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Is "truth-value" a British spelling or something?
In American English, it seems inappropriate unless used as an attributive.
("In topos theory, the truth-value object takes the place of the set of truth values.")
-- Toby 04:22 Mar 3, 2003 (UTC)
As noted in the discussion on the Intuitionistic logic, there is a common misconception that Intuitionistic Logic has three truth values. This misconception has been removed from the Intuitionistic Logic article, and hence it has been removed from here. -- FatherBrain 14:35, 22 September 2005 (UTC)
I actually prefer the problematic-but-imprecise old formulation, A simple intuitionistic logic has truth values of true, false, and unknown, to what we have now, the precise, problematic in intuitionistic logic there may be any number of truth values between true and false, related by a partial order (not introduced by FatherBrain). There's a strong reason why Heyting algebras don't give a truth-value semantics for intuitionistic logic, because the structure of the Heyting algebra for a given logic changes with the logic, and there's nothing like a Stone representation theorem to tell you how the families of theories are compounded out of (in classical logic's case) power set algebras.
And there's a good reason for this. The BHK semantics for classical logic is expressed in terms of proof conditions, not truth conditions. The case of implication, in particular, can't be reduced to truth conditions, since the value of $\phi \rightarrow \psi$ turns on what constructions there are that take argument of type φ and give result of type ψ, something that isn't compositional in terms of formulae, but instead is compositional in the sense of proof constructions.
As a side note, if we restrict the language of intuitionistic propositional logic to have no propositional variables, then it has the same theory as propositional-variable free classical logic. This is because truth tables work for intuitionistic propositional logic: if the atomic formulae are all either true or false, then we can determine the value of the whole formula by using the classical truth table.
The talk of truth values in topos theory is different than that in logic: the usual construction of a topos from intuitionistic type theory yields two truth values (up to isomorphism), i.e. two global elements of the subobject classifier, namely true and false, and this does not mean that intuitionistic logic is classical logic.
So I shall delete lots from this article. — Charles Stewart(talk) 07:54, 25 June 2009 (UTC)
Re-reading Andrej Bauer's article, I see now that what I wrote was confused/confusing. So apologies for the rash edit. But could you please expand the article to make your points more clearly? In particular, what role do truth values play in a logic, and what is it that makes a logic truth valuational, if "Having truth values in this sense does not make a logic truth valuational"? Also, I don't see how the fact that intuitionistic logic has a proof-based semantics can be used as an argument that it does not have a truth-based semantics. — Noamz (talk) 11:17, 25 June 2009 (UTC)
Putting a cleaned-up version of the above would be a bit of a personal essay: I just wanted to put a defensible case against what was being asserted in the article. A better thing would be to survey attitudes to truth values, which range from "True and false are the only things that could be truth values" through "a logic has truth values if it has a truth-functional semantics expressible as through finite matrices; truth values are then the values the rows and columns can take" through my position to "Lindenbaum algebras are algebras of truth values". — Charles Stewart(talk) 09:15, 26 June 2009 (UTC)
"Truth value" as a special case of probability?
I ran into this in Paul C. Rosenbloom, The Elements of Mathematical Logic, Dover Publications, Inc. Mineola, NY, first published 1950, Dover edition 2005. Unfortunately my edition is flawed and contains no bibliography (!), hence the empty references [ ]:
"The notion of truth value may be regarded as a special case of that of probability, which is, in turn, essentially equivalent to that of measure. (See Kolmogoroff, Grundbegriffe der Wahrscheinlichketsrechnung, Berlin 1933; Cramer, Methods of Mthematical Statistics, Princeton, 1946, Reichenbach [ ], Koopman [ ], Kleene and Evans [ ]). In most of the precise treatments of probability the concept is defined on an algebra of classes. Since propostions also form a Boolean algebra, it should be easy and desirable to treat directly the notion of the probability of a propostion." (p. 198)
I believe that this has progressed through the years into "machine learning"... but am otherwise unfamiliar with the concepts, except this: they seem to echo the philosophy of Bertrand Russell. Does anyone have any comments? Thanks, wvbaileyWvbailey 21:08, 27 June 2006 (UTC)
"Truth value" is better than "Logical value"
It was a mistake to move the article from Truth value to Logical value. The phrase truth value is the established terminology in the logic community. Frege 21:36, 21 November 2006 (UTC)
Definition of "truth value", PM, Frege, Peano, Wittgenstein, etc
This is just to hold the references:
From Principia Mathematica 2nd Edition page 7 and repeated on page 72:
"Truth-values. The "truth-value" of a proposition is truth if it is true, and falsehood if it is false*.
[In footnote:] "*This phrase is due to Frege" [PM 2nd edition p. 7]
I have not located this in Frege so far [Begriffsschrift, van Heijenoort 1967:1ff]. Frege uses the notions 'affirmation' and 'denial' in his definition of "conditionality":
"§5. If A and B stand for contents that can become judgements (§2), there are the following four possibilities:
(1) A is affirmed and B is affirmed;
(2) A is affirmed and B is denied;
(3) A is denied and B is affirmed;
(4) A is denied and B is denied.
"Now [drawing here symbolizing (~B V A)] stands for the judgment that the third of these possibilites does not take place, but one of the three others does. Accordingly, if [the sign (~B V A)] is denied, this means that the third possibility takes place, hence that A is denied and B affirmed." (van Heijenoort 1967:13-14) [see Begriffsschrift ]
Peano 1889 defines two signs ? to represent "the false" and ? to represent the true:
"[The sign ? means the true, or identity; but we never use this sign.]
"The sign ? means the false, or the absurd." (brackets in original, Peano 1889 in van Heijenoort 1967:86-87
These sign ? also "denotes the class that contains no individuals". Again in brackets, Peano states that the sign ? "denotes the class composed of all individuals under consideration" and that this sign will not be used. (page 88).
Other places to search: Russell 1903 (Principles of Mathematics), Hilbert in van Heijenoort, Dedekind. Back further to Boole.
Wittgenstein in Tractatus (in particular 4.2ff:
"4.3 The truth-possibilities of the elementary propositions mean the possibiliites of the existence and nonexistence of the atomic facts.
"4.31 The truth-possibiliites can be presented by schemata [truth tables] of the following kind ("T" means "true", "F" "false" . . .." BillWvbailey (talk) 17:18, 13 November 2012 (UTC)
Alfred North Whitehead, Bertrand Russell 1927 Principia Mathematica to *56 (PM second edition), Paperback Edition to *56, Cambridge University Press UK. No ISBN, no LCCCN.
Jean van Heijenoort, 1967 from Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931 3rd printing 1976, Harvard University Press, Cambridge, MA, ISBN:0-674-32449-8 (pbk.)
Ludwig Wittgenstein, Major Works Selected Philosophical Writings, (Tractatus Logic-Philosphicus 1918), HarperPerennial Modernthought, 2009 harperCollins Publishers, ISBN978-0-06-155024-9.
Intuitionistic logic
In this revision, the intro was moved into the body - but the sentence on intuitionistic logic was removed altogether. As there is no mention of this in the edit summary, I'm assuming that this was accidental. Because logical statements in intuitionistic logic are of course assigned truth values. I'll add something back in there.