## RS72 - Graham Priest on Paradoxes and Paraconsistent Logic

Release date: October 21 2012

Can a statement be simultaneously true and false? That might seem like sheer nonsense to you -- but not to certain modern logicians. In this episode Massimo and Julia are joined again by philosopher and logician *Graham Priest*, who explains why we have to radically revise our notions of "true" and "false." In the process, he explains classic puzzlers like the "barber paradox": "In a village, the barber shaves all men who do not shave themselves. Does he shave himself?" Follow along for an episode that really takes to heart the podcast's tagline: exploring the borderlands between reason and nonsense.

*Graham's picks: **"Dialogues Concerning Natural Religion: The Posthumous Essays of the Immortality of the Soul and of Suicide" *and *"Logic: A Very Short Introduction" *

## Reader Comments (21)

here's what i think is at the heart of julia's resistance to this idea:

the liar's paradox shows we can formulate a statement that is true and not true. but is it a statement that can conform to any reality? this it does not show.

we can use the idea of infinity to make a set that, when divided by a million, is still infinity. but can the word "infinity" ever correspond to a reality? i think not. we can have the idea, but we cant have the thing.

you can draw a shape on paper that seems to be 3 dimensional, but can never be built in 3d space. so too you can formulate sentences that, while logically coherent, are still merely collections of words that are untranslatable to the universe of "stuff", and therefore, at the very least, are non-useful, no matter how stimulating they may be.

if you say "what i am saying now is false"...notice that "what i am saying now" is nothing. nothing has actually been asserted in the place where an assertion is supposed to go. i could just as sensibly say "BLAH BLAH PLACEHOLDER BLAH is false" for all the content of that sentence. if you actually say SOMETHING, the problem vanishes. just use words that correspond to real things, like "what i said yesterday about your mothers' titties is false". that useful and meaningful statement, in the moment of speaking, either corresponds to reality (it is true) or it doesnt (it is false).

words are labels for things. but we are free to make labels for things whose existence are not established. i can define a being as omnipotent, and we can get ourselves into all kinds of trouble figuring out how that works ("can he microwave a burrito so hot he can't eat it?"), but i havent shown omnipotence to exist. so we are just having fun. i realize that at the forefront of knowledge we may entertain many ideas without knowing whether they are real or not. but maybe we should have a sense of humor about those exercises and keep a big asterisk next to any conclusions we draw, if we have the temerity to draw conclusions at all.

should we build rules of logic based on manipulation of labels for potentially unreal things? only if we don't need our logic to correspond to reality.

did M. C. Escher prove a staircase made of just fifty steps can go up forever? allow me to don my philosopher's hat, and say "yes. yes he did."

put another way, the simplest form of the liar's paradox is:

"this statement is false."

we could look at this as a mathematical equation, like:

(this statement) == false

the part in parenthesis is the part we have to resolve, all else is crystal clear. if i walk up to you and say "this statement" what have i really said? anything? i've muttered a placeholder for something, perhaps, but i haven't actually made a statement with content. certainly there is nothing there to resolve to true or false. we might as well say:

unspecified == false.

as they say, garbage in, garbage out. should we be amazed that our mind returns an error when we feed it this gibberish?

i remember hearing this many years ago:

"the following statement is true. the preceding statement was false."

if we break it up as a programmer might, we see:

[1] (the following statement) == true

[2] (the preceding statement) == false

in order to resolve line [1] we have to resolve line [2] which contains a reference back to line [1] and so in. its not a paradox, its an infinite loop. no more profound than:

function doThis() {

doThat();

}

function doThat() {

doThis();

}

sure, its great busywork for irritating computers when you want to be left alone for a while. but at best, its another "deepity"

The description for this fine episode begins with the question, "Can a statement be simultaneously true and false?" But at times, Julia, Massimo and Graham Priest used the word "something" interchangeably with "statement." "Something" suggests concreteness, "thingness," empirically observable objects or "reality" as owen says in a previous comment. Can it be simultaneously true and false that there are (were) WMD hidden in Iraq? While I'm sure some politicians might want to assert that this could be the case, it cannot, just as it cannot be simultaneously true and false that Graham Priest was paid in full for his book contract (though it's conceivable that a publisher's lawyer might try to argue otherwise).

Graham mentioned that the law of non-contradiction is not taken for granted in some Eastern philosophies. This is true, though not necessarily in the ways some modern Westerners who associate themselves with Eastern philosophies might think. As skeptics are well aware, there are many people today who appeal to Eastern philosophy and quantum physics to support ideas that seem to defy the law of non-contradiction. Some appeal to paradox (such as paradoxical koans from Japanese Zen) in an attempt to show that those who question certain assertions (about reincarnation and psychic phenomena, consciousness as eternal and ubiquitous, etc.) just "don't get" the "paradoxical nature of reality" that is "beyond" the "dualistic" categories of true and false. So it is good that Julia questions the notion that "something" can be simultaneously true and false. Some statements can be simultaneously true and false. Quantum physics may lead us to make paradoxical

statementsabout the behavior of particles, etc. But many people abuse such ideas to make claims about concrete reality that should arouse healthy skepticism.Just curious as to why this episode hasn't shown up on the iTunes feed yet.

Can anyone recommend a concise introduction to paraconsistent logic, either written by Graham Priest or another author?

Seems like you could just as easily construct a Revenge style argument against para consistent logic, ie. "This statement is either false or simultaneously true and false."

1) I think this podcast finally downloaded from the iTunes feed today. 2) Another obvious version of the Self Referential Paradox is the Christian Holy Bible - which tells of a God and is the Word of said God and is Truth and Holy by definition, according to the Bible. Atheists, who are "outside" of the referential frame of that book clearly see this paradox, but Believers, inside the framework created by the book, do not tend to see it at all and get upset when one tries to point it out.

Thanks for a very thought-provoking episode ( but then I assume that that was the idea :-) I have a number of thoughts relating to the podcast:

A contradiction, expressed simply, is essentially this:

(A & !A) is true.

which is nonsense under traditional logic. To say that this statement makes sense in a way that is useful to me in my day-to-day life brings this quote to mind:

Extraordinary claims require extraordinary evidence.Note that if you evaluate the contradiction again, you get:

(A & !A) == TRUE

(FALSE) == TRUE

FALSE

As I understand the similarity between Newtonian and relativistic physics, and traditional logic and paraconsistent logic, the following come to mind:

Newtonian physics approximately describes reality. The discrepency isn't noticable until you are traveling fast enough to measure your speed as a fraction of the speed of light, or if you are using REALLY sensitive instruments. Relativistic physics does a much better job of explaining reality for fast-moving objects, but is more complex.

Traditional logic exactly describes truth relationships, the only reason to extend logic would be to manipulate additional truth values (see below).

Paraconsistent logic seems to be an attempt to deal with the problem that in mathematics is stated as 'What do you get when you divide by zero'. It may have similarities to the problem of taking the square root of -1, which is represented by the symbol

i, and operations that useiare represented in a number plane (real + imaginary), rather than the normal real number line.The 'divide by zero' idea works well for explaining the 'conclusion explosion', which can be defined as:

x/0 = UNDEFINED, solve for x.

One possible extension of logic (I'm not sure it would qualify as a paraconsistent logic system as contradictions are still 'wrong') is as follows:

Define a new truth value called UNDEFINED (or UNSURE) (U). One way to think of it is as the Heisenberg Uncertainty Principle applied to logic. You don't know the value of U at the moment, it might end up being either T or F. Another use is for logical values representing paradoxes or contradictions.

Operations that can definitely return T or F will do so, and return U if they cannot.

The normal logical operations are as follows:

T and T == T ... T and U == U

T and F == F ... F and U == F

F and F == F ... U and U == U

`T or T == T ... T or U == T`

T or F == T ... F or U == U

F or F == F ... U or U == U

`!T == F ... !U == U`

!F == T

One way to think of the Liar's Paradox is what I call a

Moebius fallacy(each time you evaluate it you flip the logic value, and end up back where you started every two iterations, just like a Moebius strip).Another way is to state that the truth value of the Liar's Paradox is U.

To owen's Escher statement above, I would add: just not in Euclidean space.

Logic, math, and grammar are essentially a bunch of rules for making substitutions. Like a*(b+c)=a*b+a*c and a&(b|c)=(a&b)|(a&c).

You can make up whatever axioms you want, but they're useful if they model some relationships in reality. Like, I can cut out rectangles of size a x b and a x c from one paper rectangle of size a x (b+c).

If some set of rules models something better, use those rules. See the square-rectangle problem in object oriented programming.

Graham's attempt to use "revenge" on "what I am saying now is false" doesn't seem to work.

Let's say a statement has to be either true or false. If something is neither, it isn't a statement.

"What I am saying now is either false, or it isn't a statement" is not a statement because it can't be true or false.

Graham would then say "aha, so it isn't a statement, so therefore it is true because it claimed that it wasn't a statement". But it's not actually making that claim, because it isn't a statement. Graham's mistake is that he seems to forget that the last thing that I put in quotes isn't a statement.

Owen, what's a deepity?

...as in deepity chopakrita?

As to the paradox per se, here's its simplest form: "This sentence is false." Is it true? If so, it's false. Is it false? If so, it's true.

My own solution: It's not necessary that it's either, especially if it's both. Because it's simply true that it's false. Or not so simply, it's not false that it's true.

mark,

although full of deepities, deepak chopra is unrelated to the meaning. and on the off chance the question isnt rhetorical - the word was coined by daniel dennet to refer to statements that can be interpreted as (a) profound but untrue, or (b) true, but trivial.

example: "love is just a word"

...also thanks to dave p for saying what i was trying to say in a more educated way.

also, i will now have a hard time refering to deepak chopra with his actual name. thanks.

Check this for an unravelable paradox:

Any alteration of sequential change prevents the changes from returning to the inevitable track of the sequence they had previously been on. Yet all changes remain on an inevitable track.

And as to this one: "In a village, the barber shaves all men who do not shave themselves. Does he shave himself?"

There is no law of logic that requires the initial statement to be true.

Owen's answer asserts the self referential nature of the Lie Paradox is empty: that the sentence must have some object less ephemeral than itself upon which to operate, in order to have meaning.

After reflection I dislike it. Although we tend to run an iteration in our head when we're trying to check these tongue twisters, it is the technique of recursion that's being called into context by a self referential sentence. There's nothing wrong with recursion.

Recursion's a perfectly legitimate, if quirky, technique. It can be programmed up easily enough.

That an iterative evaluation of the Lie Paradox runs forever instead of returning with a fixed answer is only problematic for the computer if you demand an immutable answer. It's an infinite series that does not converge: F, T, F, T, F, etc... Is there something useful in the observation that this sequence can be viewed as a history In Time? The computer, asked at any moment, would have a definite answer! It's just time variant. There's very useful math that covers this (the z-transform) and nothing "para" about it, though it is fun to think about.

So, I don't think it's an invalid word trick to imply recursion, but I do think it's invalid to imply simultaneity: all iterations being valid at the same time. There's a mental leap from self referential to iterative execution with which I'm uncomfortable. That seems like the moment when the card trick happens. Admission of time, a time sequence of answers, resolves the paradox. Of course, to that Graham may say, "Yah, I meant at one moment in time."

Sorry all, but here's another: I call baloney on "Revenge"

Recall that paraconsistent resolution to the Lie paradox said "this is either False or Neither." (Neither true nor false.)

Let's see how that works out. You want to say, "well if we begin with neither, then it IS neither, so the sentence is ok.

However, I think that's a little sloppy. Watch this:

Iterating, starting from a presumption of Neither...

X = (F or N) --> X is Neither so the clause evaluates as True

X = (F or N) --> X is now True, then the clause is now False,

...and we're off to the TFTFTF races. The evaluation of the sentence has stumbled on it's own invention.

There are lots of good rebuttals above of Graham's claim that the 'Liar Paradox' is a genuine contradiction in formal logic, rather than a vague hand-wave involving undefined words. like 'true' and 'meaningless' (which is what it really is). I particularly like Elliot's, which sums up a fatal flaw in the paradox claim clearly and concisely.

On a different tack, one can just take a pragmatic approach. If any genuine contradiction could be deduced by a strict, formal application of the rules of classical logic (eg Second Order Predicate Logic), logicians would have abandoned that type of logic completely and we'd all be doing paraconsistent logic. The fact that paraconsistent logic is only a fringe interest pursued by very few logicians - at least one of whom appears unable to formally analyse the 'liar paradox' claim - tells us all we need to know about the claim.

And another thing ..... Graham seemed to think he'd scored a point off Julia when he asked her to define what 'true' means. In fact he is the one that needs to define 'true' (and also 'meaningless') if he wants to claim that the 'liar paradox' is anything other than a jumble of words that is grammatically correct but has no logical interpretation.

Great episode, as usual!

I'm not a logician; but it does not blow up my world to accept certain statement-like (but also statement-unlike) expressions as being neither true nor false.

Not every expression that looks like a classical factual statement functions as an expression that is either true or false (i.e., functions as a classical statement), as J.L. Austin amply demonstrated (_How to do things with words_), performatives being a case in point.

Russell's paradox itself was not delved into in the episode as far as I recall; but it can be handled procedurally.

AI philosopher Aaron Sloman (now at the University of Birmingham) touched on a related problem and argued that these supposed paradoxes can be resolved by reference to computer programming systems. If you apply a procedure able in finite time to determine the truth value of a statement and the procedure never returns a result, then it is neither true nor false. (Of course a lot hinges on whether we ascribe this ability.)

Cf:

http://www.cs.bham.ac.uk/research/projects/cogaff/crp/postscript.html

I think Sloman's argument does a good job of articulating at least my tacit intuition that an infinite regress is at play here.

Luc Beaudoin

http://cogzest.com/ http://mySleepButton.com

Although I did not hear it in this episode of the Podcast, the description for this episode describes the Barber's Paradox. The Barber does indeed shave his own head.