New thought on Dash’s comment on yesterday’s post.
Think of it this way. You are talking to someone and you are confused because he seems to be saying the opposite of what he just said. You are in this position if you are a logician: there is a contradiction and it needs to be resolved or there is not one; there is just an apparent one.
But the fellow who is just trying to understand what somebody is saying is in no position at all. He is just talking to somebody.
He is not a logician or legal counsel or someone else intruding his order into the discourse to regulate it in some way, in the name of clarity or credibility or something else.
If the conversation ends and the fellow understands what someone is saying, the issue of contradiction does not arise or is at best moot.
“Well, there was a contradiction there at one point.”
“You can say that, or you can say that there never was one, that you just did not understand what the person was saying until the end.”
“Well, there was an apparent one.”
“For someone worried about word order or precision, maybe. But he got it all out by the end. You knew what he was saying. What is the point of worrying?”
“Are you saying I should not worry about these things?”
‘Yes, not here. You can worry about them when they matter, e.g., when certain conventions make them important.”
“Well, the fact that we note that there are contradictions in conversation shows that the law of the excluded middle is independent of the subject matter to which it is applied. So Brouwer is wrong.”
“No. You’ve made it independent by introducing it independently. In the conversation an opposition seemed to occur which was resolved by the end. Had it not been resolved, there would have been no understanding."
"Hmm."
“The opposition would have been in the things opposed. They would have been opposed not because of the law, but because of the way they lined up in the conversation. The law is simply an expression of their opposition.”
(Think of this case: “he had to have been talking about two different aunts and was running the two together, and I never could get straight what he was saying there. I did not know he had any aunts in the first place. I’ll have to try to ask him again tomorrow when he is fresh and if he is in the mood for talking.” )
Think of it this way. You are talking to someone and you are confused because he seems to be saying the opposite of what he just said. You are in this position if you are a logician: there is a contradiction and it needs to be resolved or there is not one; there is just an apparent one.
But the fellow who is just trying to understand what somebody is saying is in no position at all. He is just talking to somebody.
He is not a logician or legal counsel or someone else intruding his order into the discourse to regulate it in some way, in the name of clarity or credibility or something else.
If the conversation ends and the fellow understands what someone is saying, the issue of contradiction does not arise or is at best moot.
“Well, there was a contradiction there at one point.”
“You can say that, or you can say that there never was one, that you just did not understand what the person was saying until the end.”
“Well, there was an apparent one.”
“For someone worried about word order or precision, maybe. But he got it all out by the end. You knew what he was saying. What is the point of worrying?”
“Are you saying I should not worry about these things?”
‘Yes, not here. You can worry about them when they matter, e.g., when certain conventions make them important.”
“Well, the fact that we note that there are contradictions in conversation shows that the law of the excluded middle is independent of the subject matter to which it is applied. So Brouwer is wrong.”
“No. You’ve made it independent by introducing it independently. In the conversation an opposition seemed to occur which was resolved by the end. Had it not been resolved, there would have been no understanding."
"Hmm."
“The opposition would have been in the things opposed. They would have been opposed not because of the law, but because of the way they lined up in the conversation. The law is simply an expression of their opposition.”
(Think of this case: “he had to have been talking about two different aunts and was running the two together, and I never could get straight what he was saying there. I did not know he had any aunts in the first place. I’ll have to try to ask him again tomorrow when he is fresh and if he is in the mood for talking.” )
posted by Mauberly | 12:37 PM
3 Comments:
Apparently as you were typing this post, I was adding a (very long, split over three comments) response to the earlier one. We were cosmically synchronized :-)
The "law of the excluded middle" as you are describing it here parallels what I was saying there about how and why mathematicians still sometimes do classical rather than constructive math, in practice. It's a vague short-hand for pointing towards things that can be said without it.
One slight confusion (about math) in what you are saying here is that the issue in constructivist math isn't the case of a person who says "X and not(X)" but rather of a person who says "I can show you that not(X) has no proof and therefore X must be true - it's one or the other, take it or leave it." The usual summary of the constructivist formalism is that "not(not(X)) does not imply X" but in context what that means is "Here is a proof that 'not(X)' can not be proved so let's assume X is now proved"
-t
err, that should be:
"Here is proof that not(X) can not be proved but that does not entitle us to assume X is true."
-t
The entire collection of your recent posts is very well conceived. Appreciate your showing up.
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