Talk:Propositional calculus

On the Interpretation of the Propositional Calculus

An IP has been adding On the Interpretation of the Propositional Calculus to the external links. It (appears to be) an award-winning Ph.D. thesis (which would not normally be adequate for a reference, although possibly for an external link to point to references), but even if it were a good resource, it isn't a good resource for this article. It might belong in semantics of the propositional calculus (if it existed), or possibly in one of the other articles in the semantics clade. The connection to this article is weak. — Arthur Rubin (talk) 15:25, 21 April 2011 (UTC)

I am the user who has been adding the article mentioned above. (Full disclosure: it is not an award-winning PhD thesis but an award-winning Honours thesis, submitted at the University of Sydney. It won the John Anderson Prize for Best Thesis in Philosophy, and caused me to be awarded a University Medal.)

I appreciate the civilized manner in which Arthur Rubin has made his dispute. I concede that if there were an article for the semantics of the propositional calculus, my link would be best placed there. Since there is not, however, I maintain that my link is well-placed here, and indeed better placed here than on any of the pages existing under the umbrella 'semantics'.

It is not as though the interpretation of the calculus is some out-of-the-way topic only obliquely connected to the topic of the propositional calculus. The interpretation of the calculus is a natural thing for someone to wonder about when they look at this particular Wikipedia article, and this Wikipedia article is also a natural place to go for someone who is wondering about this already.— Preceding unsigned comment added by 123.243.36.225 (talk • contribs) 09:21, April 24, 2011

Hmmm. As not being an expert in the philosophy of the propositional calculus, I decline comment as to the value of the article. It doesn't seem valuable as to modern usage in and relating to mathematical logic, which this article is mostly about. I still think it fits better in Formal semantics (logic) than in propositional calculus. I'm not really sure it's helpful there, either, because of the comparisons between meaning (not formal semantics) of natural-language expressions which are or resemble statements in the propositional calculus.

Because of the relatively low-level (an honours thesis is less "reliable" than a Ph.D. thesis, in general), it couldn't be used as a reference in Wikipedia unless it is referenced favourably in peer-reviewed literature. It may meet WP:EL guidelines, but

Also, per WP:COI, you really shouldn't be adding pointers to your own material. I've asked (in WT:MATH) whether it's proper for me to reference my parents' books and papers in relevant subjects in set theory and mathematical logic, and consensus seems to be it was OK, as they and I are recognized experts in (at least some subfields) of set theory, but I did have to ask.

— Arthur Rubin (talk) 15:56, 24 April 2011 (UTC)

I would argue that the propositional calculus, in addition to being of technological and purely mathematical interest, is also of interest from a philosophical point of view. For a great many students of introductory logic in philosophy departments, this aspect is important. Insofar as this is true, my article is relevant. —Preceding unsigned comment added by 123.243.36.225 (talk) 10:00, 26 April 2011 (UTC)

123.243.36.225 (talk) has made similar edit in articles Philosophy of logic and Philosophy of language : seemingly self-publicity by a student, TG Haze. — Philogos (talk) 23:57, 28 April 2011 (UTC)

Admittedly I do have a double-interest in adding these resources, but I do not think there is a *conflict* of interest. -TH.

Have you read Wikipedia:External links and Wikipedia:Identifying reliable sources? Would it not be more dignified to have your work published in a peer-reviewed academic journal before providing a link to it in these articles? — Philogos (talk) 03:12, 30 April 2011 (UTC)

"...if there were an article for the semantics of the propositional calculus, my link would be best placed there. Since there is not, however, I maintain that my link is well-placed here, and indeed better placed here than on any of the pages existing under the umbrella 'semantics'." There's no requirement for your paper to be a reference anywhere on Wikipedia. If it is needed to reference something asserted in the article, then use it. If it doesn't, then it has absolutely no place. 203.27.72.5 (talk) 03:52, 19 June 2012 (UTC)

Wishing there was more information about the history and development of propositional calculus

I was interested in this topic, found by following other links, and was disappointed to find no information regarding the "history" of propositional calculus. I am left not knowing if these concepts were developed in the 20th century, or in the 19th century (or earlier). Who "invented" it? When was it invented? Why was it invented (what problems does it solve)? What limitations does it have? What impact did it have on other branches of math? What was missing or refined after it's first invention? Are there alternate symbols or systems representing the same ideas? These are all items that I think would be relevant in an encyclopedia, even more than several sections which (to me) would be better served in a math text book on the topic. I'm not suggesting what is there is removed. Anybody out there familiar with this topic capable of adding this content? — Preceding unsigned comment added by 76.9.200.130 (talk) 18:55, 22 July 2011 (UTC)

I have not studied the history, but I have learned a little bit about it along the way. The Laws of Thought by George Boole applies arithmetic (simple algebra on real numbers) to develop the basis of classical propositional logic and Boolean algebra (which are pretty much two different ways of looking at the same thing). Aristotle improved syllogisms, a progenitor of predicate logic of which propositional logic is a simplification (by removing quantifiers over objects). Also see Charles Babbage and the development of computers, a closely related subject. JRSpriggs (talk) 05:48, 23 July 2011 (UTC)

Also see History of logic.

There are very many versions of logic. I suggest you use the templates and categories listed at the bottom of the article to search for them. JRSpriggs (talk) 05:56, 23 July 2011 (UTC)

History section

I've planted the seeds for a history section. I have a bit more to add to it though. If i recall correctly, the stoic propositional calculus was lost and then 'rediscovered' (independently?) by Peter Abelard. I'll have to take a look at my medieval philosophy text before i make this claim though. Ideally, i'd also like to bring in DeMorgan, Boole, Wittgenstein (for truth tables), and maybe pierce. I'd like to end it by introducing Frege's calculus and how it combines propositonal and term based statements. (since i introduced aristotle in the beginning.) — Preceding unsigned comment added by Xenfreak (talk • contribs) 21:41, 1 January 2012 (UTC)

A must-read in this context is Emil Post's 1921 Theory of Elementary Propositions in van Heijenoort 1967:264ff in particular section 2 "Truth table development⁶" with footnote 6 "Truth values, truth functions" on page 267. Whether or not you give Wittgenstein (Tractatus) or Post credit for the modern notion of "truth tables", Post traces the notion back to Jevons and Venn via Lewis 1918. He also mentions Boole and Schroeder. But contemporary symbolic logic relies on the symbolic-logic path from Dedekind, Peano, Frege to Russell 1903 (in particular). Then truth tables, then Veitch and finally Karnaugh's mapping and simplfication methods together with de Morgan's theorems. A destinction between algebraic and symbolic logic is important. They both have their roles to play. Bill Wvbailey (talk) 03:49, 2 January 2012 (UTC)

If you'd like, you can say that truth tables were hinted at by those philosophers, although they were developed by Wittgenstein. I'm not sure how far i'd be willing to go, but feel free to add anything you think is relevant. But remember that we're trying to limit the scope to propositional logic, so i doubt i'd include anyone like godel, cantor, Russell, or anyone else unless they made substantial contributions to symbolic logic. I only wanted to introduce frege to sort of imply 'this is where propositional calculus ends, and where first-order begins.' But again, feel free to add anything you feel is relevant. Xenfreak (talk) 00:18, 5 January 2012 (UTC)

RE Wittgenstein and Post: W didn't "develop" truth tables beyond their meager presentation in Tractatus, in fact as I recall he became an architect for his sister, for a while, then abandoned philosophical logic altogether. To quote from the wikipedia biography: "In his lifetime he published just one book review, one article, a children's dictionary, and the 75-page Tractatus Logico-Philosophicus (1921)." Post, on the other hand, made significant contributions after his PhD paper (the one reprinted in van Heijenoort); he was the mentor and teacher of e.g. Martin Davis, and he contributed to mathematical logic until the 1950's.

However, the truth of the matter will be found in in the bibliographies of the important papers. Here's a start:

Bibligraphy: I have cc's of the original papers of Karnaugh, Veitch, and Shannon and the book of Couturat 1914. Not a one of their footnotes or bibliographic references directly include either Post or Wittgenstein. Here are their references, working backwards:

Maurice Karnaugh 1953, The Map Method for Synthesis of Combinatorial Logic Circuits, A.I.E.E, [I had to pay for this puppy]:

1. The Design of Switching Circuits (book), William Keister, a. E. Ritchie, S. H. Washburn, D. van Nostrand Compnay, New York, N.Y. 1951, chap. 5.

2. Synthesis of Electronic Computing and Control Circuits (book), Staff of the Harvard Computation Laboratory. Harvard University Press, Cambridge, Mass, 1951, chap 5.

3. A Chart Method for Simplifying Truth Functions, E. W. Veitch. Proceedings, Association for Computing Machinery, Pittsburgh, Pa., May 2, 3, 1952.

E. W. Veitch 1952, A Chart Method for Simplifying Truth Functions, Proceedings of the 1952 ACM Annual Conference/Annual Metting, ACM, NY:

Footnote 1: C.E. Shannon, A symbolic Analysis of Relay and Switching Circuits, Trans. A.I. E. E. vol. 57 (1938), pp. 713-723.

Footnote 2: Harvard Computation Laboratory staff, Synthesis of Electronc Computing and Control Circuits, Harvard University Press, 1951

Footnote 3: W. V. Quine [1952] The Problem of Simplifying Truth Function, unpublished paper to appear in the american Mathematical Monthly.

Claude E. Shannon 1938, A Symbolic Analysis of Relay and Switching Circuits, Transactions American Institute of Electrical Engineers, vol 57, 1938, Washington D.C. Reprinted in Claude Elwood Shannon, Collected Papers, IEEE Press, New York:

1. "A complete bibliography of the literature of symbolic logic is given in the Journal of Symbolic Logic, volume 1, number 4, December 1936. Those elementary parts of the theory that are useful in connection with relay circuits are wll treated in the two following references."

2. The Algebra of Logic, Louis Couturat. The Open Court Publishing Company.

3. Universal Algebra, A. N. Whitehead. Cambridge at the University Press, volume I, book III, chapters I and II, pages 35-42.

4. E. V. Huntington, Transactions of the American Mathematical Society, volume 35, 1933, pages 274-304. The postulates refered to are the fourth set, given on page 280.

Edward V. Huntington 1933, New Sets of Independent Postulates for the algrebra of Logic, with Special Reference to Whitehead and Russell's Principia Mathematica, http://www.ams.org/journals/tran/1933-035-01/S0002-9947-1933-1501684-X/S0002-9947-1933-1501684-X.pdf.

This comes with a huge bibliography on the first two pages of the document, the following are the names of the authors (with repeats, in the order of presentatation):

Schröder, A. N. Whitehead, E. V. Huntington, Schröder, A. Del Re, Sheffer, B. A. Bernstein, L. L. Dines, B. A. Bernstein, B. A. Bernstein, J. G. P. Nicod, N. Wiener, C. I. Lewis, H. M. Sheffer, A. N. Whitehead and B. Russell, Principia, H. M. Sheffer, Paul Bernays, B. A. Bernstein, D. Hilbert and W. Ackermann, Alfred Tarski, Kurt Gödel, Kurt Gödel, J. Lukasiewicz and A. Tarski, J. Lukasiewicz, A. B. A. Bernstein, Jörgen Jjárgensen, E. V. Huntington, P. Henle, B. A. Bernstein, C. I. Lewis and C. H. Langford.

B. A. Bernstein 1929 Whitehead and Russell's Theory of Deduction as a Mathematical Science http://www.ams.org/journals/bull/1931-37-06/S0002-9904-1931-05191-0/S0002-9904-1931-05191-0.pdf

Bernstein converts what he calls the "theory" of PM to his notion of a mathematical "science" in the way that he converted the "theory of deduction" in Boolean (algebraic logic) to a mathematical science that consists of a system (K, ', +) [he intentionally orrows the signs ' and + from Boolean logic: (K, ~, V) would be the symbols of PM]. The system consists of an undefined class K of elements p, q, r . . . , a unary operation symbolized by ' on the K-elements and a binary operation on the K-elements p, q. There is the "supposition" of a unique element that he calls it "1" (aka "truth" cf p. 487). There is another symbol " = " which he asserts "is not [part of the theory], and so is free from the criticsm expressed in the preceding section" (p. 487, on p. 484 he discusses this notion). From this system of symbols he then derives the 8 postulates that correspond to the postulates found in PM, the first of which is a symbolic version of modus ponens. He believes he has clearly separated the ideas inside the theory and the ideas that are outside it (i.e. " = "). With regards to this notion of "ideas and propositions thare are outside [the theory]" he cites two of his papers in a footnote on page 484. These are the only two cites in the paper excepting PM 2nd edition.

E. J. McCluskey 1965 Introduction to the Theory of Switching circuits, McGraw-Hill Book Company, LCCCN: 65-17394.

Chap 3 (Switching Algebra) references 15 publications including:

1. Shannon 1938

Chap 4 (Simplification of Switching Functions) references 16 publications including these:

1. Veitch 1952

2. Karnaugh 1953

3. Quine, W. V.: The Problem of Simplifying Truth Functions, Am. Math. Monthly, vol. 59 no. 8, pp. 521-531, October 1952

Interestingly, Shannon's treatment is mostly Boolean, with a smattering of symbolic logic. In fact the postulates on page 471 are given as Boolean algebra e.g. 1.a 0*0=0, 1.b. 1+1=1, [etc]. He goes on to discuss Analogue with the Calculus of Propositions (page 474) where he states: "E. V. Huntington⁴ gives the following set of postulates for symbolic logic [etc]". De Morgan's theorems appear in a mixed Boolean-Symbolic form (page 474) e.g. he uses X+Y, XY, and X' (logical NOT). Shannon is using the symbolism of Couturat 1914 i.e. the X+Y, XY, X'.

Couturat 1914 can be downloaded from www.books.google.com. With regards to the influence on Couturat who influenced Shannon who influenced Veitch who influenced Karnaugh, here's a quote re the history of symbolic logic, from Couturat (but who influenced Quine? of the Quine-McClusky method) -- the whole preface is quite useful, actually:

"LEIBNIZ thus formed projects of both what he called a characteristica universalis, aud what he called a calculus rationator; it is not hard to see that these projects are interconnected, since a perfect universal characteristic would comprise, it seems, a logical calculus. LEIBNIZ did not publish the incomplete results which he had obtained, and consequently his ideas had no continuators, with the exception of LAMBERT and some others, up to the time when BOOLE, DE MORGAN, SCHRODER, MacCoLL, and others rediscovered his theorems. But when the investigations of the principles of mathematics became the chief task of logical symbolism, the aspect of symbolic logic as a calculus ceased to be of such importance, as we see in the work of FREGE and RUSSELL. FREGE'S symbolism, though far better for logical analysis than BOOLE'S or the more modern PEANO'S, for instance, is far inferior to PEANo's symbolism in which the merits of internationality and power of expressing mathematical theorems are very satisfactorily attained- in practical convenience. RUSSELL, especially in his later works, has used the ideas of FREGE, many of which he discovered subsequently to, but independently of, FREGE, and modified the symbolism of PEANO as little as possible. Still, the complications thus introduced take away that simple character which seems necessary to a calculus, and which BOOLE and others reached by passing over certain distinctions which a subtler logic has shown us must ultimately be made." (Couturat 1914: Preface pages VI-VII)

There are others in the footnotes e.g. 1 Cf. A. N. WHITEHEAD, A Treatise on Universal Algebra with Applications, Cambridge, 1898; Venn, Peirce, Ladd-Franklin, etc etc.

BillWvbailey (talk) 03:10, 6 January 2012 (UTC)

If you go to the wikipedia article for truth table you'll see that the very last sentence is " Ludwig Wittgenstein is often credited with [truth table's] invention in the Tractatus Logico-Philosophicus" which is cited. I didn't intend for this to be a big debate. I don't think Wittgenstein's contributions to symboic logic were that great compared to his predecessors or contemporaries. However, as someone who's uses truth tables, and who's wondered who invented them, i think it's worthy to include wittgenstein in the history, as minute as his contribution may be. — Preceding unsigned comment added by Xenfreak (talk • contribs) 02:03, 6 January 2012 (UTC)

I agree it seems a minor quibble, but I'm not going to accept a rumor. Just because wikipedia states it is not a cause for celebration. I may have to edit that page to set the record straight: here are the facts -- Post's PhD thesis was finished in 1920 and published in 1921 (cf van Heijenoort), Tractatus appeared in 1921 and was translated in the English language in 1922 by Ogden and Ramsey (cf Monk 2005). The apparent (as-yet-to-be verified) truth is that the development was independent and simultaneous. I'm not convinced either were the first, per the quote from Post. And I've seen the tables somewhere in the late 1800's Venn or Jevons or somewhere (I'll search). Plus the now-huge bibliography above mentions neither. And really until further notice the point is moot -- I haven't been able to convince myself that either had influence on the development of symbolic logic in the 20th C. The only citation of either that I know of, tenuous as it is, is via the bibliography in the 2nd edition of Principia Mathematica where Russell cites Wittgenstein (his erstwhile student and bugbear). But this is in the context of the axiom of reducibility, not "truth tables". I know little about Tarski and Quine, so connections may be there. My advice to both of us is to leave W and Post out of the article until we know more. It appears that far more influential is de Morgan and Couturat + Huntington via Shannon; for example, in a table on page 475 he equates the 'Interpration in relay circuits' to 'Interpretation of the Calculus of Propositions'. Bill Wvbailey (talk) 03:10, 6 January 2012 (UTC)

sorry, i just saw this comment. I'll edit wittgenstein out (for now.) But to keep him out, you'll have to convince me: why doesn't the reference in the truth-tables article that wittgenstein being considered the father of truth-tables suffice? Xenfreak (talk) 03:40, 6 January 2012 (UTC)

Here's why I discourage us from crediting either until we understand better. I just found three references: (1) Hans Reichenbach 1947 Elements of Symbolic Logic, Dover Publications, Inc. NY, ISBN:0-486-24004-5, (2) Alfred Tarski 1946/1961 Introduction to Logic and to the Methodology of Deductive Sciences, Dover Publications Inc, NY, ISBN: 0-486-28462-X (pbk.), and Kleene 1952 (see full citation below). The first two ultimately attribute the notion of truth functions (tables?) to Peirce; Kleene attributing 2-value tables to Post 1921. The first two authors are trivializing "truth tables", considering them to be just manifestations of "truth functions":

(1) From Reichenbach:

"[footnote 1] ¹Truth tables were used by L. Wittgenstein, Tractatus Logic-Philosophicus, Harcourt, Brace, New York, 1922, p. 93, and by E. L. Post, Amer. Journal of Math., XLIII, 1921, p. 163. Materially, the definition of propostional operations in terms of truth and falsehood was used earlier, for instance in B. Russell and A. N. Whitehead, Principia Mathematical, Vol. I., 1910, p. 6-8. Furthermore, C. S. Peirce employed this definition; cf footnote on p. 30." (page 27 in Reichenbach 1947)

(2) From Tarski:

"Chapter 13. Symbolism of sentential calculus; truth functions and truth tables

"There exists a certain and simple and general method, called METHOD OF TRUTH TABLES OR MATRICES, which enabls us, in any particular case, to recognize whether a given sentence from the domain of the sentential caluclus us true, and whether, therefore it can be counted among the laws of this calculus.⁵" [footnote 5: "This method originitates with PEIRCE (who has already been cited at an earlier occasion; cf. footnote 2 on p. 14." (but this reference only mentions that CH. S. PEIRCe (1839-1914) was 'an outstanding American logician and philospher"). (page 38, both text and footnote; capitals in the original).

(3) Kleene 1952 Introduction to Meta-Mathematics, North-Holland Publishing Company, Amsterdam NY, ISBN 0-7204-2103-9, credits Lukasiewicz 1920 for 3-variable tables, and Post 1921 for two variable tables (cf p. 140). Post 1921 (same as the above) is referenced by Kleene on page 531; Wittgenstein is not to be found in the extensive references. BillWvbailey (talk) 04:41, 6 January 2012 (UTC)

i was linked to this article:

http://frege.brown.edu/heck/pdf/unpublished/TruthInFrege.pdf

In it it says

"Frege has sometimes been credited with the discovery of truth-tables (Kneale and Kneale, 1962, pp. 420, 531; Wittgenstein, 1979b, pp. 135ff), and something akin to truth-tables is indeed present in Frege’s early work."

This are sources from Wittgenstein and Kneale, who we include. It says this at the bottom of page 9, Sentential Connectives as Truth Functions. Later in the page, however, the section concludes:

"That said, so far as we know, nowhere in his later writings does Frege give the sort of 'tabular' account that Wittgenstein and the Kneales mention, so there is no real basis for attributing the discovery of truth-tables to Frege. 13" (emphasis in original text.)

And in the footnote on the bottom of page 10, it says:

"Moreover, Frege never considers truth-tables for arbitary formulae, but only for the simplest cases, and there is no indication that he realized, as both Wittgenstein and Post (1921) did, that truth-tables can be used to determine the validity of an arbitrary propositional formula. As is now widely recognized, then, it is Wittgenstein and Post who deserve the real credit for the discovery of truth-tables."

So perhaps, the best thing to say would be to say that the seeds of truth tables were in Frege's work, but the actual tables themselves were developed by Wittgenstein and Post. Xenfreak (talk) 05:23, 6 January 2012 (UTC)

EDIT

i was also linked to this article

http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1219&context=russelljournal

It mentions Russell's use of truth-tables which predates Post and Wittgenstein. I think it would be best to mention all four people (russell and frege in the "pre-truth-tables" and Post and Wittgenstein for their tabular development with the quote that the article concludes with

"It is far from clear that anyone person should be given the title of "inventor" of truth-tables"

This is how i'll develop the article for now. If any further information arises, let me know, and i'll alter the history section accordingly. (and you're free, and more than welcome, to do so yourself if you'd like. Xenfreak (talk) 06:15, 6 January 2012 (UTC)

I put all the considerations for truth tables, and their controversy in the section. I cited all the major claims. What i didn't do, however, was copy the citations from the PDF i used as he (Shosky) cited them. I just ended up using the PDF itself (i.e. instead of citing Anscombe, Quine, etc. I just cited the PDF.) This is fine for me, but if anyone in particular is feeling very persnickety about citing my claims in this way, let me know. It won't take more than a moment to copy over the actual citations. Xenfreak (talk) 07:05, 6 January 2012 (UTC)

---

What you wrote looks quite good and it addresses the problem well. This content should also should go into the article Truth tables perhaps in more detail, e.g. the cites I found + yours. Since you started I'll let you finish and then do I'll some editing and make additions. Your sources are interesting, to say the least.

Distinguish "truth table" versus truth functions: Authors seem to be mixing "truth table" with an "algebra of switching"; see McCluskey reference above and below. What I do think you/me should do is be sure to clearly separate out the graphical expedient of a "truth table" for evaluation of truth functions, as opposed to the notion of truth function itself (as it developed historically), i.e. a propositional function that evaluates to {truth, falsity} (this being Russell's usage in PM, that he cites to Frege).

Boolean Algebra of classes versus a logic of truth-functions: Boole developed a theory of classes as opposed to [who? Venn in his 1881 Symbolic Logic? Peirce? Frege? Russell 1903? Russell-Whitehead PM? All?] developing symbolic (bivalent) logic, i.e. with truth and falsity. For Boole: "Let the symbols 1 and 0 be respectively used to denote the Universe [of discourse] and Nothing" (p. 89) and "If then we construct an Algebra in which the only particular symbols of number shall be 0 and 1 and in which every general symbol as x, y etc. shall be understood to admit only of the above special determination . . ." (p. 91*).

• From his On the Foundations of the Mathematical Theory of Logic' to be found in Grattan-Gunness and Gerard Bornet (ed.) 1997 George Boole: Selected Manuscripts on Logic and its Philosophy, Birkhauser Verlag, , Basel Switzerland, ISBN 3-7643-5456-9.

The more I read about this the more important this particular distinction seems, although without a look at e.g. Quine and the problem of mixing "truth" and "falsity" into a symbol-system, I need to get cc's of Peirce.

Confusion of conceptual notions, cf Huntington's 1931 basis set of axioms for both "Boolean Algebra of classes" and "Symbolic Logic of truth functions": This confusion of systems (logic of class membership versus logic of truth-functions) needs to be woven into the article in some detail and developed, because it is from the Boolean notation via Couturat [to Shannon to Veitch to Karnaugh that we now have what we engineers were taught how to manipulate truth tables, and why many of us learned with the Boolean notation, not the symbolic-logic notion. From Shannon via Huntington we got the mixing of the two theories. Here's Huntington:

"Three sets of independent postulates for the algebra of logic, or Boolean algebra, were published by the present writer in 1904. . . . In the meantime, the primitive propositions of Section A of the Principia Mathematica (1910) were expressed in terms of a class called the class of "elementary propositions," a binary operation called "disjunction," and a unary operation called "negation" ; and Bernstein has recently shown (June, 1931) how these primitive propositions can be expressed in abstract mathematical form in terms of (K, +, '). Since the relation between the theory of the Principia and the theory of Boolean algebra has been the subject of some discussion, it becomes a matter of interest to construct a set of independent postulates for Boolean algebra explicitly in terms of (K, +,'), for comparison with the Principia." (Huntington 1931:274ff).

So it looks like Bernstein is very important here. [Note added 17 January 2012: Yes, this is a signficant paper. But I can't find the formation rules, i.e. the syntactic rules. These seem tacit. See more at the references listed above. Wvbailey (talk) 20:22, 17 January 2012 (UTC)]

A confusion of notations: No wonder students get confused and mix the two up. The notations have been used interchangeably, the algebraic xy, x+y, bar-x or x' appears in symbolic truth-functional logic and so does , x & y, x V y, ~x which is clearly symbolic. From what I can see the path from Couturat to the engineers via Shannon to Veitch to Karnaugh is to blame for this: all the notations in these 4 are Boolean in nature. All my engineering texts are in algebraic notation, Notably my E. J. McCluskey 1965. McCluskey was a student of Quine; reference [1] is to you guessed it -- Shannon 1938:

"This algebra will here be called switching algebra. It is identical with a Boolean algebra and was originally applied to switcing circuits [1] by reinterpreting Boolean algebra in terms of switching circuits rather than by developing a switching albebra directly, as will be done here." (p. 66)

When and how did the separation of symbolisms occur? Where did usage of &, V, ~ (or bent-bar) come from? The symbolic notations { V, &, ➙ } is used by Kleene 1952, Reichenbach 1947, and Tarski 1941. The V is used by Herbrand 1930 Investigations in proof theory: the properties of true propositions uses {V, ~ }, Skolem 1928 On Mathematical Logic uses the Boolean set. Goedel 1930 The Completeness of the Functional Calculus uses { V, overbar for negation, &, ➙ }, Goedel 1931 uses { V, ~, ⊃ for implication , & }. So something happened in about 1928-1930 (a need to separate the algebraic +, - and * from the logical?) that led to a clear distinction between "Boolean Algebra of classes" and "Symbolic logic of (bivalent) truth-functions".

Wvbailey (talk) 18:20, 6 January 2012 (UTC)Bill

Sorry for my delay. Unfortunately, i expect to be very busy starting as soon as tomorrow, and probably will lack the time to make these edits. Feel more than free to add in anything you fee is relevent. If i can find some time later today, i'll do the same

Best

-xenfreak — Preceding unsigned comment added by Xenfreak (talk • contribs) 16:16, 8 January 2012 (UTC)

Classical and Non-Classical Propositional Calculus

So this article, as well as the Rule of replacement articles, treat the Propositional Calculus as inherently classical. There is mention of Intuitionist logic on this page, but not, say, the pages for Transposition (logic) or De Morgan's laws. (It does get mentioned on Double negative elimination.) I wanted to know if this was deliberate and well-motivated, or an oversight.

Is there any reason I (or others, if they like) should not go ahead and point out places where classical logic differs from prominent non-classical logics, and add the adjective classical in appropriate places?

Notapipe (talk) 04:50, 5 June 2012 (UTC)

You speak as if those other logics are just as good as classical logic. They are not. Classical logic is the true logic. The others such as intuitionistic logic are just crippled variants which might better be described as logic-like algebras. Any competent person uses classical logic when dealing with serious problems.

If you want to add a brief paragraph at the end of those articles saying how things are different in this or that non-classical logic, then you may do so. But I would not advise changing the bulk of the articles, especially the leads. JRSpriggs (talk) 07:51, 5 June 2012 (UTC)

Is there an error in the last item "Conditional proof (conditional introduction)" there seem to be one too many implies symbols. In other words, shouldn't it say:

"That is, $(p\vdash q)\vdash (p\to q)$." ? This seems to translate what is written in words above.

[If not, then some further explanation is required here.]