Talk:Judgment (mathematical logic)

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The contents of the Logical assertion page were merged into Judgment (mathematical logic) on 7 May 2018. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page.

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)[reply]

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.

${\displaystyle \phi \to \psi }$

that is clear, ${\displaystyle \to }$ is part of the object language.

${\displaystyle \Gamma \vdash \phi }$

that is clear, this is a full-citizen of the metatheory, ${\displaystyle \vdash }$ is not part of the object language, it can be used rather when saying metatheorems.

${\displaystyle {\frac {\langle \Gamma ,\phi \mid \psi \rangle }{\langle \Gamma \mid \phi \to \psi \rangle }}}$

this is a rule of inference in natural deduction, introducing ${\displaystyle \to }$ (implication). I use the sign horizontal line for separating premises from conclusion in the rule of inference. Ordered pair separated with vertical bar ${\displaystyle \langle \Gamma \mid \phi \rangle }$ is used here to form syntactically a judgment. It is more familiar to write ${\displaystyle \Gamma \vdash \phi }$ instead, but I want to avoid here using the same sign ${\displaystyle \vdash }$ in both usages.

What is the difference between ${\displaystyle \langle \Gamma \mid \phi \rangle }$ and ${\displaystyle \Gamma \vdash \phi }$, 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 ${\displaystyle {\frac {\dots \;\;\;\;\;\;\;\;\dots }{\dots }}}$

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)[reply]

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. {{cite book}}: Check date values in: |year= (help); External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help) 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)[reply]

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)[reply]

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)[reply]

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)[reply]

I find the explanation of "judgment" as clear as mud. Not one single sentence on the page makes sense to me. I have been seeing this word "judgment" in the logic pages for the last week, and nowhere is it clear what is supposed to mean, especially on the "judgment" page itself.

Is "judgment/judgement" a synonym for "assertion"? Guessing by all the contexts in which I have seen it on wikipedia, I would say that the answer is almost certainly "yes". If so, I think the authors of these web pages should stop obfuscating and use plain English. Every mathematician and every reasonably well educated person knows what an "assertion" is. I have not seen the word "judgment" in any of my 45 mathematical logic books. Nor have I seen it in any of the other 173 mathematics books on my bookshelf. The only place I have ever seen it is on wikipedia. I think that if wikipedia is supposed to explain things to people who are not already specialist experts in each topic, then surely the comprehensible word "assertion" should be used instead of the incomprehensible word "judgment".

The first paragraph of the "judgment" article lists a few things that a "can be", and none of these "can be" options is comprehensible to me. The second paragraph muddies the waters even further. The 3rd and 4th paragraphs are just way off the end of the obfuscation meter.

If there is some subtle or not-subtle difference between "judgment" and "assertion", then surely the page on "judgment" should clarify this. My guess is that the difference is purely of interest to certain sections of the philosophical community. It certainly has no interest for mathematicians, in my opinion. Kantian splitting of hairs in the undertones of words is not really what mathematicians are into, in my opinion. And outside the mathematical community, I think most people will be even more perplexed by "judgment", especially after they have read this wikipedia page on the subject!! If even mathematicians will find this page incomprehensible, what hope does anyone else have?

Suggested actions:

1. Add clarification of the difference, if any, between "judgment" and "assertion" on the "judgment" page.
2. Explain if the difference, if any, has any implications at all for mathematical logic or mathematics.
3. Add clarification of the spelling variations "judgment" and "judgement". They are both used within this limited philosophical mathematical logic context. (The links which I have followed all use "judgement".)
4. Remove the word "judgment" as far as possible from all wikipedia pages about mathematical logic, and replace it with "assertion", which will be understood by 99% of readers instead of less than 1%. In my opinion, there will be no harm at all in doing a search/replace for "judgment" or "judgement" with "assertion" in the mathematical logic pages. And this will do a large amount of good! (I mean "good" in the sense of plain English, not in the Kantian sense.)

--Alan U. Kennington (talk) 07:20, 15 June 2014 (UTC)[reply]

Well ... the situation is confusing. I only have a few books on logic. None of them use the word "judgment". Well, they also don't use the words "assertion" or "logical assertion", either. (I just looked). So, actually I'm befuddled as to what either of these could be. ... However, the article on natural deduction uses the word "judgment" a lot. There, it is used in a highly type-theoretical sense (type theory), in the style "thing x is a thing of type T" so for example, the following are judgments: "t is a term" (as in term algebra or term (logic)), or "v is a variable" (as in first order logic) or "p is a proposition" (as in propositional logic). So, at least for natural deduction, it would appear that "judgment" is a word with a fairly clear meaning. It also seems that, in type theory, "judgment" is also in common circulation. The problem is that this article does not explain what a judgment actually is, what this word means, in any effective sense. (It also tries to explain what a "logical assertion" is, and completely fails to do that, in any coherent fashion). I'm slapping "expert attention required" on this article.

To argue with myself: saying that "t is a term" does seem like a kind-of assertion on it's face, but it cannot be, it has to be a meta-assertion. That is "t is a term" is a required ingredient to define first-order logic. So it can't be a "logical assertion", because we are in the process of defining the logic, to begin with. So "t is a term" has to be a kind-of meta-logical assertion. Per Martin-Lof seems to say that its "something you know", but "knowing something" is a part of modal logic, and so if you are trying to define what it means to know or believe, you can't use the words "know" or "belief" to define themselves, as that would be circular. Thus, "judgment" is used to avoid circularity. 67.198.37.16 (talk) 16:04, 18 December 2018 (UTC)[reply]

As raised above, judgment is indeed a synonym for assertion (see Martin-Löf citation). Further, the content at logical assertion is essentially a special case of the content of this article. Hence I propose logical assertion be merged into Judgment (mathematical logic) article. Quiddital (talk) 22:45, 13 August 2016 (UTC)[reply]

 Done Klbrain (talk) 15:22, 7 May 2018 (UTC)[reply]

It's not at all clear that this merge was correct. One can only make a "logical assertion" if one already knows what a logic is. But until one defines a logic, there's no such thing as a "logical assertion". By contrast, judgments seem to be used to define logics. In particular, after reading natural deduction, it would appear that a judgment should be understood as a "type declaration". For example "t is a term (logic)" or "v is a variable" or "p is a proposition" (propositional logic). So, for example, the last statement is a "type judgment" used to define what "p" is. We need to efine what "p" is in order to define what propositional logic is. Only after defining what propositional logic is, can we start to define what a logical assertion is. Right? I'm confused, as this article is very poorly written. 67.198.37.16 (talk) 21:41, 18 December 2018 (UTC

The current article contains this text:

For example, if p = "x is even", the implication

${\displaystyle (\vdash p)\rightarrow (x{\pmod {2}}\equiv 0)}$

which is clear-as-mud and/or wrong. I suspect the parenthesis is mis-placed. I suspect it should be

${\displaystyle \vdash (p\rightarrow (x{\pmod {2}}\equiv 0))}$

If I try to read the former, with the bad paren placement, I start with ${\displaystyle (\vdash p)}$ which can be readily recognized as a tautology: from nothing at all, from thin air, I can prove that p is true. Since its a tautology, p is always true. So this reduces to ${\displaystyle true\rightarrow (x{\pmod {2}}\equiv 0)}$, which is blatently wrong, unless x is restricted to be a member of the set of even natural numbers. For example, x must not belong to the set of quadruped furry animals, because ${\displaystyle x{\pmod {2}}}$ is not even well-defined for furry animals.

The second form with re-arranged parenthesis, makes slightly more sense, but is still ill-defined. There, the reading starts with ${\displaystyle p\rightarrow (x{\pmod {2}}\equiv 0)}$, so that p is now some proposition, might be true, might be false, who knows, but whenever p is true, then if follows that ${\displaystyle x{\pmod {2}}\equiv 0}$. Presumably, it works out that whenever p is true, then x is even.

Oh, hang on. It also says: ''For example, if p = "x is even",... and so perhaps this is meant to be the definition of what p is? In that case, simple substitution works. That is, perform the substitution ${\displaystyle p/(x\!\!\!\!{\pmod {2}}\equiv 0)}$ aka ${\displaystyle [p:=(x\!\!\!\!{\pmod {2}}\equiv 0)]}$. The first formula gives

${\displaystyle (\vdash (x{\pmod {2}}\equiv 0))\rightarrow (x{\pmod {2}}\equiv 0)}$

which is insane because ${\displaystyle (\vdash (x{\pmod {2}}\equiv 0))}$ is not a tautology. The second form gives

${\displaystyle \vdash ((x{\pmod {2}}\equiv 0)\rightarrow (x{\pmod {2}}\equiv 0))}$

which obviously is a tautology, and totally acceptable. So I'm editing the article to correct what seems to be an obvious error. 67.198.37.16 (talk) 15:48, 18 December 2018 (UTC)[reply]

No, I'm just gonna bulk remove that. This content appeared due to a merge that should not have been performed in the first place. Its crazy gobbly-gook; it was incoherent from the get-go. 67.198.37.16 (talk) 22:17, 18 December 2018 (UTC)[reply]