I'm currently at Chapter 4, vol. 1 and 1st ed. I have to ask this question because the most important thing about this book is in its minute details. Thanks.
Take *3.3 for example. Acording to this site, I think Proof would be more appropriate.
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$\begingroup$ Can you cite passages in the text using those terms? $\endgroup$ – dfeuer Jan 6 '14 at 4:47
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$\begingroup$ I've only encountered Dem. so far, but I think Proof would be more appropriate. $\endgroup$ – George Chen Jan 6 '14 at 4:52
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4$\begingroup$ The question seems to presuppose that "proof" and "demonstration" are not synonymous in mathematics. It would be useful to know what difference in meaning you have in mind, because I'm not aware of any difference. $\endgroup$ – Andreas Blass Jan 6 '14 at 4:56
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$\begingroup$ Proof is generic; demonstration can be used for one instance only. It is possible that PM does not distinguish the two. I just want to make sure that I don't miss anything important. $\endgroup$ – George Chen Jan 6 '14 at 5:03
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4$\begingroup$ @GeorgeChen, that is not a standard use of the term in modern mathematical usage; it would not have been in Russell and Whitehead's time either. It might have been back in the 1600s—I don't know. $\endgroup$ – dfeuer Jan 6 '14 at 5:45
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In standard mathematical terminology, "proof" and "demonstration" mean the same thing. Therefore Russell and Whitehead's use of language was appropriate.
Quod erat demonstrandum.
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$\begingroup$ I agree with the above answers that standard mathematical practice does not distinguish between "proof" and "demonstration". It can be an interpretative issue if W&R's PM has different meanings. From pag.87 : "The subjects treated in Part I [Math Log] may be viewed in two aspects: (1) as a deductive chain depending on the primitive propositions, (2) as a formal calculus. Taking the first view first: We begin, in $*1$, with certain axioms as to deduction of one proposition or asserted propositional function from another." 1/2 $\endgroup$ – Mauro ALLEGRANZA Jan 8 '14 at 10:36
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$\begingroup$ So we may expect that they will use deduction as the name of the "formal" object in the object language (like modern use of derivation). A proof is a "normal" mathematical proof (or "demonstration") like those used in the meta-theory (e.g.the proof of Deduction Theorem). From pag.100 (the first "formal proof": $*2.01$, I suppose) : "The proof is as follows (where "Dem" is short for "demonstration")". I think that there is NO deep issue involved in the choice of "Dem" (instead of "Proof"). The issue can be : why they do not use "deduction" ? 2/2 $\endgroup$ – Mauro ALLEGRANZA Jan 8 '14 at 10:41
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$\begingroup$ My last comment doesn't make sense. @Mauro, Very good question. "Mathematics is deductive logic" is what PM set out to prove. "Deduction" would be right on point. $\endgroup$ – George Chen Jan 10 '14 at 8:22
