Logic

[Goliath]

And since my premise was untrue, it follows that my conclusion was also untrue: you should not all bow down and worship me.

Actually that is not so. It is perfectly well possible to arrive at a true conclusion from a false premise via logic. Your conclusion that we should all bow down and worship you could still be true (in the minimalist interpretation: it could still be possible that we nod our heads and say "yes indeed, we should bow down and worship Camelopard") even if you are not God. It just does not follow logically from the previous statements.

You too have not read closely enough.

What I said was not: "You should all bow down and worship me."

What I said was: "Therefore, you should all bow down and worship me."

"Therefore" means "from that fact or reason or as a result." In other words: you should all bow down and worship me, from the fact or reason or as a result of my being God.

It may be true that you should all bow down and worship me. But it is not true that you should bow down and worship me by reason of my godhood.

 

[Shaw]

Knowledge of set theory is no substitute for attentive reading.

Grouping statements in a list, specially with an ill defined element, was a poor choice and had nothing to do with my reading of it. If he wanted three independent statements, why list them numerically? More to the point, why not be specific about which Richmond in the first two lines?

As for attentive reading, why didn't you quote his list with the numerical elements? That was part of the original statement. And I never said anything about Sacramento, that wasn't the part I had issue with (it was the ill define Richmonds).

Read, read, read.

 

[Plain Simple]

I can assure you that as a professional mathematician I'm quite familiar with set theory and logic...

In what field of study?

My background is in differential geometry and differential topology, but I was still expected to have an adequate background in things like logic, set theory, group theory, etc.

How in the world can you do proofs without knowing logic? How did you get through Modern Algebra or Real Analysis without an understanding of logic?

What is your definition of a professional mathematician? Granted, I was a professional mathematician when I was being paid by the National Science Foundation and the Department of Energy to do math (I was doing research in tight immersions of surfaces, both smooth and polyhedral)... but I still hold myself (and others) to a pretty high standard of what counts as a mathematician (and I fall short of that currently).

Sorry... but I'm not seeing it here from you in this topic. Here is a link so that you have some idea about my background (for full disclosure).

I don't really feel the need to disclose much about my personal life here, but in short: I work in a department of mathematics at a university (hence professional mathematician) in the field of analysis. I'm not claiming I'm a genius nor that I live up to your high standard of what a mathematician should be (probably not, I don't think I'm anywhere near 'up there'), but fact is that my daily life is concerned with mathematics and someone pays me for that. In addition I have a Ph.D. in math and some papers published in mathematical journals, so peer review seems to be satisfied with my use of logic.

How did you come to the apparent conclusion that I don't have an adequate background in things like logic, set theory, group theory or that I do proofs without knowing logic or any of that? I only said that I'm not a logician and so I'm not familiar with all the intricacies of the field of logic. Seems reasonable, doesn't it? Logic is a big field of research, why would I know anything about those details if I'm not in that field (and even people who are in that field are probably only familiar with a little specialised part of it).

If I anywhere wrote something in this thread that shows that I do not have an adequate knowledge of logic, please show me. If not, let's shift the topic back from me to logic.

I started this thread not to ask a question about logic, but to transfer the discussion from another thread. In my original and second post in the current thread I refrained as much as possible from incorporating my own views on the matter, because the question was to recap the discussion in the previous thread.

And since my premise was untrue, it follows that my conclusion was also untrue: you should not all bow down and worship me.

Actually that is not so. It is perfectly well possible to arrive at a true conclusion from a false premise via logic. Your conclusion that we should all bow down and worship you could still be true (in the minimalist interpretation: it could still be possible that we nod our heads and say "yes indeed, we should bow down and worship Camelopard") even if you are not God. It just does not follow logically from the previous statements.

You too have not read closely enough.

What I said was not: "You should all bow down and worship me."

What I said was: "Therefore, you should all bow down and worship me."

"Therefore" means "from that fact or reason or as a result." In other words: you should all bow down and worship me, from the fact or reason or as a result of my being God.

It may be true that you should all bow down and worship me. But it is not true that you should bow down and worship me by reason of my godhood.

I seem to have misinterpretated your example, although I'm still not exactly sure what it is you're saying. Is this the structure of your argument:

If we have the following statements A, B, and C,

A="Camelopard is God"
B="You should all bow down and worship God"
C="You should all bow down and worship Camelopard"

then the statement is

(A and B) implies C

The truth value of this statement is determined by the truth values of A, B, and C. If "A" is not true, then "(A and B)" is not true, and therefore "(A and B) implies C" is true irrespective of the truth value of C.

Or should I not interpret your "therefore" as a logical implication?


It seems we encounter here first hand one of the reasons why mathematical logic prefers the use of symbols over words: less or no chance for misinterpretation.

 

[Shaw]

How did you come to the apparent conclusion that I don't have an adequate background in things like logic, set theory, group theory or that I do proofs without knowing logic or any of that?

Well, you said...

"As logic I take first order (predicate) logic, albeit in a less formalised way for the sake of discussion (so we can discuss in words, not symbols)..."​

But in logic the symbols are generally a shorthand for the words, so I would think that had you had a background in this area you would have wanted a more formalized means of discussion. The more formal, the less likely that there would be misunderstandings. And it seems like we can post using symbols from what I can tell (assuming you can read these on your system: ∩, ∪, ⊂, ⊃, ⊆, ⊇, ∈, ⊄, ∉, ¬, ∧, ∨, ∴, ∀, ∃).

But I'll take this time to repeat my apology... Sorry, I just didn't see that in what you'd presented here.

I don't really feel the need to disclose much about my personal life here, but in short: I work in a department of mathematics at a university (hence professional mathematician) in the field of analysis. I'm not claiming I'm a genius nor that I live up to your high standard of what a mathematician should be (probably not, I don't think I'm anywhere near 'up there'), but fact is that my daily life is concerned with mathematics and someone pays me for that. In addition I have a Ph.D. in math and some papers published in mathematical journals, so peer review seems to be satisfied with my use of logic.

I wasn't asking for personal information, I was curious as to what your area is. And yes you meet my standard... which (as I stated) I currently don't.

But honestly I was hoping that you specialized in geometry or topology.

 

[Plain Simple]

^Alas, I don't. My interest is broad and definitely includes those fields, but since it's hard enough to practice mathematics on a reasonable level in ones' own field, I do not find the time to keep up with many other fields beyond a basic understanding of what they're trying to do. (Perhaps I would have if I didn't engage in discussions like this one. )

I can read the symbols you wrote (does everyone?) but I didn't want this to turn into a too technical thread, since it was not my intention to limit this discussion to mathematicians only. So that's why I tried to convey the ideas in words.

If your interested you should read some of the final pages of the thread I linked to in my OP, to see just what kind of discussion was going on there. That might clarify what I was trying to do in this thread --- although that should ideally be clear from my posts in this thread; if it's not I'll accept the blame for that.

The apology is appreciated.

 

[Smiley]

What I'm getting from the early discussion is that calling something illogical is too vague. An argument is either sound, valid but not sound, or invalid. Logical and illogical can work to describe the first and third cases, respectively, but they do not describe the second case precisely.

 

[Captain Robert April]

I'm just finding it gratifying to see a discussion like this on a Star Trek board instead of the usual donnybrook regarding phaser emitter limits or warp core locations.

 

[Augustus]

so is "logical" a point of view then?

 

[Goliath]

Read, read, read.

That was the most unconvincing reply I've read in a long time.

A simple "You're right" would have sufficed.

 

[Shaw]

A simple "You're right" would have sufficed.

Maybe, but a simple "You're wrong" would have been more accurate.

 

[Jadzia]

I would say that logic is using a correct method of deduction.

Logic does not care nor question of validity of the assumptions used to form those corollaries. The truth/validity of those corollaries is not relevant to whether the deduction is logical.

If you are given that B follows from A, and C follows from B, then to say that C follows from A is always logical, independent of what A, B and C are.

Logical is when we have made only correct applications of deductive axioms.

 

[Goliath]

I seem to have misinterpretated your example, although I'm still not exactly sure what it is you're saying. Is this the structure of your argument:

If we have the following statements A, B, and C,

A="Camelopard is God"
B="You should all bow down and worship God"
C="You should all bow down and worship Camelopard"

then the statement is

(A and B) implies C

The truth value of this statement is determined by the truth values of A, B, and C. If "A" is not true, then "(A and B)" is not true, and therefore "(A and B) implies C" is true irrespective of the truth value of C.

Or should I not interpret your "therefore" as a logical implication?

Not as I understand it.

Your final statement--C--seems to be missing an important part of the sentence. "Therefore" is more than just a logical implication: it's a word in a sentence which means, roughly, "for these reasons" or "it follows from this that". Take that word out, and you change the sentence.

This becomes clear if we consider the two sentences in isolation. "You should all bow down and worship me" makes sense by itself, whether you agree with it or not. But "Therefore, you should all bow down and worship me" makes no sense by itself.

This is because "therefore" is what linguists call a deictic expression: it's a word that points to something. In the case of my original syllogism, it points to my premises. If there's nothing to point to, then it loses its meaning.

At the risk of sounding like I'm making a joke, the logical implication in my syllogism is implicit rather than explicitly stated. What is explicit about my statement is that what I say is true, for the reasons already mentioned. And if those reasons are false, how can what I say be true?

It seems to me that, instead of translating what I said properly, you wrote something like this: "If I am God, and you should all bow down and worship God, then you should all bow down and worship me." Or: "The statements 'I am God' and 'You should all bow down and worship God' together imply that you should all bow down and worship me."

And you're right: those are true statements, whether I'm God or not, and whether you should bow down and worship me or not. They're just not what I said, exactly.

It seems we encounter here first hand one of the reasons why mathematical logic prefers the use of symbols over words: less or no chance for misinterpretation.

Yes: but mathematical logic is itself a language, and as with all languages, there are problems when translating the one into the other. I actually encountered this problem recently while writing a paper on historical counterfactuals.

In English, we can say things like, "The cancellation of Operation Sealion was caused by Germany's defeat in the Battle of Britain." But it's actually quite difficult to translate that into a logical conditional: it comes out sounding like, "If Germany was defeated in the Battle of Britain, then Operation Sealion was cancelled." And while that means roughly the same thing, some of the sense of the original has been lost.

The opposite is also true. ""If Germany was defeated in the Battle of Britain, then Operation Sealion was cancelled" implies that "If Operation Sealion was not cancelled, then Germany was not defeated in the Battle of Britain". "If A then B" clearly implies "if not-B then not-A". Simple.

But in "plain" English, that comes out something like, "For the Germans to not cancel Operation Sealion, they would at least have had to win the Battle of Britain first."

 

[Jadzia]

The implication does not require an immediate carrying forward of information:

correct usage: true implication:
(any true statement) implies (any true statement)
(any false statement) implies (any true statement)
(any false statement) implies (any false statement)

incorrect usage: false implication
(any true statement) implies (any false statement)


So it is a true implication to say these things:
"1+1=2" implies "2+2=4"
"2+0=3" implies "1+2=3"
"1+3=5" implies "2+3=-12"

A false implication would be:
"3+3=6" implies "0+1=2"

What I was taught in undergraduate logic is that as a connective word, "implies" and "is implied by" can be used interchangeably with words like "is necessary for / is sufficient for", "if / only if", "therefore / because"



What the implication basically does, as you can see, is to not allow truth to be eroded, only added to.

But if you start out with even one untruth in a set of statements, then logic becomes permissive of any untruths being added to that set, since they are all implied by the atom of untruth. The system of truth falls apart at that point.

Realise then that doubt is carried forward through an implication, it is never erased or diluted further down the line. Undermining the foundations of deduced knowledge with just one atom of untruth (or doubt), will bring down the entire castle that is built upon it... down to the same level of doubt.

Deduced knowledge is that fragile.

Perhaps the most insightful thing to learn from logic, is that whatever "facts" we deduce, they are no more certain and no more knowing than the assumptions upon which they are based.

And since the root assumptions are our sensory interpretations of the human experience, well... it means that all facts (mathematical and empirical) are merely tautologies of our senses. So it is incorrect to grant them the status of being objective/universal/cosmic/divine truths.

 

[Plain Simple]

It seems we encounter here first hand one of the reasons why mathematical logic prefers the use of symbols over words: less or no chance for misinterpretation.

Yes: but mathematical logic is itself a language, and as with all languages, there are problems when translating the one into the other. I actually encountered this problem recently while writing a paper on historical counterfactuals.

In English, we can say things like, "The cancellation of Operation Sealion was caused by Germany's defeat in the Battle of Britain." But it's actually quite difficult to translate that into a logical conditional: it comes out sounding like, "If Germany was defeated in the Battle of Britain, then Operation Sealion was cancelled." And while that means roughly the same thing, some of the sense of the original has been lost.

The opposite is also true. ""If Germany was defeated in the Battle of Britain, then Operation Sealion was cancelled" implies that "If Operation Sealion was not cancelled, then Germany was not defeated in the Battle of Britain". "If A then B" clearly implies "if not-B then not-A". Simple.

But in "plain" English, that comes out something like, "For the Germans to not cancel Operation Sealion, they would at least have had to win the Battle of Britain first."

Indeed, the logical implication does not equal a causal relation. Causality is a difficult concept anyway imo. In 'the real world', how can we ever find out if there's a causal relation between two events? The only way I can see (actually the only way to ever say anything about the 'real world') is within the context of a theory. But then, if that theory has a 'logical' structure, there cannot be any causality in there, since we only have the implication.

Now I do seem to remember once reading or hearing something about attempts to construct some logic capable of defining causal relations, or at the very least come closer to the meaning of 'therefore' as you used it above. I'm not quite sure what the current status of that is (it might very well be very old news).

So it is incorrect to grant them the status of being objective/universal/cosmic/divine truths.

I would already be impressed if you could tell me what it means for a truth to be objective/universal/cosmic/divine. Objective and universal, perhaps, but cosmic and divine?

 

[Goliath]

What I was taught in undergraduate logic is that as a connective word, "implies" and "is implied by" can be used interchangeably with words like "is necessary for / is sufficient for", "if / only if", "therefore / because."

Then I think you were taught wrongly about those last two. I think those are clearly not synonyms for the previous six.

 

[Jadzia]

So it is incorrect to grant them the status of being objective/universal/cosmic/divine truths.

I would already be impressed if you could tell me what it means for a truth to be objective/universal/cosmic/divine. Objective and universal, perhaps, but cosmic and divine?

ok.

It is the degree of esteem and universality with which we hold them in our minds. Mathematical truth is often presented in a way that makes it "feel" like it is something handed to use from the gods, and having a spiritual quality. But what I'm saying that it is incorrect to put logical deductions upon such a pedestal, since their foundations (the assumptions which they are derived from) are very mundane.

What I was taught in undergraduate logic is that as a connective word, "implies" and "is implied by" can be used interchangeably with words like "is necessary for / is sufficient for", "if / only if", "therefore / because."

Then I think you were taught wrongly about those last two. I think those are clearly not synonyms for the previous six.

Well in mathematics, words often mean something different from their common usage. There's that old adage from Alice in Wonderland, "When I use a word, it means what I want it to mean. Nothing more nor less."

"therefore" colloquially means "consequence of something just stated", but it's just semantics Camel. I think the choice of words helps the reader to see where a statement is deduced from. But it isn't hazardous to a theory if we use "therefore" instead of "implies".

 

[Goliath]

"therefore" colloquially means "consequence of something just stated", but it's just semantics Camel. I think the choice of words helps the reader to see where a statement is deduced from. But it isn't hazardous to a theory if we use "therefore" instead of "implies".

First, it's clearly not correct to dismiss the ordinary meaning of 'therefore' as a mere colloquialism. The word 'therefore' is routinely used in that sense in both formal speech and writing. Indeed, you are much more likely to encounter the word 'therefore' in formal speech than you are in informal speech. If anything, the usage you describe strikes me as much more colloquial than ordinary usage.

Second: I don't believe it is just semantics, and I refer you to the section in Locke's Essay Concerning Human Understanding on the abuse of words. If logicians want to use that word loosely and freely to mean the things you mentioned, I suppose that's their business. But so long as they do, their translations of English statements into logical statements will remain just as loose and free.

 

[Jadzia]

I'd say it's a different language, utilizing the same word list. And although there is much overlap and similarity, it is important to not forget that a translation must be made before interpreting a statement made in another language.

I use "therefore" as defined in the Queen's English. I also use "therefore" as defined in logic. It may look and sound the same, but the words have different meaning depending what language I'm talking in.

People should make an effort to understand what language is being spoken in beforehand. Meaning that the confusions here are largely an issue of semantics.

Which language is the OP talking in? Do you know?

 

[Plain Simple]

So it is incorrect to grant them the status of being objective/universal/cosmic/divine truths.

I would already be impressed if you could tell me what it means for a truth to be objective/universal/cosmic/divine. Objective and universal, perhaps, but cosmic and divine?

ok.

It is the degree of esteem and universality with which we hold them in our minds. Mathematical truth is often presented in a way that makes it "feel" like it is something handed to use from the gods, and having a spiritual quality. But what I'm saying that it is incorrect to put logical deductions upon such a pedestal, since their foundations (the assumptions which they are derived from) are very mundane.

I do agree that most people, sometimes even scientists so it seems, add too much 'extra truth' to scientific statements which goes way beyond what it is that is actually stated. Mathematical statements and truths are different all together since they a priori only exist within the logical construct (ideally, they also exist on paper or as bits on my hard drive), so most people will not very often encounter them in pure mathematical form (without reference to the 'real world') anyway.

What I was taught in undergraduate logic is that as a connective word, "implies" and "is implied by" can be used interchangeably with words like "is necessary for / is sufficient for", "if / only if", "therefore / because."

Then I think you were taught wrongly about those last two. I think those are clearly not synonyms for the previous six.

Well in mathematics, words often mean something different from their common usage. There's that old adage from Alice in Wonderland, "When I use a word, it means what I want it to mean. Nothing more nor less."

"therefore" colloquially means "consequence of something just stated", but it's just semantics Camel. I think the choice of words helps the reader to see where a statement is deduced from. But it isn't hazardous to a theory if we use "therefore" instead of "implies".[/QUOTE]

If logicians want to use that word loosely and freely to mean the things you mentioned, I suppose that's their business. But so long as they do, their translations of English statements into logical statements will remain just as loose and free.

I'd hardly say that the definitions that logicians use are "loose and free". If anything they are strict and precise. I think that's one of the problems (among many others probably) for those logicians who try to formalise human languages. Or I'm misunderstanding your meaning.

Which language is the OP talking in? Do you know?

Dutch, quite often. (Although English is taking over actually...)

That of course depends on the context (for example, in my profession as a mathematician I'm used to thinking and writing in the logical language), but for the sake of this thread I'd say that the possible misunderstandings that can arise when this translating occurs are of interest, so we shouldn't restrict ourselves to one language here, just be clear in what language we make our statements.

 

[Goliath]

I'd say it's a different language, utilizing the same word list.

What--you mean, "English as spoken by mathematicians"? I don't think you'd find many people who would agree with you on this point.

It's a different sociolect, at most. It's English, with jargon--like the lit-crit-speak of literary theorists. And jargon is merely the slang of the educated.

So there's no need for me to 'translate' your use of the word 'therefore'. This is not an issue of two different languages. It's an issue of private usage v. public usage.

You assigned a private meaning to this word--in much the same way that Foucauldian literary theorists have assigned a private meaning to the word 'discourse'--and then dismissed its ordinary, public meaning as 'colloquial'.

Some of those aforementioned theorists have argued (ironically, in my opinion) that technical vocabularies exist primarily to reinforce the social status and exclusivity of the groups that use them: that, for example, the 'discourse of mathematics' exists primarily to increase the status-authority of mathematicians.

I'm not ordinarily a fan of such speculation. But based on what you've written here, I wonder if there isn't a grain of truth to what they say.

I'd hardly say that the definitions that logicians use are "loose and free". If anything they are strict and precise.

Ordinarily, I would agree with you. Just not in this case. Please note that I said "that word."