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Sense Impressions, Beliefs, Formal Systems, Then Conceptualize

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Posted by Harvey on December 27, 2001 03:44:39 UTC

Hi Paul,

>>>It is only the extremely rare Euclid, Peano, or Russell who comes along and shows us that we can set aside some of these beliefs and build logical structures of inferences that are not only independent of the beliefs, but which also give us considerable insight into the subject matter of the beliefs. Dick has done a similar thing by asking us to set aside our beliefs about how or what we know about our universe and to see what logical inferences we can draw about what we can know about "reality" in the abstract.>For example, if I am considering some issue of Geometry, I might visualize familiar examples of lines or triangles in order to get some intuitive ideas. But, when it comes to writing out a proof, I must shy away from those examples and stick with the logical statements of the axioms and proved theorems. And, of course in all of this, we still harbor certain beliefs about lines and triangles that we can't really avoid. We just need to make sure we don't allow those beliefs to unintentionally contaminate our logic.>>H: just as we acknowledge that whatever we hold as an undefined primitive is a belief and not yet logical fact. P: This, I think, is your stumbling block, Harv. When we choose undefined terms as part of an axiomatic system, it is probably unavoidable that the concept which is to be symbolized by that term has been inspired or suggested by some belief of ours. But the rigor of the axiomatic process demands that those beliefs must play absolutely no part in the process of deduction. There must be no "meaning" attached to the term which is in any way involved in the part that term plays in any statement in which it is used. It is only after the logical edifice has been constructed, by pure logical deduction, that we may step back and identify some of the primitive terms with intuitive ideas, or beliefs, in order to use the logical consequences of the development to suggest predictions in "reality".>So, in the process of building the logical structure, we *cannot* "acknowledge that whatever we hold as an undefined primitive is a belief". We cannot acknowledge those beliefs any more than a student can say to the Geometry teacher, "I know what you mean by the primitive term 'line'. You mean tightly stretched spider webs, don't you?" The teacher's response to this would have to be, "No! No! No! We are going to have to go back and start all over.">H: For something to be a logical fact we must first define what we mean by logic and determine criteria which makes something logical and also criteria which makes something a fact. P: You are reading too much into the symbol 'logical fact'. In our ordinary world of communication, of course we try to use standard denotations and connotations of words in order to understand one another. So it is natural, when seeing the pair of words 'logical fact', to assign the customary "meaning" in an attempt to understand. In this situation, you are right:, we would "first define what we mean by logic and determine criteria which makes something logical and also criteria which makes something a fact." This would be tantamount to, when seeing the term 'line' appear in geometry, of assuming that it referred to a strand of spider web.>Dick has used various word combinations in addition to 'logical fact' to identify the primitive concept of "information about the universe". Two such examples are "totally undefined data" and "a set of numbers". The quandary he is in is that no matter what words he chooses, they will bring along with them some preconceived notions, which he asks you to set aside and not consider at all. (The second example is an exception because he does acknowledge that he presupposes the properties of numbers in his development.)>>H: What I want to say is that in order to distinguish something as logical, we must understand what this means within the language context by which we use these terms. D: Yes, it is pretty clear that this is what you wanted to say, Harv. It is also pretty clear that, although this idea is of great interest in a subject such as philosophy, Dick has no interest in it at all in the context of discussing the ideas in his paper.

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