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This baby step from ❋3.01 to ❋4.5 is so tiny that I can barely see the difference. Please kindly explain why it is so important to distinguish the two. What is the philosophical importance of this distinction? Thanks.

❋3.01 is a definition: 3.01 https://i.stack.imgur.com/5Ujin.png

❋4.5 is an assertion of equivalence: 4.5 https://i.stack.imgur.com/ZQboI.png

George Chen
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  • Here a sentence from the introduction: "For a definition is concerned wholly with the symbols, not with what they symbolise. Moreover it is not true or false, being the expression of a volition, not of a proposition." Probably I need to mull on this to feel it. – George Chen Jan 08 '14 at 08:57
  • What is "the expression of a volition?" – George Chen Jan 08 '14 at 09:01
  • What is the difference between a volition and a proposition in this context? – George Chen Jan 08 '14 at 09:13

1 Answers1

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In general, a definition like $*3.01$ [in modern notation : $p \land q := \lnot (\lnot p \lor \lnot q)$] must be understood as an abbreviation.

I.e., we can replace in every context where we have used $p \land q$ the original formula without "altering" the set of theorems derivable.

Theoretically, it is unnecessary ever to give a definition ... the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous. (Chapter 1. Page 12, Merchant Books, 1910)

The usefulness of definition arises from two causes:

First, a definition usually implies that the definiens is worthy of careful consideration. Hence the collection of definitions embodies our choice of subjects and our judgment as to what is most important. Secondly, when what is defined is (as often occurs) something already familar, such as cardinal or ordinal numbers, the definition contains an analysis of a common idea, and may therefore express a notable advance. (Chapter 1. Page 12, Merchant Books, 1910)

In $*4.5$ they merely reformulate the above definition as an equivalence. The comment is :

The following formulae are due to De Morgan [...]. The first of them, it will be observed, merely embodies our definition of the logical product.

The context of $*4.5$ is :

we shall be concerned with rules analogous, more or less, to those of ordinary algebra. It is from these rules that the usual "calculus of formal logic " starts. [...] Thus two propositions are equivalent when they have the same truth-value.

The fundamental difference between *3.01 and *4.5 is that the latter asserts a proposition and the former does not assert any proposition. A definition does not always lead to the equivalence between the definiendum and the definien. Take *30.01 and *30.22 for example:

*30.01 $ R‘y =(℩x)(xRy) $ Df

*30.22 $ E!R‘y.≡.R‘y =(℩x)(xRy) $
Note that we do not necessarily have $R‘y =(℩x)(xRy)$ which is only true when $E!R‘y$.

This is because when an incomplete symbol appears in a proposition, the definiendum is the proposition, instead of the incomplete symbol in isolation. Thus, "the present King of France is the present King of France" is false. See *14.28 for further illustration.

There does not appear to be a general condition in PM that enables a definition to imply an equivalence. Such implications have to be proved in particular cases. This is why *3.01 cannot take the place of *4.5.

George Chen
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Mauro ALLEGRANZA
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  • Thanks, @Mauro. And what do they mean by "volition" in this context? Is there a modern term for this? – George Chen Jan 08 '14 at 09:33
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    Assuming that I'm NOT a scholar of W&R's Principia, I think that they are simply stating (in an "old fashioned" language) that a definition regards only the symbols, i.e.the language, and NOT the "real" world ("a definition is concerned wholly with the symbols, not with what they symbolise"). So, must be regarded as an abbreviation, i.e.a stipulation made by the "speakers" of the language ("it is not true or false, being the expression of a volition, not of a proposition"). 1/2 – Mauro ALLEGRANZA Jan 08 '14 at 09:39
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    Because it is a stipulation, it does not express a fact about the world; so it is not a proposition (only propositions express true or false facts). 2/2 – Mauro ALLEGRANZA Jan 08 '14 at 09:40
  • Thanks, @Mauro. So "the earth is round" is a proposition because when I say it I express my belief and my belief can be verified by facts. Could you give me an example of an expression of volition? – George Chen Jan 08 '14 at 09:45
  • Here is just a guess: when I recite a multiplication table, I'm expressing a volition, because there is no events corresponding to 1 X 2 = 2. – George Chen Jan 08 '14 at 09:49
  • I have not found the locus of your citation from PM Intro. You must thake into account the complex evolution of PM's philosophical part . See Gregory Landini, Russell's hidden substitutional theory (1998), and Bernard Linsky, The Evolution of Principia Mathematica (2011). Between the first and 2nd ed of PM, Russell met Wittgenstein. The Intro to the 2nd ed was due to Russell only and reflect the influence of W's Tractatus. – Mauro ALLEGRANZA Jan 08 '14 at 09:51
  • Thanks for the reference. I'll definitely check it out. The citation is near the bottom of page 11, 1st edition. – George Chen Jan 08 '14 at 10:00
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    Proposition is a difficult concept in Russell's philosophy and logic (see Landini's book). Today, mathematical logic avoid (mainly under Quine's influence) the complex issue involved with "abstract" entities like proposition : often the locution "sentential logic" is preferred against the traditional "propositional logic", in order to convey the idea that it is concerned with language more than "abstract" entities. You must read first Frege, then Russell and Wittgenstein. Frege's theory of sense and reference was rejected by Russell. 1/2 – Mauro ALLEGRANZA Jan 08 '14 at 10:05
  • For Frege, roughly speaking, the relation of "meaning" between language and world involves three terms: a sentence (of the language), its sense : an "abstract" object (it is a "real" object) conveying its "conceptual content", and its reference : its truth value (again an object). For Russell, the relation between language and world has only two terms : a sentence (in language) and a proposition (in the world) that is the "meaning" of the sentence. 2/2 – Mauro ALLEGRANZA Jan 08 '14 at 10:11
  • In Russell's latter work, An Inquiry into Meaning and Truth, a sentence expresses a belief; a belief is a mental state; the mental state refers to facts or expectations, feelings, etc. I don't remember he mentioned volition. – George Chen Jan 08 '14 at 10:18
  • I can look it up again to find out. – George Chen Jan 08 '14 at 10:18
  • You are right. @Mauro. In Russell's Our Knowledge of the External World, there was still the concept of sense-data that was similar to Frege's idea but was abandoned by Russell in his latter works. PM was definitely written in Russell's sense-data period. – George Chen Jan 08 '14 at 10:26
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    Thanks for page num. Here you can find: "the definitions are no part of our subject, but are, strictly speaking, mere typographical conveniences. Practically, of course, if we introduced no definitions, our formulae would very soon become so lengthy as to be unmanageable; but theoretically, all definitions are superfluous." The usefulness of definition "arises from two causes. First, a definition usually implies that the definiens is worthy of careful consideration. Hence the collection of definitions embodies our choice of subjects and our judgment as to what is most important." – Mauro ALLEGRANZA Jan 08 '14 at 10:27
  • Thanks for your help @Mauro – George Chen Jan 08 '14 at 10:39
  • Actually I skipped much of Into. Looks like I need to finish intro first. – George Chen Jan 08 '14 at 10:40
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    I think it can be a recommended practice ... I'm joking: your effort of studying PM is worthy of the highest respect: I never had the courage to try it. P.S. A last word: Linsky's book is very useful (and not hard to read); Landini's one is much harder (for me). – Mauro ALLEGRANZA Jan 08 '14 at 10:47
  • It's hard to finish intro at once. I'll have to go back and forth. – George Chen Jan 13 '14 at 00:25
  • I think it is a very difficult book. Not only the symbolism is "outdated", but the book mix up (according to my understanding) at least three topics : (i) a treatise on Math Log, (ii) a discussion about philosophy of logic and mathematics (and this has implications regarding ontology and epistemology, also), (iii) the foundations of math, i.e. what today is treated under the heading of set theory : cardinals, ordinals, etc. – Mauro ALLEGRANZA Jan 13 '14 at 07:15
  • This answer partially explained how words can help thinking. Simple truth. – George Chen Sep 08 '14 at 17:16