# Can Logic Be Rationally Revised?

August 25, 2013 9:55 AM Subscribe

Here is a video of the philosopher Graham Priest giving a talk at the 2012 Conference on Paradox and Logical Revision. He addresses three questions. Can logic be revised? If so, can it be revised rationally? If so, how?

In the talk, Priest makes a couple of unimportant errors that audience members point out immediately. The first error is a mistake about the mood of the syllogistic form called Darapti. Wikipedia has a nice discussion of types of syllogism. For lots more on syllogistic and its history, see (of course) the SEP article on Medieval Theories of the Syllogism.

The second error was a goof about the Wason selection task. Here is a nice interactive example of how the task works, followed by some discussion.

Timothy Williamson is the "Tim" that Priest refers to several times in the talk -- mostly in order to disagree with him.

In the talk, Priest makes a couple of unimportant errors that audience members point out immediately. The first error is a mistake about the mood of the syllogistic form called Darapti. Wikipedia has a nice discussion of types of syllogism. For lots more on syllogistic and its history, see (of course) the SEP article on Medieval Theories of the Syllogism.

The second error was a goof about the Wason selection task. Here is a nice interactive example of how the task works, followed by some discussion.

Timothy Williamson is the "Tim" that Priest refers to several times in the talk -- mostly in order to disagree with him.

Nice post! Metafilter readers might also be interested in Graham Priest's New York Times Opinionator piece on true contradictions. To my mind, it's one of the best Opinionator pieces and should be a model for all others.

posted by painquale at 10:25 AM on August 25, 2013 [4 favorites]

posted by painquale at 10:25 AM on August 25, 2013 [4 favorites]

Hey, is the problem with "Darapti" that the class you're talking about can be empty?

posted by zscore at 12:00 PM on August 25, 2013 [2 favorites]

posted by zscore at 12:00 PM on August 25, 2013 [2 favorites]

Yes. Aristotelian syllogistic presupposes that every class has at least one member. With that assumption made explicitly, Darapti is just fine.

posted by Jonathan Livengood at 1:05 PM on August 25, 2013 [3 favorites]

posted by Jonathan Livengood at 1:05 PM on August 25, 2013 [3 favorites]

Oh man, it finally got cool at the very end!

posted by zscore at 3:27 PM on August 25, 2013

**Jonathan Livengood**(or others), do you have any material that outlines model-theoretic and proof-theoretic validity?posted by zscore at 3:27 PM on August 25, 2013

The Times piece is fun, but I have to say that I'm baffled as to why people are baffled by the "Liar Paradox".

There are numerous other paradoxes that (it seems to me) also fail on the same issue - for example, the Unexpected Hanging.

Some of it seems to rely on the fact that statements individuals make about the future have some absolute truth or falsity value before that future happens... but of course, they don't. I can say, "I promise, I won't throw up in the cab," but no matter how trustworthy I appear, you won't know for sure if this is true until I've actually exited the cab.

So if I announce that everything I say is going to be false, you have no way of knowing this is true until I've actually died.

So far, so good - but what I you restrict the extent of the issue? Suppose I make a series of statements and then end with, "...And all of these statements are false." As the Times article points out, that's not good enough - you can define another possibility, that the last statement to be "meaningless", so the true paradox comes up if you say, "...And all of these statements are false or meaningless."

The article claims a contradiction there, as follows: "If this were true or false, we would be in the same bind as before. And if it’s meaningless, then it’s either false or meaningless, so it’s true."

But you're being scammed. And a lot of scam has to do with the idea of "meaningless". This is logician bullshit(*) - they are so convinced of the universality of their snake oil that if any sentence isn't either 100% true or 100% false, it's meaningless.

Of course, in the real world this simply isn't so. If you're on a first date someone and they tell you, "Oh, everything I say is a lie," this is by no means a "meaningless" sentence, even though it can't be resolved as true or false. (For most of us, if someone said this and "meant" it, it would be a cue to make an excuse and leave...)

The correct concept isn't "meaningless" - the correct concept "doesn't have a well-defined truth or falsity value" - in other words, "is neither true nor false".

With this, the paradox vanishes. Now you have to make the statement, "Everything I say is either false, or it is neither true nor false." Then if I then emit a pack of lies, all those statements are false, and the original statement is still neither true nor false - "doesn't have a well-defined truth or falsity value".

Isn't it just "true"? No, the "paradox" actually

Another way to think of this is to ask the following question: "If someone makes a statement, and there is no refutation for it, is it necessarily true?" Logicians say, "Yes" but there is no refutation for lots of real-world statements that aren't "true". Just because there is no refutation for the statement, "Everything I say is either a lie, or is neither true nor false," doesn't mean the statement is "true".

Again, it's back to the unreliability of humans. The idea that we're little logic machines emitting statements to which God or Logic attaches one of two tags, TRUE or FALSE, is patently ridiculous. Indeed, people who answer, "Yes," to questions like, "Could you get me a glass of water?" are universally considered to be terminally annoying.

Which reminds me of a joke (**). Three logicians are at a bar, and the bartender asks, "Do any of you want a glass of water?" The first logician says, "I don't know," the second logician says, "I don't know," and the third one says, "No."

(* - I took a couple of graduate-level mathematical logic classes and got high marks, so I have chops to say this and at least be wrong in an interesting way...)

(** - I know this was linked from something here a few weeks ago but I've loved this joke for ages and I have so few chances to tell it.)

posted by lupus_yonderboy at 5:11 PM on August 25, 2013 [2 favorites]

There are numerous other paradoxes that (it seems to me) also fail on the same issue - for example, the Unexpected Hanging.

Some of it seems to rely on the fact that statements individuals make about the future have some absolute truth or falsity value before that future happens... but of course, they don't. I can say, "I promise, I won't throw up in the cab," but no matter how trustworthy I appear, you won't know for sure if this is true until I've actually exited the cab.

So if I announce that everything I say is going to be false, you have no way of knowing this is true until I've actually died.

So far, so good - but what I you restrict the extent of the issue? Suppose I make a series of statements and then end with, "...And all of these statements are false." As the Times article points out, that's not good enough - you can define another possibility, that the last statement to be "meaningless", so the true paradox comes up if you say, "...And all of these statements are false or meaningless."

The article claims a contradiction there, as follows: "If this were true or false, we would be in the same bind as before. And if it’s meaningless, then it’s either false or meaningless, so it’s true."

But you're being scammed. And a lot of scam has to do with the idea of "meaningless". This is logician bullshit(*) - they are so convinced of the universality of their snake oil that if any sentence isn't either 100% true or 100% false, it's meaningless.

Of course, in the real world this simply isn't so. If you're on a first date someone and they tell you, "Oh, everything I say is a lie," this is by no means a "meaningless" sentence, even though it can't be resolved as true or false. (For most of us, if someone said this and "meant" it, it would be a cue to make an excuse and leave...)

The correct concept isn't "meaningless" - the correct concept "doesn't have a well-defined truth or falsity value" - in other words, "is neither true nor false".

With this, the paradox vanishes. Now you have to make the statement, "Everything I say is either false, or it is neither true nor false." Then if I then emit a pack of lies, all those statements are false, and the original statement is still neither true nor false - "doesn't have a well-defined truth or falsity value".

Isn't it just "true"? No, the "paradox" actually

*proves*that it isn't just "true" (by contradiction), and you're left with only "doesn't have a well-defined truth or falsity value".Another way to think of this is to ask the following question: "If someone makes a statement, and there is no refutation for it, is it necessarily true?" Logicians say, "Yes" but there is no refutation for lots of real-world statements that aren't "true". Just because there is no refutation for the statement, "Everything I say is either a lie, or is neither true nor false," doesn't mean the statement is "true".

Again, it's back to the unreliability of humans. The idea that we're little logic machines emitting statements to which God or Logic attaches one of two tags, TRUE or FALSE, is patently ridiculous. Indeed, people who answer, "Yes," to questions like, "Could you get me a glass of water?" are universally considered to be terminally annoying.

Which reminds me of a joke (**). Three logicians are at a bar, and the bartender asks, "Do any of you want a glass of water?" The first logician says, "I don't know," the second logician says, "I don't know," and the third one says, "No."

(* - I took a couple of graduate-level mathematical logic classes and got high marks, so I have chops to say this and at least be wrong in an interesting way...)

(** - I know this was linked from something here a few weeks ago but I've loved this joke for ages and I have so few chances to tell it.)

posted by lupus_yonderboy at 5:11 PM on August 25, 2013 [2 favorites]

Hmm, now I look at the article more closely I seem to be a dialetheian. I should have published my opinions decades ago...

Also, I don't suppose there's a transcript of the video? 42 minutes and change is too long to watch, and I'm much more comfortable reading a text when it comes to logic, you can pull all sorts of things over people when you talk....

posted by lupus_yonderboy at 5:14 PM on August 25, 2013

Also, I don't suppose there's a transcript of the video? 42 minutes and change is too long to watch, and I'm much more comfortable reading a text when it comes to logic, you can pull all sorts of things over people when you talk....

posted by lupus_yonderboy at 5:14 PM on August 25, 2013

zscore,

I'm not sure what you want from an outline or exactly what level of material you are after, but here goes. You can tell me how badly I've missed the mark, and we'll go from there. (Also if I make any mistakes, someone please point them out!)

The main ideas for model theoretic and proof theoretic validity are these.

Model Theoretic Validity: An argument {P1, P2, ..., Pn} |= C is valid just in case every model of the premisses {P1, P2, ..., Pn} -- that is, every interpretation of the premisses that makes them all true (that's what a model is) -- is also a model of the conclusion C.

Proof Theoretic Validity: An argument {P1, P2, ..., Pn} |- C is valid just in case there is a sequence such that every element in the sequence is either one of the premisses or derived from previous elements in the sequence according to a good rule of inference.

That a rule of inference is a good one in the proof theoretic framework is supposed to be something that we can read off of the meaning of the logical constants, e.g. connectives like AND. The thought is that you understand what AND means in virtue of seeing that the introduction rule for AND -- if you have P and you have Q, then you may infer P AND Q -- is a good rule. (Some theorists also think you need to see that the elimination rule is a good one, for example if you have P AND Q, then you may infer P and you may infer Q. Gentzen thought that the elimination rules could be derived from the introduction rules.)

For much more detail on model theory, see the SEP articles on Model Theory and on First-Order Model Theory.

For much more detail on proof theory, see the SEP articles on Proof-Theoretic Semantics and on The Development of Proof Theory.

You might also find the SEP article on Logical Consequence helpful, especially Section 3. In fact, you would probably be well-served to begin with the article on Logical Consequence.

posted by Jonathan Livengood at 5:59 PM on August 25, 2013 [1 favorite]

I'm not sure what you want from an outline or exactly what level of material you are after, but here goes. You can tell me how badly I've missed the mark, and we'll go from there. (Also if I make any mistakes, someone please point them out!)

The main ideas for model theoretic and proof theoretic validity are these.

Model Theoretic Validity: An argument {P1, P2, ..., Pn} |= C is valid just in case every model of the premisses {P1, P2, ..., Pn} -- that is, every interpretation of the premisses that makes them all true (that's what a model is) -- is also a model of the conclusion C.

Proof Theoretic Validity: An argument {P1, P2, ..., Pn} |- C is valid just in case there is a sequence such that every element in the sequence is either one of the premisses or derived from previous elements in the sequence according to a good rule of inference.

That a rule of inference is a good one in the proof theoretic framework is supposed to be something that we can read off of the meaning of the logical constants, e.g. connectives like AND. The thought is that you understand what AND means in virtue of seeing that the introduction rule for AND -- if you have P and you have Q, then you may infer P AND Q -- is a good rule. (Some theorists also think you need to see that the elimination rule is a good one, for example if you have P AND Q, then you may infer P and you may infer Q. Gentzen thought that the elimination rules could be derived from the introduction rules.)

For much more detail on model theory, see the SEP articles on Model Theory and on First-Order Model Theory.

For much more detail on proof theory, see the SEP articles on Proof-Theoretic Semantics and on The Development of Proof Theory.

You might also find the SEP article on Logical Consequence helpful, especially Section 3. In fact, you would probably be well-served to begin with the article on Logical Consequence.

posted by Jonathan Livengood at 5:59 PM on August 25, 2013 [1 favorite]

Jonathan Livengood: really nice summary!

As a quibble, "just in case" has another meaning in "regular English" so I try to use "if and only if" or "exactly if" when I'm talking to people because they get confused - heck, I get confused if I don't put my math hat on.

But I came to ask this:

> Gentzen thought that the elimination rules could be derived from the introduction rules.

How could that be? The introduction rule for AND says, "the statement A and the statement B imply the statement A AND B". But this rule would also be true of OR. Surely it's the elimination rule for AND that distinguishes it from OR...?

posted by lupus_yonderboy at 6:06 PM on August 25, 2013

As a quibble, "just in case" has another meaning in "regular English" so I try to use "if and only if" or "exactly if" when I'm talking to people because they get confused - heck, I get confused if I don't put my math hat on.

But I came to ask this:

> Gentzen thought that the elimination rules could be derived from the introduction rules.

How could that be? The introduction rule for AND says, "the statement A and the statement B imply the statement A AND B". But this rule would also be true of OR. Surely it's the elimination rule for AND that distinguishes it from OR...?

posted by lupus_yonderboy at 6:06 PM on August 25, 2013

Nice question. I'm not very certain about what I'm going to say, but for what it's worth, I think the difference is supposed to be this. The introduction rule for AND requires that you have both A written down at some place in your proof sequence and B written down at some place in your proof sequence. But the introduction rule for OR has a different form. I don't remember exactly what Gentzen's rule looks like, but here are a couple of options:

1. If you have A written down, you may write down A OR B, where B is arbitrary.

2. If you have ~A -> B written down, you may write down A OR B.

While it is true that if you have both A and B written down, you could also write A OR B, the introduction rules for AND and OR do not give you all and only the same inferences.

posted by Jonathan Livengood at 6:14 PM on August 25, 2013

1. If you have A written down, you may write down A OR B, where B is arbitrary.

2. If you have ~A -> B written down, you may write down A OR B.

While it is true that if you have both A and B written down, you could also write A OR B, the introduction rules for AND and OR do not give you all and only the same inferences.

posted by Jonathan Livengood at 6:14 PM on August 25, 2013

Aha, got it. So he has a slightly different introduction rule than what I might expect, which differentiates the two. Seems quite slick to me...

posted by lupus_yonderboy at 6:37 PM on August 25, 2013

posted by lupus_yonderboy at 6:37 PM on August 25, 2013

lupus_yonderboy: "

I'm rather partial to "precisely when", rolls nicely off the tongue.

posted by Proofs and Refutations at 7:23 PM on August 25, 2013 [1 favorite]

*"*

As a quibble, "just in case" has another meaning in "regular English" so I try to use "if and only if" or "exactly if" when I'm talking to people because they get confused - heck, I get confused if I don't put my math hat on.

As a quibble, "just in case" has another meaning in "regular English" so I try to use "if and only if" or "exactly if" when I'm talking to people because they get confused - heck, I get confused if I don't put my math hat on.

I'm rather partial to "precisely when", rolls nicely off the tongue.

posted by Proofs and Refutations at 7:23 PM on August 25, 2013 [1 favorite]

On the main topic, I feel compelled to point to a wonderful paper in the form of a short story that illustrates some of Graham's ideas on contradictions: Sylvan's Box.

posted by Proofs and Refutations at 7:41 PM on August 25, 2013

posted by Proofs and Refutations at 7:41 PM on August 25, 2013

Proofs and Refutations: I imagine you're referring to Proofs and Refutations with your name? What a great book - and how very appropriate to this thread...

posted by lupus_yonderboy at 8:02 PM on August 25, 2013

posted by lupus_yonderboy at 8:02 PM on August 25, 2013

lupus_yonderboy: Yes, I took my name from Lakatos' marvelous dialogue. Even better, I first encountered the book as a set text in a course on the philosophy of mathematics co-taught by Graham!

posted by Proofs and Refutations at 8:19 PM on August 25, 2013 [2 favorites]

posted by Proofs and Refutations at 8:19 PM on August 25, 2013 [2 favorites]

Good work up above! Sheesh, there is a reason I gave up on philosophy after I managed to squeak out an M.A. But I did get to take a class in metalogic/set theory from Raymond Smullyan, which was worth the price of admission, so I guess I got my money's worth, even if the only way I managed to pass was by memorizing the entire proof for Goedel's incompleteness theorem without understanding it. My utmost respect to you.

posted by JimInLoganSquare at 8:47 PM on August 25, 2013 [1 favorite]

posted by JimInLoganSquare at 8:47 PM on August 25, 2013 [1 favorite]

Whoa, studying logic with Smullyan like saying you studied painting with Picasso!

Trivia note: did you know that Gödel also had a

EDIT: that link for non-standard analysis isn't really very enlightening - I couldn't find a better one...?

posted by lupus_yonderboy at 9:13 PM on August 25, 2013

Trivia note: did you know that Gödel also had a

*Completeness*Theorem - which is only somewhat distantly related to the Incompleteness Theorem and has consequences that almost as trippy?EDIT: that link for non-standard analysis isn't really very enlightening - I couldn't find a better one...?

posted by lupus_yonderboy at 9:13 PM on August 25, 2013

Smullyan seems like the ideal professor for this. At least you'd be entertained as the remnants of your intellectual ego were ground up.

posted by thelonius at 9:18 PM on August 25, 2013

posted by thelonius at 9:18 PM on August 25, 2013

*> How could that be? The introduction rule for AND says, "the statement A and the statement B imply the statement A AND B". But this rule would also be true of OR. Surely it's the elimination rule for AND that distinguishes it from OR...?*

Livengood did explain that the introduction rule for conjunction is different from that of disjunction. I don't know what you mean by Gentzen being "slick." The way Livengood described disjunction introduction is the way disjunction works regardless of if you're doing proof-theoretical research, or teaching undergraduates basic propositional logic.

As for the later point about the elimination rules following from the introduction rules: I believe this response has become the default response to give. I don't believe Gentzen ever argued this point, but later big proof-theoreticians like Prawitz and Dummett continued this line of thought from Gentzen. So, I'm sure there are those out there who say that introduction rules follow from the elimination rules. Although, it may be that whichever way doesn't matter.

In any case, the way this is supposed to work, is that given an introduction rule, there exists an elimination rule that will satisfy particular conditions. I suppose an introduction to this line of thought without having to dig into a book, would be here and here.

posted by SollosQ at 11:21 PM on August 25, 2013

*Trivia note: did you know that Gödel also had a Completeness Theorem - which is only somewhat distantly related to the Incompleteness Theorem and has consequences that almost as trippy?*

Yes; in fact, the semester immediately prior to studying the Incompleteness Theorem with Smullyan, I studied the Completeness Theorem with a different professor. I believe it was considered "warming up."

posted by JimInLoganSquare at 7:15 AM on August 26, 2013

*Smullyan seems like the ideal professor for this. At least you'd be entertained as the remnants of your intellectual ego were ground up.*

True. On the first day of class, he did card tricks and told jokes. On Day Two, the grinder was turned on.

posted by JimInLoganSquare at 7:18 AM on August 26, 2013 [1 favorite]

For what it's worth, one of the jokes Smullyan told on that first day of metalogic was the old "comedians convention" joke. The Straight Dope boards had a fun thread about that joke.

posted by JimInLoganSquare at 7:10 PM on September 5, 2013

posted by JimInLoganSquare at 7:10 PM on September 5, 2013

« Older Let’s not complicate things unnecessarily. | Cat belts out Journey's "Don't Stop Believing" Newer »

This thread has been archived and is closed to new comments

posted by anotherpanacea at 10:13 AM on August 25, 2013