,

Here's some inconsistent geometry: — Banno

I didn't say anything about truth values (semantics) for paraconsistent logics.

I am not expert, but, if I am not mistaken, the main point about a paraconsistent system is that it does not have EFQ. There is nothing stopping us from having premises that are a contradiction, and using a paraconsistent system to derive those premises trivially by the rule of placing a premise on a line. And those premises might be contentual axioms. So we would have theorems in contradiction with one another. And if the system has the rule of adjunction, then we can have the conjunction of the two contradicting premises. The important point though is that we can't use EFQ. But, of course, there are widely different kinds of paraconsistent systems, so I don't intend a complete generalization.

That's syntax (proof system). As to semantics, if I am not mistaken, not all paraconsistent systems accommodate dialetheism, but some do. Indeed the SEP article states that every dialetheistic approach must have a paraconsistent syntax . So, since the set of dialetheistic semantics is not empty, there must be paraconsistent systems (syntax) that accommodate dialetheism (semantics),

In a chart:

Exist. Paraconsistent syntax with dialetheistic semantics.
Exist. Paraconsistent syntax with non-dialetheistic semantics.
Exist. Dialetheism (which is semantics) with paraconsistent syntax.
Not Exist. Dialetheism (which is semantics) with non-paraconsistent syntax.

Paraconsistent logic does not allow contradictions; it does not allow (A & ~A) to be true. — Banno

If I'm not mistaken, that is incorrect as a generalization over all paraconsistent systems, as I mentioned above. As a rough generalization, paraconsistency does not "frown" on deriving a contradiction, and some paraconsistent approaches do not frown on having true contradictions. Rather, all paraconsistent systems don't have EFQ. (By saying that they don't have EFQ, I mean that they don't have "For all sentences P and Q, {P ~P} |- Q)".

a certain way of talking about the world is found in many places. There are, after all, languages without much by way of number. Would you say that the folk who speak them have failed to notice an aspect of reality, or would you say that they have no use for a particular process, a certain way of speaking? — Banno

I still think Betrand Russell nailed it in a nutshell. 'Physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover.' You can trace that all the way back to the Greeks. They discerned that numerical qualities were persistent, stable, and directly knowable by reason, unlike sense-able things which are always mutating and never stable. From there is also derived Galileo's 'book of nature is written in mathematics'. That's why I could make sense of @Joshs passage about Husserl and Heidegger - they both analyse the 'history of ideas' through that lens - even though I can understand your perpexity.

Of course, I also think modern physics has gone way overboard with their reduction to mathematical qualities, but that is tangential to this thread. However the point about the implications of mathematical realism is still a good one.

Hence an appropriate response to @Olivier5's question, I thought! (I had Godel Escher Bach on my bookshelf for decades but must admit never more than skimmed it.)

Ok, I should have said not all paraconsistent systems allow contradiction.

The point of the post was to point out that paraconsistent systems avoid inconsistency by redefining it as other than (A & ~A), usually by adding a third truth value.

I still think Betrand Russell nailed it in a nutshell. — Wayfarer

Nail a nutshell and it will crack.

Fair enough, although you make it sound as if Wittgenstein wanted to invite as many contradictions as possible. — Janus

According to the principle of explosion, if you accept one contradiction in a logical system, you accept them all.

That's a good one!

Graham Priest seems the go-to for all this kind of material. See this review. Somehow, Banno, I think this might be of interest to you.

I'm not being snide. It is a genuine issue amongst mathematics teachers. See the Wiki article on 0.99... It's on a par with kids who are not able to see three dots as three.

Folk who think 0.99...<>1 have missed a vital aspect of mathematics. — Banno

I can meet you halfway and say that I can see that for formal purposes, the conceptualized infinite series of fractions we are discussing equals 1. But since it is impossible to instantiate an infinite series, what we are really talking about is a rounding off, a "for all intents and purposes". I don't understand why you related this to the 'three dots' example; who would not be able to see that three dots are three? Someone who actually couldn't count, maybe?

Life is too short — Olivier5

Yeah, you really got that flippant dismissive thing down.

Anyway, better that life is too short than it be too long.

I tend to find SEP unreliable — Olivier5

I have not found problems (though, of course, no source is perfect). You asked about paraconsistency in context of engineering. I told you of a place where there is a real nice writeup (much more eloquent than, by your account, my own postings) about use in data systems; one could see how that could be adapted to an engineering context too.

So, my explanations are not eloquent enough for you, but you still are interested, but not interested enough to take a few moments from your too short life to read a professional account much better than I would write. Such a case you are.

it approaches 1, but it never quite does become 1, though, does it? — Janus

Again, that reflects a fundamental misunderstanding of what '.999...' stands for.

...instantiate... — Janus

What's that, exactly?

Here's an instantiated infinite series for you:
1,2,3...

who would not be able to see that three dots are three? — Janus

A three-year-old. They have to count. They have not moved from the process of counting to recognising the number when they see it. Just like those who cannot move from seeing the process of adding 9/10, 9/100, and so on, to seeing 1.

It's an issue of pedagogy, not maths.

Are you going to make a cogent argument as to how a series that is approaching one and could do so forever without actually reaching it is the "very same as I" — Janus

Again, it's not about a sequence "reaching" anything. '.999...' stands for the limit of a certain sequence. There is no "becomming" or "reaching". Simply, the limit is 1.

Such a case you are. — TonesInDeepFreeze

I'm here to please.

paraconsistent systems avoid inconsistency by redefining it as other than (A & ~A), usually by adding a third truth value. — Banno

That is not my understanding, which is: Certain paraconsistent systems do not avoid inconsistency; rather they avoid explosion. But, yes, in the semantics, three truth values is one way. Also, in syntax one way is to use three values in truth tables and take derivation rules based on those truth tables. But, if I'm not mistaken, there can be dialetheistic semantics for paraconsistent systems.

I don't think you are disagreeing with me. Rather there is a disagreement as to the use of "inconsistency".

A set S of formulas is inconsistent iff there is a formula P such that both P and ~P are members of S.

As far as I know, that is the presumed mathematical definition.

I'm here to please. — Olivier5

Then hurry up and take our lunch orders.

t is impossible to instantiate an infinite series — Janus

We instantiate them all the time. I instantiated the series in question earlier in this thread.

A set S of formulas is inconsistent iff there is a formula P such that both P and ~P are members of S. — TonesInDeepFreeze

Shh... not in front of the children...

yes, in paraconsistent logic

The role often played by the notion of consistency in orthodox logics, namely, the most basic requirement that any theory must meet, is relaxed to the notion of coherence:... Simple consistency of a theory (no contradictions) is a special case of absolute consistency, or non-triviality (not every sentence is a part of the theory).

(SEP)

.

Of course, I don't begrudge adding a rubric 'absolute consistency' that way, though I like the term 'non-trivial' better.

A set S of formulas is non-trivial iff there is a formula P that is not a member of S.

With that definition, with a paraconsistent logic, a set of sentences can be closed under deduction while being inconsistent while being non-trivial. Pretty much another way of saying that EFQ does not obtain.

Yes, but the other point is that if you added .9, .09, .009, .0009. .00009, and so on forever the sum of what you had written could never equal !, and that was literally all I was saying.

By "instantiate an infinite series" I mean write it down as a full series, not in a shorthand form. I'm no mathematician, but that logical distinction is clear.

Here's an instantiated infinite series for you:
1,2,3... — Banno

LOL, no that's just a finite set of marks on the screen. I get that it stands for an infinite series but it is not itself actually an infinite series. Are you getting it yet?

LOL, no that's just a finite set of marks on the screen. — Janus

You misread it.

As I said, it's an issue of pedagogy. You need learnin'.

(the bit where I said I wan't going to do this...!)

You misread it.

As I said, it's an issue of pedagogy. You need learnin'.

(the bit where I said I wan't going to do this...!) — Banno

You just don't want to admit the distinction I pointed to, so you default to patronising mode instead. I understand the concept of "1,2,3,..." denoting an infinite series, but it is not itself an infinite series. As I said, I'm not a mathematician, but that logical distinction is simple enough; there can be no actual infinite quantities of anything.

I understand the concept of "7" denoting seven, but it is not itself a seven.

End of the test. you get 0/10.

if you added .9, .09, .009, .0009. .00009, and so on forever — Janus

And we don't do that.

By "instantiate an infinite series" I mean write it down as a full series, not in a shorthand form. I'm no mathematician, but that logical distinction is clear. — Janus

The word 'instantiate' has a certain meaning in mathematics. What you mean though - to type out in finite time and space individually all the members of an infinite set - is of course impossible. But that doesn't entail that the limit of an infinite sequence is not rigorously defined.

The word 'instantiate' has a certain meaning in mathematics. What you mean though - to type out in finite time and space individually all the members of an infinite set - is of course impossible. But that doesn't entail that the limit of an infinite sequence is not rigorously defined. — TonesInDeepFreeze

OK, thanks, I'm not familiar with the mathematics specific use of 'instantiate'. I haven't claimed that the limit of an infinite series is not rigorously defined, and you agree that an infinite series cannot be instantiated in the sense I meant it; so it seems we are not disagreeing about anything.

there can be no actual infinite quantities of anything. — Janus

Of course, if one doesn't countenance infinite sets, then one might not countenance the classical notion of convergence to a limit. But that doesn't change that what ordinary mathematics means by '.999...' is not some kind of "reaching" but rather convergence to a limit.

it seems we are not disagreeing about anything — Janus

To be in agreement, you'd have to agree that the limit of the sequence is rigorously defined.

Of course, if one doesn't countenance infinite sets, then one might not countenance the classical notion of convergence to a limit. But that doesn't change that what ordinary mathematics mean by '.999...' is not some kind of "reaching" but rather convergence to a limit — TonesInDeepFreeze

I am not opposed to the idea of infinite sets. I can even accept that some infinite sets are larger than others. Logically I see convergence to a limit as analogous to an asymptotic approach, but I can also accept that the mathematical concept may be different, even though I don't have the background to properly grasp the difference.

My original point was only that what we understand intuitively can be understood in ways inconsistent with that in mathematics.