statement S is purely satisfied if and only if it is satisfied without being satisfied in some more specific way, that is, if it is satisfied without being true, or obeyed, or correctly answered… . In this it differs from (1), which is obeyed as a whole: if either conjunct is disobeyed, the conjunction is disobeyed. If the antecedent of (3) is true and the consequent is disobeyed, then it is unsatisfied. If the antecedent is false, the conditional is satisfied[1], and the consequent is not so much disobeyed as void. Being disobeyed, then, is more like negative satisfaction than a mere lack of satisfaction.
[1] If we held it to be unsatisfied if the antecedent is false, then the conditional would be satisfied if and only if the antecedent is true and the consequent is obeyed, making it a mixed conjunction—and a conditional “If A , then B” which turns out to be equivalent to “A and B” is a very poor conditional in my eyes—properly speaking, it isn’t a conditional at all.
Saturday, November 18, 2006
A Brief Sketch of an Expressivist Theory of Vagueness
How should we interpret vague language? All the current accounts I know of seem to presuppose that vague language is an attempt, perhaps failed, to get at the truth; that in such assertions as "John is bald", or that a certain collection of grains of sand is "a heap", we have a problem in that such assertions seem perfectly meaningful, and so apparently truth-evaluable, in spite of the fact they are vague, and so apparently have vague truth conditions if they have any at all. If we are not to posit vagueness in nature, we face the problem of accounting for vague language in wholly precise terms. Instead, I propose that vague language serves primarily to express our attitudes towards the magnitude of some quality, as well as to recommend, promote, or regulate certain aspects of behavior. Instead of focusing on the sorites and related arguments, I think we would do better if we asked ourselves for what conceivable purpose we would talk about “heaps”,"tallness", "baldness" and such in normal, everyday circumstances. If we do, I think we’ll find it has nothing to do, e.g. with specifying some precise number of grains of sand.
My sketch of the theory goes something like this: The first idea is that the correspondence between language and thought can be one-many or many-one rather than one-one. We must reject the notion that behind any one sentence, even one sentence as uttered in a particular context, there stands a single thought. Instead, there corresponds to vague sentences a multiplicity of thoughts: Some are propositions which are either true or false, and others non-truth evaluable suggestions or recommendations concerning possible courses of action the speaker’s audience might take.
Suppose Bill is visiting Samantha’s cottage in the mountains, and Samantha remarks “It often snows here.” How often is “often”? Once a day? Once a week? Once a month (even in July)?[1]My proposal is that Samantha’s assertion actually conveys a cluster of related meanings which we conflate consciously, but can tease out on reflection. First, there is the truth condition “It has snowed here more than once.” If it has not snowed there more than once, the truth-evaluable portion of what she says will simply be false. In addition there are an indefinite number of (understood) recommendations which will vary depending on the specific context: “Expect it to snow here again”, “You should buy a heavy coat”, “Be sure to find some shelter”, “Don’t be surprised if you feel cold at times”, “Don’t be surprised if it snows more here than you expect”, and so on.
At this point, I should stress that the meanings thus conveyed need not, even implicitly, be things that Samantha entertains or asserts. What I have above called "meanings" could more accurately be called "functions". Instead of talking about “meaning”, with all its attendant vagueness and confusion, we could identify the meaning of a word with its function, the “function” of a word being whatever it is that prevents it from dropping out of the language. Our memory for lexical items is a finite resource, and so at some point words have to “pay their way” to earn a place in the lexicon. Unlike our DNA, which has vast sequences of nucleotides that most biologists consider to be meaningless “junk”, we do not memorize arbitrary sequences of letters that play no role in our language. Words that have no relevant function will disappear from the lexicon. This idea bears obvious affinities to Wittgenstein’s notion of meaning as “use”. Given the vagueness of the word “use”, we could view “function” as a somewhat more specific prescification .
Retuning to the above scenario, the recommendations may, instead being part of the “asserted content” of the utterance “It often snows here”, be nothing more than inferences the speaker makes based on that utterance, and that in general, it is part of the meaning—the function—of such utterances to produce such inferences in one’s audience. If the utterances of “It often snows here” stopped generating such inferences as that "It is rational to buy a heavy coat" in one’s audience, they would lose their point, and would either acquire some new meaning or drop out of the language altogether. So long as they can succeed in modulating behavior, such utterances have a rationale, and this rationale will not involve anything like specifing some precise number of snowings that occur in a given interval of time.
The key point is that in asserting “It often snows here”, what Samantha is really saying is that it snows here often enough; often enough that certain precautions ought to be taken or that certain kinds of behavior become rational. In effect, she is taking a certain attitude towards the frequency of "snowings" in her area, namely, that it is frequent enough to be salient for certain purposes. I think this point can be extended to cover most other cases of vague terms: To say someone counts as ‘tall’ in a given context is to say they are tall enough that their (above average) height is salient, or ‘bald’ if they are bald enough that their lack of hair becomes salient. To say these things does not presuppose that there is some height which all tall persons exceed, nor that there is some number of hairs which all bald persons fail to exceed. It is rather a stance or attitude that we express towards their actual height or number of hairs, whatever these may be. When a speaker describes a person as tall, they are doing neither more nor less than indicating that they consider that persons actual height, whatever it is, to be far enough above average as to be salient. People can disagree about whether a given person is ‘tall’ without attributing different heights to that person, just as two people can disagree about whether a given room is “too hot” or “too cold” without disagreeing about what temperature the room is. In the former dispute only the saliency of the person’s height is in question, just as in the latter only the disputants’ level of comfort is in question. As we might say in normal life, different people have different ideas of “tall”. Once the saliency of an attribute such as height, or the frequency of snowings in a certain area, is brought to our attention, our minds can begin their work of deducing the appropriate course of action, and this effect that gives vague language its rationale.
[1] And all this in addition to the problem of individuating “snowings”!
My sketch of the theory goes something like this: The first idea is that the correspondence between language and thought can be one-many or many-one rather than one-one. We must reject the notion that behind any one sentence, even one sentence as uttered in a particular context, there stands a single thought. Instead, there corresponds to vague sentences a multiplicity of thoughts: Some are propositions which are either true or false, and others non-truth evaluable suggestions or recommendations concerning possible courses of action the speaker’s audience might take.
Suppose Bill is visiting Samantha’s cottage in the mountains, and Samantha remarks “It often snows here.” How often is “often”? Once a day? Once a week? Once a month (even in July)?[1]My proposal is that Samantha’s assertion actually conveys a cluster of related meanings which we conflate consciously, but can tease out on reflection. First, there is the truth condition “It has snowed here more than once.” If it has not snowed there more than once, the truth-evaluable portion of what she says will simply be false. In addition there are an indefinite number of (understood) recommendations which will vary depending on the specific context: “Expect it to snow here again”, “You should buy a heavy coat”, “Be sure to find some shelter”, “Don’t be surprised if you feel cold at times”, “Don’t be surprised if it snows more here than you expect”, and so on.
At this point, I should stress that the meanings thus conveyed need not, even implicitly, be things that Samantha entertains or asserts. What I have above called "meanings" could more accurately be called "functions". Instead of talking about “meaning”, with all its attendant vagueness and confusion, we could identify the meaning of a word with its function, the “function” of a word being whatever it is that prevents it from dropping out of the language. Our memory for lexical items is a finite resource, and so at some point words have to “pay their way” to earn a place in the lexicon. Unlike our DNA, which has vast sequences of nucleotides that most biologists consider to be meaningless “junk”, we do not memorize arbitrary sequences of letters that play no role in our language. Words that have no relevant function will disappear from the lexicon. This idea bears obvious affinities to Wittgenstein’s notion of meaning as “use”. Given the vagueness of the word “use”, we could view “function” as a somewhat more specific prescification .
Retuning to the above scenario, the recommendations may, instead being part of the “asserted content” of the utterance “It often snows here”, be nothing more than inferences the speaker makes based on that utterance, and that in general, it is part of the meaning—the function—of such utterances to produce such inferences in one’s audience. If the utterances of “It often snows here” stopped generating such inferences as that "It is rational to buy a heavy coat" in one’s audience, they would lose their point, and would either acquire some new meaning or drop out of the language altogether. So long as they can succeed in modulating behavior, such utterances have a rationale, and this rationale will not involve anything like specifing some precise number of snowings that occur in a given interval of time.
The key point is that in asserting “It often snows here”, what Samantha is really saying is that it snows here often enough; often enough that certain precautions ought to be taken or that certain kinds of behavior become rational. In effect, she is taking a certain attitude towards the frequency of "snowings" in her area, namely, that it is frequent enough to be salient for certain purposes. I think this point can be extended to cover most other cases of vague terms: To say someone counts as ‘tall’ in a given context is to say they are tall enough that their (above average) height is salient, or ‘bald’ if they are bald enough that their lack of hair becomes salient. To say these things does not presuppose that there is some height which all tall persons exceed, nor that there is some number of hairs which all bald persons fail to exceed. It is rather a stance or attitude that we express towards their actual height or number of hairs, whatever these may be. When a speaker describes a person as tall, they are doing neither more nor less than indicating that they consider that persons actual height, whatever it is, to be far enough above average as to be salient. People can disagree about whether a given person is ‘tall’ without attributing different heights to that person, just as two people can disagree about whether a given room is “too hot” or “too cold” without disagreeing about what temperature the room is. In the former dispute only the saliency of the person’s height is in question, just as in the latter only the disputants’ level of comfort is in question. As we might say in normal life, different people have different ideas of “tall”. Once the saliency of an attribute such as height, or the frequency of snowings in a certain area, is brought to our attention, our minds can begin their work of deducing the appropriate course of action, and this effect that gives vague language its rationale.
[1] And all this in addition to the problem of individuating “snowings”!
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