Talk:Judgment (mathematical logic)

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[edit] Judgment as a philosophic issue?

Am confused. Is this the sort of topic that Bertrand Russell and Willard Quine devoted much attention to -- i.e. the nature of an assertion as an "objective truth" i.e. observable by others? Lemme know, thanks, Bill Wvbailey (talk) 18:45, 3 January 2008 (UTC)

Unfortunately, I lack the knowledge yet for such overview. Till then, word "judgment" is for me just a comprehensive concept of the many ways the various deduction systems use a strange auxiliary concept in their foundation. I used it yet only as a syntactic construct, which is part of the foundation of the big machinery of a deduction sytem.

\phi \to \psi
that is clear, \to is part of the object language.
\Gamma \vdash \phi
that is clear, this is a full-citizen of the metatheory, \vdash is not part of the object language, it can be used rather when saying metatheorems.
\frac{\langle\Gamma, \phi \mid \psi\rangle}{\langle\Gamma \mid \phi \to \psi\rangle}
this is a rule of inference in natural deduction, introducing \to (implication). I use the sign horizontal line for separating premises from conclusion in the rule of inference. Ordered pair separated with vertical bar \langle\Gamma \mid \phi\rangle is used here to form syntactically a judgment. It is more familiar to write \Gamma \vdash \phi instead, but I want to avoid here using the same sign \vdash in both usages.

What is the difference between \langle\Gamma \mid \phi\rangle and \Gamma \vdash \phi, if any? I do not know.

I regard the difference analogous to another question: What is the difference between

  • a metatheorem claiming "If … and …, then surely …",
  • a rule of inference of the form \frac{\dots \;\;\;\;\;\;\;\; \dots}{\dots}

they "say the same", but somehow seem for me to be used differently. A metatheorem is a full-citizen part of the metatheory. A rule of inference, together with all metasigns (space between premises, line between sequence of premises and conclusion), is something auxiliary thing for establishing the foundation of the deduction system.

Thus, unfortunately my knowledge is far from enabling me grasping the semantics of these notions, I just used them separately as syntactic auxiliary constructs.

Physis (talk) 20:36, 3 January 2008 (UTC)


Dear Wvbailey,

I tried to find sources which discuss these questions.Till now, I found Pfenning, Frank (2004 Spring). "Natural Deduction", 15-815 Automated Theorem Proving.  It seems to fit here, but I shall have to read through it thoroughly yet. I shall check Kneale & Kneale The Development of Logic, and Curry Combinatory Logic in such questions tomorrow.

Best wishes,

Physis (talk) 04:37, 4 January 2008 (UTC)

Thank You for making me interested in the ontology of the concept "judgment", transcending regarding it only on a technical level. I have found also

Martin-Löf, Per (1983). On the meaning of the logical constants and the justifications of the logical laws. Siena Lectures.

I hope it addresses such questions. I have just found it, I have not read it through yet. It seems to introduce a lot of new things for me: proposition and judgment are different concepts; a proposition can be true, a judgment can be evident. Best wishes, Physis (talk) 03:14, 5 January 2008 (UTC)


Dear Wvbailey,

As I said, I know nothing about the ontology of the concept "judgment". Maybe, my argumentation above is entirely erranous. There is a thread on Talk:Hilbert-style deduction system#Concept of “judgment” as a more characteristic difference among various deduction systems, the answers I received there to my questions may ellucidate the problem better than the above thoughts of mine.

Best wishes,

Physis (talk) 12:59, 7 January 2008 (UTC)

Bertrand Russell and "judgment":
In Bertrand Russell's 1912 The Problems of Philosophy (Oxford U Press, 1997 edition) there appears a chapter VII called "On Our Knowledge of General Principles" in which he includes his expression of implication:
"In other words, 'anything implied by a true propostion is true', or 'whatever follows from a true propostion is true.' (p. 71)
On the next page he mentions the three: (1) The law of identity, (2) The law of contradiction, (3) The law of excluded middle, and states:
"The three laws are samples of self-evident logical principles, but are not really more fundamental or more self-evident than varous other similar principles: for instance, the one we considered just now, which states that what follows from a true premisse is true." (p.72-73)
There follows a discussion of "a priori" knowledge (what he says he doesn't want to call "innate" but I would have chosen to call "innate knowldege" i.e. "built-in" or "by-design" or "hardwired"). By chapter XI "On Intuitive Knowledge" Russell is discussing "judgements":
"In addition to general principles, the other kind of self-evident truths are those immediately derived from sensation. We will call such truths 'truths of perception', and the judgments expressing them we will call 'judgments of perception.' (p. 113)
Russell's discussion in his 1912 closely parallels his discussion in the 1913-1927 Principia Mathematica, Chapter III of his introduction titled "Definition and Systematic Ambiguity of Truth and Falsehood" (p. 41-47 of my edition). Here he again discusses "judgment of perception":
"In fact, we may define truth, where such judgments are concerned, as consisting in the fact that there is a complex corresponding to the discursive thought which is the judgment. That is, when we judge "a has the relation R to b," our judment is said to the true when there is a complex "a-in-the-relation-R-to-b," and is said to be false when this is not the case. This is a definition of truth and falsehood in relation to judgments of this kind." (p. 43)
There is more, both preceding and following this, that I believe is very good. My belief, based on my inquiries into "consciousness" is that he is correct, but he is using some archaic notions such as "a priori" that are nowadays better described as "buit-in-by-design" (here I mean: selected by Darwinian evolution as the best answers to the challenges of a merciless world). This is all I've been able to find in my books about "judgement" in context of truth and falsity -- none of the books have the word in their indexes. Bill Wvbailey (talk) 15:55, 7 January 2008 (UTC)