Of all the interesting things I learned at our Routleyfest ("Remembering Richard Sylvan") event last week, perhaps the most striking was about the history of deep ecology:
The famous last man thought experiment was first devised and proposed by Routley, in 1973.
This very dramatic experiment asks you to imagine that you are the last person on earth. No other people can or will ever be born. In the time you have remaining, alone, you decide it would be fun to go around cutting down all the trees. Just for the hell of it. Question: what, if anything, would be wrong about this?
In his talk on Wednesday, Roger Lamb presented Routley's answer to the question. There does not need to be a valuer -- a sentient being according something value -- in order for something to have value. At least, there does not need to be any actual valuer. In an attempt to split the difference between realist and subjectivist accounts, Routley suggests that as long as there is a valuer at some possible world, then things like trees have value. And this accounts for why it would be wrong to chop them down.
Now, it seems to me that this gets the intuition behind "the last man" experiment wrong. The idea is to isolate the trees from the projects of humankind. The point is that the trees have value completely independently of the people. So invoking people -- even people somewhere else, merely possible people -- concedes that the trees have no deeply intrinsic value. Therefore I propose instead (as a counterexample to Lamb's version of Routley's solution) the modal last man:
You are the last possible person. Every other possible world is empty of people; your world, the actual world, has only you left. Since synchronizing time across worlds is problematic, we simply posit that you are the only possible person. (And, as it happens, contingently so.) Admittedly, this is a strange picture of modal space. But it seems legitimate enough to entertain, for the purposes of the experiment.
And so, again, you the last modal man choose to spend the last time any possible person will ever live going around killing all the trees. Is it wrong?
It seems to me that, if you think the original last man's decision was wrong, then you should also think the modal last man is wrong. But then, there is not -- because there cannot be, ex hypothesi -- a valuer to confer on the trees any worth over and above their own intrinsic worth. So Routley's solution misses the point.
I wonder what Routley/Sylvan would have said. That there are valuers at other impossible worlds? Perhaps...but then we could iterate my response, to the ultramodal last man--you are the only person in any world, possible or not....
In any case, it was a thrill to find out that a man I already credit with so many advances in philosophy (he was the first true dialetheic set theorist, I think), is also the mind behind one of the most famous and striking ideas in environmental philosophy, too. In so many cases, Routley was the first.
Sunday, July 31, 2011
Tuesday, May 24, 2011
Monday, April 25, 2011
Important Announcement!
I have been inducted into the Logicians Liberation League, the shadowy organization founded by Robert K. Meyer in 1969, when he read out the Manifesto.
A list of the LLL is now maintained by Jc Beall on his website, where you will notice today a new member:
The Cardinal of Comprehension.
(That's me.)
A list of the LLL is now maintained by Jc Beall on his website, where you will notice today a new member:
The Cardinal of Comprehension.
(That's me.)
Monday, April 11, 2011
The Recurring Question
In the past few weeks I've been listening and speaking at a lot of different places around the northeastern United States and Scotland. And the question that keeps coming up -- about dialethism and paraconsistency -- is the following. It is simple to state and hard to answer:
How much inconsistency is too much?
Classically speaking, of course, the answer is easy. Any inconsistency at all is too much. But there are too many things we want to talk about, to know, to understand, for consistency to stand in the way: even to formulate sensible statements about all the truths, all the sets, all the properties, etc (not to mention vague predicates...) requires going over the line. Some inconsistency is needed, maybe even welcome.
But how much is too much?
Absurdly speaking, the answer is also easy. There is no limit; let us have all the inconsistency and more. But let's set such an extremum aside -- or, better, take it as an obvious limiting condition on our answer.
Some, but not all, inconsistency. That's what we're looking for in a good theory. Where to find the line, though? How much noise is too much noise? My best guess so far: It depends on what kind of music you like.
Perhaps next time someone asks me this question, a more precise answer will follow....
How much inconsistency is too much?
Classically speaking, of course, the answer is easy. Any inconsistency at all is too much. But there are too many things we want to talk about, to know, to understand, for consistency to stand in the way: even to formulate sensible statements about all the truths, all the sets, all the properties, etc (not to mention vague predicates...) requires going over the line. Some inconsistency is needed, maybe even welcome.
But how much is too much?
Absurdly speaking, the answer is also easy. There is no limit; let us have all the inconsistency and more. But let's set such an extremum aside -- or, better, take it as an obvious limiting condition on our answer.
Some, but not all, inconsistency. That's what we're looking for in a good theory. Where to find the line, though? How much noise is too much noise? My best guess so far: It depends on what kind of music you like.
Perhaps next time someone asks me this question, a more precise answer will follow....
Monday, March 14, 2011
And here our travels begin...
Along with Vicki and Oskar, I'm heading off on a big trip this week to talk and learn with Jc Beall and many others for the next few months.
I'm looking forward to finding out what goes on in the Northern Hemisphere these days, both philosophically and...not.
Some major dates (titles subject to change):
-March 25, University of Connecticut: "Paraconsistent Vagueness"
-March 31, St Andrews (Arche): "Paraconsistent Vagueness"
-April 1, Paradox and Logic Revision (Arche): "Between Finite and Infinite"
-April 4, NYU: "Paraconsistent Vagueness"
-April 7, MIT: "Paraconsistent Vagueness"
-April 13, UMass (Dartmouth): "Paraconsistent Vagueness"
-April 15, UConn Logic Group: "Between Finite and Infinite"
-April 22, UConn Logic Group: "Godel's Theorems"
-April 27, University of Massachusetts (Amherst): "Paraconsistent Vagueness"
-April 29, CUNY: "Consistency points in the number line"
-TBA: New England Logic and Language Colloquium: "Godel's Theorems"
I'm looking forward to finding out what goes on in the Northern Hemisphere these days, both philosophically and...not.
Some major dates (titles subject to change):
-March 25, University of Connecticut: "Paraconsistent Vagueness"
-March 31, St Andrews (Arche): "Paraconsistent Vagueness"
-April 1, Paradox and Logic Revision (Arche): "Between Finite and Infinite"
-April 4, NYU: "Paraconsistent Vagueness"
-April 7, MIT: "Paraconsistent Vagueness"
-April 13, UMass (Dartmouth): "Paraconsistent Vagueness"
-April 15, UConn Logic Group: "Between Finite and Infinite"
-April 22, UConn Logic Group: "Godel's Theorems"
-April 27, University of Massachusetts (Amherst): "Paraconsistent Vagueness"
-April 29, CUNY: "Consistency points in the number line"
-TBA: New England Logic and Language Colloquium: "Godel's Theorems"
Monday, February 21, 2011
Is paraconsistent arithmetic categorical?
Is naive set theory capable of describing and pinning down the intended model of arithmetic?
On the one hand, it doesn't seem very likely. With so many more sets out there to cause trouble, and easy ways to make isomorphisms fail (the totality of the ordinals isn't even isomorphic to itself, for instance), the odds of showing that any two peano relations coincide are low.
On the other hand, taking naive set theory as our base opens some intriguing possibilities. In particular, we can stay in a first order language, while at the same time, noting that the axiomatic equivalence
x is a member of X iff Xx
essentially grants us all the expressive power of second (and higher) order logic. And we know that classical second order theories can establish (classical) categoricity.
And so the question of the month is from Peter Smith's Godel book (p. 186). He's asking rhetorically, but in the paraconsistent context it's an open problem:
"If second order arithmetic does pin down the structure of the natural numbers, then -- given that any arithmetic sentence makes a determinate claim about this structure -- it apparently follows that this theory does enough to settle the truth value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic..."
Well, it's an open problem if you leave out that "consistency" part...
On the one hand, it doesn't seem very likely. With so many more sets out there to cause trouble, and easy ways to make isomorphisms fail (the totality of the ordinals isn't even isomorphic to itself, for instance), the odds of showing that any two peano relations coincide are low.
On the other hand, taking naive set theory as our base opens some intriguing possibilities. In particular, we can stay in a first order language, while at the same time, noting that the axiomatic equivalence
x is a member of X iff Xx
essentially grants us all the expressive power of second (and higher) order logic. And we know that classical second order theories can establish (classical) categoricity.
And so the question of the month is from Peter Smith's Godel book (p. 186). He's asking rhetorically, but in the paraconsistent context it's an open problem:
"If second order arithmetic does pin down the structure of the natural numbers, then -- given that any arithmetic sentence makes a determinate claim about this structure -- it apparently follows that this theory does enough to settle the truth value of every arithmetic sentence. Which makes it sound as if there can after all be a (consistent) negation-complete axiomatic theory of arithmetic..."
Well, it's an open problem if you leave out that "consistency" part...
Tuesday, January 25, 2011
Upcoming Workshop: Constructive Mathematics
"What constructive mathematicians actually do"
University of Melbourne
Maarten McKubre-Jordens, University of Canterbury, New Zealand
I) Introduction to constructive mathematics
11:00am, Monday 14 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)
In the first of this two-part workshop, constructive mathematics is characterised as reasoning from positive definitions.
Several different schools (or models) of constructive mathematics are outlined, together with a common base. Some important non-constructive principles are outlined and and their impact evaluated, from the point of view of the practising mathematician. A small amount of familiarity with classical logic and mathematics will be assumed.
II) Constructive mathematics in action
11:00am, Wednesday 16 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)
What is the constructive role of examples, and counterexamples, in mathematics? What are some common techniques for proving theorems constructively? The second part of this workshop surveys how the role of examples in mathematical thought sets apart constructive thought, and will briefly discuss some typical strategies for proving theorems. Time permitting, we will investigate the idea of reverse mathematics as practised constructively.
III) Strange encounters and the importance of constructive thought
11:00am, Friday 18 February 2011, Old Quad Moot Court
Everyone has heard of some variation of the infinite monkey theorem:
Theorem. In an infinite collection of monkeys, given typewriters and a suitable amount of time, there is a monkey that with probability 1 reproduces the entire collected works of Shakespeare.
Of course this is not rigorously worded, but you get the idea. Now, how do you find the cheeky monkey that actually did it? In this seminar we take a look at some of the stranger and more intriguing aspects of constructive analysis — such as strictly increasing bounded sequences that do not converge, and well-defined sets in R^n from which we cannot calculate distances — and discuss their impact.
University of Melbourne
Maarten McKubre-Jordens, University of Canterbury, New Zealand
I) Introduction to constructive mathematics
11:00am, Monday 14 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)
In the first of this two-part workshop, constructive mathematics is characterised as reasoning from positive definitions.
Several different schools (or models) of constructive mathematics are outlined, together with a common base. Some important non-constructive principles are outlined and and their impact evaluated, from the point of view of the practising mathematician. A small amount of familiarity with classical logic and mathematics will be assumed.
II) Constructive mathematics in action
11:00am, Wednesday 16 February 2011, Old Arts-227 (Cecil Scutt Collaborative Teaching Room)
What is the constructive role of examples, and counterexamples, in mathematics? What are some common techniques for proving theorems constructively? The second part of this workshop surveys how the role of examples in mathematical thought sets apart constructive thought, and will briefly discuss some typical strategies for proving theorems. Time permitting, we will investigate the idea of reverse mathematics as practised constructively.
III) Strange encounters and the importance of constructive thought
11:00am, Friday 18 February 2011, Old Quad Moot Court
Everyone has heard of some variation of the infinite monkey theorem:
Theorem. In an infinite collection of monkeys, given typewriters and a suitable amount of time, there is a monkey that with probability 1 reproduces the entire collected works of Shakespeare.
Of course this is not rigorously worded, but you get the idea. Now, how do you find the cheeky monkey that actually did it? In this seminar we take a look at some of the stranger and more intriguing aspects of constructive analysis — such as strictly increasing bounded sequences that do not converge, and well-defined sets in R^n from which we cannot calculate distances — and discuss their impact.
Wednesday, January 19, 2011
2011 Preview: Beyond the Possible
This year is set to be a busy and productive one, with visitors, workshops, conferences, and travels. And, of course, writing up some results. A quick preview:
>> July 27 - 29: Beyond the Possible: A Conference in Memorial of Richard Sylvan, who died 15 years ago. Some big guests for this one -- more info to follow on this soon.
>> Maarten McKubre-Jordens, a constructive mathematician from Christchurch and collaborator on our paraconsistency project, will visit in February to run some workshops and give a talk.
>> Colin Caret (St Andrews) will visit in July, for work on relevant information, paradox solution, and hyperintensionality.
>> We're also looking at visits from Walter Carnielli, Juliana Bueno-Soler, and John Bell.
>> And I'll be off to the University of Connecticut, USA!, from mid March until May, as the first "Scholar of Consequence," to work with Jc Beall and the logic group there. And to give some talks at other schools in the area (including attending a Paradox and Logic Revision workshop at St Andrews -- which is "in the area" relative to Melbourne).
Contact me if with questions, ideas, interest ... and stay tuned.
>> July 27 - 29: Beyond the Possible: A Conference in Memorial of Richard Sylvan, who died 15 years ago. Some big guests for this one -- more info to follow on this soon.
>> Maarten McKubre-Jordens, a constructive mathematician from Christchurch and collaborator on our paraconsistency project, will visit in February to run some workshops and give a talk.
>> Colin Caret (St Andrews) will visit in July, for work on relevant information, paradox solution, and hyperintensionality.
>> We're also looking at visits from Walter Carnielli, Juliana Bueno-Soler, and John Bell.
>> And I'll be off to the University of Connecticut, USA!, from mid March until May, as the first "Scholar of Consequence," to work with Jc Beall and the logic group there. And to give some talks at other schools in the area (including attending a Paradox and Logic Revision workshop at St Andrews -- which is "in the area" relative to Melbourne).
Contact me if with questions, ideas, interest ... and stay tuned.
Tuesday, December 14, 2010
It's been a good (finite) year.
Since I started in March, while chasing compactness results in real analysis with my excellent co-author, Maarten McKubre-Jordens (Christchurch), and building up recursive functions towards a paraconsistent version of Godel's theorem, I've been doing a lot of work too on my old favourite: transfinite cardinals.
Maybe ironically, thinking about the size of infinite sets has got me focused on finitude.
I've argued for a few years now that there is an injective function from any set into the singleton {On}, where On = {all the ordinals}. Because of the Burali-Forti paradox, we know that
(i) On is an ordinal, and therefore
(ii) On < On. And therefore
(iii) On < On < ... < On < ... for as long as you want.
So {On} is very structurally rich -- a wellfounded subset of the ordinals that any set can map into, just by dint of a constant function, f(x) = On.
For a long time, I thought this meant that the singleton, {On}, is in fact infinite, appearances of being size 1 notwithstanding. In fact, I was convinced {On}, and indeed the singleton of any inconsistent set, is absolutely infinite, meaning it has the same cardinality as the entire universe, V. After all, an injection from V to {On} -- which exists, by that constant function -- shows that |{On}| is greater than or equal to |V|. I called this the "swelling lemma." Graham Priest suspected that I did because I thought it was, well, swell.
But then I thought some more.
It turns out that, by naive comprehension, every set has an inconsistent subset. (Let A be a set and let A[R] be the set of all members of A, to the degree that the Russell set is self-membered. Are members of A members of A[R] or not? Errr....) Okay, this proves straight away that every set is dedekind infinite, because these inconsistent subsets A[R] are proper subsets. That's not so bad. But by swelling, we also have
|V| < |A[R]| < |A|
which by transitivity implies that |A|=|V|, for any A at all. All sets are the size of the universe? Probably not.
In working (hard) to understand this problem, I've gotten thinking that swelling is the wrong direction for looking at this. We have that any set A at all injects into {On}. Well, then in some sense, |A| is less than |{On}|. That is, instead of thinking that {On} is so big, maybe this means that, no matter what A is, it is rather small.
Finite, even.
Read x < y as "y is bigger than x" and you get swelling. Read it "x is less than y" and you get soothing. I'll spare you the details for now of how to break up the notion of finitude into distinct parts (trans-finitude?) that make sense of this, so that arbitrary sets are (still) not necessarily less than or equal to 1, for instance; all that is in the final paper.
But after thinking for so long about numbers and sets so large, there is something comforting -- beautiful even -- about returning to the notion that everything is so small. Finite, even.
Wait -- everything finite?! How could this be? Well, the idea is that sets only look big, sometimes, because they are inconsistent. That's how, if we're thinking too consistently, we perceive contradictions: as infinity. (The same goes for vagueness: there's no real indeterminacy, only inconsistency. We just weren't expecting that.) So yes, infinity is real, but only as real as contradiction.
Because: the work I'm doing on Godel's theorems at the moment bears out a prediction of Priest, and, inadvertently, of Stewart Shapiro. With a semantically closed arithmetic that represents all its own recursive functions, there is a contradiction at the level of finite numbers.
And once you hit the first inconsistency in the numberline, then, like infinity, they're all inconsistent after that. So it looks like there is some natural number n = |V|. What a world.
What a pleasantly small world. (No wonder there are contradictions, with everything so crowded together.)
Finitism 2011, anyone?
While we head into the New Year, notice that if the universe is finite -- not even countably infinite, but finite--then it shouldn't take long at all to look under every last rock,
and find the Truth, wherever it's hiding.
Maybe ironically, thinking about the size of infinite sets has got me focused on finitude.
I've argued for a few years now that there is an injective function from any set into the singleton {On}, where On = {all the ordinals}. Because of the Burali-Forti paradox, we know that
(i) On is an ordinal, and therefore
(ii) On < On. And therefore
(iii) On < On < ... < On < ... for as long as you want.
So {On} is very structurally rich -- a wellfounded subset of the ordinals that any set can map into, just by dint of a constant function, f(x) = On.
For a long time, I thought this meant that the singleton, {On}, is in fact infinite, appearances of being size 1 notwithstanding. In fact, I was convinced {On}, and indeed the singleton of any inconsistent set, is absolutely infinite, meaning it has the same cardinality as the entire universe, V. After all, an injection from V to {On} -- which exists, by that constant function -- shows that |{On}| is greater than or equal to |V|. I called this the "swelling lemma." Graham Priest suspected that I did because I thought it was, well, swell.
But then I thought some more.
It turns out that, by naive comprehension, every set has an inconsistent subset. (Let A be a set and let A[R] be the set of all members of A, to the degree that the Russell set is self-membered. Are members of A members of A[R] or not? Errr....) Okay, this proves straight away that every set is dedekind infinite, because these inconsistent subsets A[R] are proper subsets. That's not so bad. But by swelling, we also have
|V| < |A[R]| < |A|
which by transitivity implies that |A|=|V|, for any A at all. All sets are the size of the universe? Probably not.
In working (hard) to understand this problem, I've gotten thinking that swelling is the wrong direction for looking at this. We have that any set A at all injects into {On}. Well, then in some sense, |A| is less than |{On}|. That is, instead of thinking that {On} is so big, maybe this means that, no matter what A is, it is rather small.
Finite, even.
Read x < y as "y is bigger than x" and you get swelling. Read it "x is less than y" and you get soothing. I'll spare you the details for now of how to break up the notion of finitude into distinct parts (trans-finitude?) that make sense of this, so that arbitrary sets are (still) not necessarily less than or equal to 1, for instance; all that is in the final paper.
But after thinking for so long about numbers and sets so large, there is something comforting -- beautiful even -- about returning to the notion that everything is so small. Finite, even.
Wait -- everything finite?! How could this be? Well, the idea is that sets only look big, sometimes, because they are inconsistent. That's how, if we're thinking too consistently, we perceive contradictions: as infinity. (The same goes for vagueness: there's no real indeterminacy, only inconsistency. We just weren't expecting that.) So yes, infinity is real, but only as real as contradiction.
Because: the work I'm doing on Godel's theorems at the moment bears out a prediction of Priest, and, inadvertently, of Stewart Shapiro. With a semantically closed arithmetic that represents all its own recursive functions, there is a contradiction at the level of finite numbers.
And once you hit the first inconsistency in the numberline, then, like infinity, they're all inconsistent after that. So it looks like there is some natural number n = |V|. What a world.
What a pleasantly small world. (No wonder there are contradictions, with everything so crowded together.)
Finitism 2011, anyone?
While we head into the New Year, notice that if the universe is finite -- not even countably infinite, but finite--then it shouldn't take long at all to look under every last rock,
and find the Truth, wherever it's hiding.
Tuesday, November 30, 2010
Wednesday, October 13, 2010
Replying to "Inadequacy"
A critical note has just appeared in the Review of Symbolic Logic:
"The Inadequcy of a Proposed Paraconsistent Set Theoy," by Frode Bjørdal. This paper claims that the set theory I used in "Transfinite Numbers in Paraconsistent Set Theory" has
for all x, ~(x = x)
as a theorem. That would be pretty lousy news. My reply to Bjørdal's note will appear shortly in the RSL. Here's the short version:
It is not a theorem that ~(x = x) for all x.
Here's a slightly longer version.
In my last post, I indicated that there is now almost overwhelming evidence that the system in TFNPST is not trivial. Still, a system could be logically non-trivial , but still be mathematically useless. Everything being not self identical is an example of the latter.
But here's all that Bjørdal has pointed out. There is a set X such that every x both is and is not in X. Almost anyone who has worked on this kind of crazy set theory knows that. Arief Daynes adds it as an axiom. Just take {x: L}, where L is your favorite true contradiction.
Unfortunately, Bjørdal wants to define identity, x=y, to mean that x and y are in all and only the same sets. Now if x isn't even in all the same sets as x itself, well then, yes, ~(x=x), for every x. Insist on writing identity signs in the wrong places, and you get wrong statements about identity.
This emphasizes -- and it's what I argue in full paper length elsewhere -- that this isn't the right way to define identity in intensional contexts. We already knew that, because Clark Kent is Superman, even if one guy wears glasses and the other does not.
It is true that identical objects are in all and only the same sets. It is, according to the rule of substitution. It's just that Leibniz' law can't be the definition of identity.
I'm reminded of some critiques of dialetheism that go like this. Let's define the zero place sad-face connective :( for the disjunction of all contradictions, and agree to pronounce the connective "so bad!". Now suppose the Russell paradox were a true contradiction; then :( follows. But :( is so bad! Therefore dialetheism is inadequate. Hmm....
"The Inadequcy of a Proposed Paraconsistent Set Theoy," by Frode Bjørdal. This paper claims that the set theory I used in "Transfinite Numbers in Paraconsistent Set Theory" has
for all x, ~(x = x)
as a theorem. That would be pretty lousy news. My reply to Bjørdal's note will appear shortly in the RSL. Here's the short version:
It is not a theorem that ~(x = x) for all x.
Here's a slightly longer version.
In my last post, I indicated that there is now almost overwhelming evidence that the system in TFNPST is not trivial. Still, a system could be logically non-trivial , but still be mathematically useless. Everything being not self identical is an example of the latter.
But here's all that Bjørdal has pointed out. There is a set X such that every x both is and is not in X. Almost anyone who has worked on this kind of crazy set theory knows that. Arief Daynes adds it as an axiom. Just take {x: L}, where L is your favorite true contradiction.
Unfortunately, Bjørdal wants to define identity, x=y, to mean that x and y are in all and only the same sets. Now if x isn't even in all the same sets as x itself, well then, yes, ~(x=x), for every x. Insist on writing identity signs in the wrong places, and you get wrong statements about identity.
This emphasizes -- and it's what I argue in full paper length elsewhere -- that this isn't the right way to define identity in intensional contexts. We already knew that, because Clark Kent is Superman, even if one guy wears glasses and the other does not.
It is true that identical objects are in all and only the same sets. It is, according to the rule of substitution. It's just that Leibniz' law can't be the definition of identity.
I'm reminded of some critiques of dialetheism that go like this. Let's define the zero place sad-face connective :( for the disjunction of all contradictions, and agree to pronounce the connective "so bad!". Now suppose the Russell paradox were a true contradiction; then :( follows. But :( is so bad! Therefore dialetheism is inadequate. Hmm....
Sunday, August 29, 2010
Non-triviality and Coherence
It has been an open problem whether the logic I used in "Transfinite Numbers in Paraconsistent Set Theory" (which does seem to be getting read by people, according to Review of Symbolic Logic site...) is non-trivial for a naive comprehension scheme. The system seems pretty strong; it can prove the axiom of choice, after all. Can it prove everything? That would be bad.
Ross Brady has proved that logics very-much-like it are safe -- there are models satisfying all the theorems of the set theory, in which some (absurd) sentences are not satisfied. But I needed the additional counterexample principle:
A, ~B; therefore ~(A --> B)
and it was unclear whether this could fit into Brady's construction. For the proof in his 2006 book "Universal Logic," the counterexample principle definitely does not fit. (Indeed, there's a counterexample.) So, TFNPST could have been fated to be a much-less-famous Fregean Grundgesetze.
Very happily, we noticed last week that one of Brady's older papers, "The Non-Triviality of Dialectical Set Theory" in Paraconsistent Logic: Essays on the Inconsistent (1989) proves non-triviality for a much stronger logic. That logic turns out to contain (in rule form) both the above counterexample principle, and its contrapose:
(A --> B); therefore ~A or B.
So unless ZF and similar systems are themselves inconsistent (...hmm...) it follows that the work in TFNPST is demonstrably coherent.
=======================================================
Speaking of coherence, Greg Restall recently wrote a paper to appear in Analysis, where he argues that non-classical logicians should stay away from using two little connectives: t, the conjunction of all truths, and u, the disjunction of all untruths. Having these little letters around allows us to define the following connective:
A > B := A&t --> B v u
where --> is an arrow that obeys modus ponens and a few other sensible properties. Restall shows that the new arrow > can make a lot of trouble, in the form of Curry paradoxes, strengthened liars, and general malaise.
So Jc Beall, Graham Priest and I wrote a reply. We claim, as dialetheists, that the disjunction of all untruths, little u, is itself true. After all, some of the conjuncts of little t are false; they have true negations. And this defuses most of that trouble and general malaise. (We also claim that a truth value gapper can say something analogous about little t.) Dave Ripley and I noticed that it boils down to this derivation:
1. u
2. t --> u (from 1, by t-introduction)
3. A&t --> u (from 2, antecedent strengthening)
4. u --> u v B (or-introduction)
5. A&t --> u v B (3, 4, conjunctive syllogism)
6. A > B (definition of >)
Line 6 is completely general, any A and B at all. So this means that the arrow > really isn't an arrow at all. It certainly shouldn't obey modus ponens, contra Restall. And so the world is (doubly) safe for paraconsistency once again.
Or so the claim goes. It all depends on how to understand little u, and whether or not logical space can be partitioned in particular ways. What are the untruths? What work is "un" doing here?
Ross Brady has proved that logics very-much-like it are safe -- there are models satisfying all the theorems of the set theory, in which some (absurd) sentences are not satisfied. But I needed the additional counterexample principle:
A, ~B; therefore ~(A --> B)
and it was unclear whether this could fit into Brady's construction. For the proof in his 2006 book "Universal Logic," the counterexample principle definitely does not fit. (Indeed, there's a counterexample.) So, TFNPST could have been fated to be a much-less-famous Fregean Grundgesetze.
Very happily, we noticed last week that one of Brady's older papers, "The Non-Triviality of Dialectical Set Theory" in Paraconsistent Logic: Essays on the Inconsistent (1989) proves non-triviality for a much stronger logic. That logic turns out to contain (in rule form) both the above counterexample principle, and its contrapose:
(A --> B); therefore ~A or B.
So unless ZF and similar systems are themselves inconsistent (...hmm...) it follows that the work in TFNPST is demonstrably coherent.
=======================================================
Speaking of coherence, Greg Restall recently wrote a paper to appear in Analysis, where he argues that non-classical logicians should stay away from using two little connectives: t, the conjunction of all truths, and u, the disjunction of all untruths. Having these little letters around allows us to define the following connective:
A > B := A&t --> B v u
where --> is an arrow that obeys modus ponens and a few other sensible properties. Restall shows that the new arrow > can make a lot of trouble, in the form of Curry paradoxes, strengthened liars, and general malaise.
So Jc Beall, Graham Priest and I wrote a reply. We claim, as dialetheists, that the disjunction of all untruths, little u, is itself true. After all, some of the conjuncts of little t are false; they have true negations. And this defuses most of that trouble and general malaise. (We also claim that a truth value gapper can say something analogous about little t.) Dave Ripley and I noticed that it boils down to this derivation:
1. u
2. t --> u (from 1, by t-introduction)
3. A&t --> u (from 2, antecedent strengthening)
4. u --> u v B (or-introduction)
5. A&t --> u v B (3, 4, conjunctive syllogism)
6. A > B (definition of >)
Line 6 is completely general, any A and B at all. So this means that the arrow > really isn't an arrow at all. It certainly shouldn't obey modus ponens, contra Restall. And so the world is (doubly) safe for paraconsistency once again.
Or so the claim goes. It all depends on how to understand little u, and whether or not logical space can be partitioned in particular ways. What are the untruths? What work is "un" doing here?
Sunday, August 1, 2010
Paraconsistent maths workshop begins!
Monday, August 2
9am Introduction - Zach Weber
10 Naive comprehension, circularity and arithmetic in fuzzy
logic- Shunsuke Yatabe
11 TEA
11.30 Groupthink -- Set theory and arithmetic
1 pm. LUNCH
2.30 Paraconsistent Set Theory and Metatheory - Graham Priest
3.30 Relevant Logic and Classical Proofs - Ed Mares
4.30 TEA
5 Paraconsistent arithmetic with binary quantifiers - Sam Butchart
6 DINNER
Tuesday, August 3
9am Coformulas - Greg Restall
10 Church/Turing Thesis - Koji Tanaka
11 TEA
11.30 Co-inductive-like definition in fuzzy truth theory - Shunsuke
Yatabe
1pm LUNCH
2.30 Groupthink -- Proofs and Computation
3.30 Groups, Inconsistency and The Routley Functor - Chris Mortensen
4.30 TEA
5 Conclusion - Open Problems
6.15 Barry Taylor Memorial Lecture (Jon Bigelow)
9am Introduction - Zach Weber
10 Naive comprehension, circularity and arithmetic in fuzzy
logic- Shunsuke Yatabe
11 TEA
11.30 Groupthink -- Set theory and arithmetic
1 pm. LUNCH
2.30 Paraconsistent Set Theory and Metatheory - Graham Priest
3.30 Relevant Logic and Classical Proofs - Ed Mares
4.30 TEA
5 Paraconsistent arithmetic with binary quantifiers - Sam Butchart
6 DINNER
Tuesday, August 3
9am Coformulas - Greg Restall
10 Church/Turing Thesis - Koji Tanaka
11 TEA
11.30 Co-inductive-like definition in fuzzy truth theory - Shunsuke
Yatabe
1pm LUNCH
2.30 Groupthink -- Proofs and Computation
3.30 Groups, Inconsistency and The Routley Functor - Chris Mortensen
4.30 TEA
5 Conclusion - Open Problems
6.15 Barry Taylor Memorial Lecture (Jon Bigelow)
Thursday, June 3, 2010
Interview in "The Reasoner"
This month's issue of "The Reasoner" features me interviewing Greg Restall.
Ever wonder why Greg used to believe in true contradictions, but doesn't now? Find out...
www.thereasoner.org
Ever wonder why Greg used to believe in true contradictions, but doesn't now? Find out...
www.thereasoner.org
Thursday, March 4, 2010
Paraconsistent Mathematics Workshop
August 2 – 3, 2010
University of Melbourne
To begin our new ARC project, we are hosting a workshop, to present ideas and determine directions for research. Core topics include:
- Set Theories
- Arithmetic
- Recursive Functions and Computability
- “Higher” mathematics – geometry, topology, real analysis, etc.
Questions to discuss include: What logic(s) are appropriate for these theories? How are basic and ubiquitous mathematical notions to be formally expressed? What should we aim to prove, and how will we prove it?
Interested parties are invited to contact me at zweber [at] unimelb.edu.au
University of Melbourne
To begin our new ARC project, we are hosting a workshop, to present ideas and determine directions for research. Core topics include:
- Set Theories
- Arithmetic
- Recursive Functions and Computability
- “Higher” mathematics – geometry, topology, real analysis, etc.
Questions to discuss include: What logic(s) are appropriate for these theories? How are basic and ubiquitous mathematical notions to be formally expressed? What should we aim to prove, and how will we prove it?
Interested parties are invited to contact me at zweber [at] unimelb.edu.au
Monday, February 1, 2010
Aims and Background
At the turn of the 20th century there was a revolution in logic. New technologies, associated with the names of Frege and Russell, radically changed philosophy, because formal logic lends incredible precision to the asking and answering of old questions. A discipline called foundations of mathematics emerged, casting new light on the nature and meaning of rationality and mathematical truth. In foundational research, philosophical issues and mathematical results develop in tandem, each informing the other. Epochal results came out of early investigations, such as the modern theory of computing as developed by Gödel, Turing and Church.
The goal of the initial foundational project, as articulated by Hilbert (reviving an idea of Leibniz’s), was a method for reliably deciding whether or not a given sentence is true. All approaches to foundations of the last hundred years, from Russell’s to Hilbert’s to Brouwer’s, are beholden to formal consistency. Beset on one side by serious paradoxes, like Russell’s contradiction, and on the other by Gödel’s incompleteness theorems, it is now known that a consistent foundation is impossible to obtain.
Starting in the 1970s, developments in logic produced a novel kind of tool, paraconsistent logic. These logics accommodate inconsistency in a sensible fashion. From the start, many logicians saw the relevance and possible applications of these logics to foundational matters. Techniques have now developed to a level of sophistication where it is possible to renew a serious investigation of foundations. The driving thought here is that one need not founder on the paradoxes that halted older foundational projects. One simply recasts, accepts and even studies some contradictions, controlling any pernicious effects with a paraconsistent logic.
Our project is to construct, for the first time, a fully articulated foundation for mathematics. This will involve the construction of paraconsistent versions of the major pillars of foundational studies:
• Foundations are tested on arithmetic, the basic tool for counting and quantity. The functions that operate over the numbers are the province of recursion theory, which ties into computation.
• The bridge between language and mathematical structures is provided by model theory, which has become an indispensable tool for philosophers and logicians alike. Consistent model theory is not able to take in the full range of what mathematicians express in language. A model theory built on paraconsistent principles will be radically more powerful and illuminating, a major achievement.
• Conducting foundational studies of proof and truth in their natural setting contributes to our understanding of knowledge and beliefs, by providing a simple and credible account of learning.
What will emerge is a wholly new perspective on the nature of mathematical thought and truth, striking results in inconsistent mathematics itself, and a reliable platform for more general theories that no strictly consistent foundation can support.
Formally, we define and describe the objects of mathematics, such as numbers and functions; we prove that there are such objects and that they have the right properties; and we do this in such a way as to allow for some inevitable inconsistencies, using paraconsistency to protect the basic integrity of the theory. Philosophically, we work from that foundation to understand truth, proof and belief in terms of what they are—rational but paradoxical ways that humans navigate through our world.
The goal of the initial foundational project, as articulated by Hilbert (reviving an idea of Leibniz’s), was a method for reliably deciding whether or not a given sentence is true. All approaches to foundations of the last hundred years, from Russell’s to Hilbert’s to Brouwer’s, are beholden to formal consistency. Beset on one side by serious paradoxes, like Russell’s contradiction, and on the other by Gödel’s incompleteness theorems, it is now known that a consistent foundation is impossible to obtain.
Starting in the 1970s, developments in logic produced a novel kind of tool, paraconsistent logic. These logics accommodate inconsistency in a sensible fashion. From the start, many logicians saw the relevance and possible applications of these logics to foundational matters. Techniques have now developed to a level of sophistication where it is possible to renew a serious investigation of foundations. The driving thought here is that one need not founder on the paradoxes that halted older foundational projects. One simply recasts, accepts and even studies some contradictions, controlling any pernicious effects with a paraconsistent logic.
Our project is to construct, for the first time, a fully articulated foundation for mathematics. This will involve the construction of paraconsistent versions of the major pillars of foundational studies:
• Foundations are tested on arithmetic, the basic tool for counting and quantity. The functions that operate over the numbers are the province of recursion theory, which ties into computation.
• The bridge between language and mathematical structures is provided by model theory, which has become an indispensable tool for philosophers and logicians alike. Consistent model theory is not able to take in the full range of what mathematicians express in language. A model theory built on paraconsistent principles will be radically more powerful and illuminating, a major achievement.
• Conducting foundational studies of proof and truth in their natural setting contributes to our understanding of knowledge and beliefs, by providing a simple and credible account of learning.
What will emerge is a wholly new perspective on the nature of mathematical thought and truth, striking results in inconsistent mathematics itself, and a reliable platform for more general theories that no strictly consistent foundation can support.
Formally, we define and describe the objects of mathematics, such as numbers and functions; we prove that there are such objects and that they have the right properties; and we do this in such a way as to allow for some inevitable inconsistencies, using paraconsistency to protect the basic integrity of the theory. Philosophically, we work from that foundation to understand truth, proof and belief in terms of what they are—rational but paradoxical ways that humans navigate through our world.
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