McGee's counterexample to the Modus Ponens

[Speakpigeon]

Here is an interesting example to try your wits...

This is one philosopher, Vann McGee, who in 1986 proposed a counterexample to the Modus Ponens, no less!

Here is the thing:

McGee's counterexample
https://link.springer.com/article/10.1007/BF00355293

Almost 10 years ago, Vann McGee pushed philosophical doubt beyond another frontier. His attempt to show that modus ponens is not a valid form of inference- and to show this by help of a counterexample and not by envisaging an evil demon confusing us - is proof of the ingenuity of a philosopher's ability to doubt. Other philosophers might be less impressed. They criticize McGee's counterexample, thinking it either rests on confusions or can, in some other way, easily be circumvented. I argue in this paper that such a reaction is unjustified. McGee's counterexample withstands the criticisms raised against it. Should we thus abolish modus ponens? It depends, I think, on what the right theory of conditionals is, and though I will provide some material for deciding this question, in the end, this material will be indecisive.

McGEE'S CLAIM ABOUT MODUS PONENS​

It is sufficient to focus our discussion on one of McGee's counterexamples - others follow the same recipe.

Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.​

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.​

This example shows that modus ponens is not an entirely reliable rule of inference. Sometimes the conclusion of an application of modus ponens is something we do not believe and should not believe, even though the premises are propositions we believe very properly. (McGee 1985, pp. 462f.)

Anyone understands what's going on here?
EB

 

[Jokodo]

Here is an interesting example to try your wits...

This is one philosopher, Vann McGee, who in 1986 proposed a counterexample to the Modus Ponens, no less!

Here is the thing:

McGee's counterexample
https://link.springer.com/article/10.1007/BF00355293

Almost 10 years ago, Vann McGee pushed philosophical doubt beyond another frontier. His attempt to show that modus ponens is not a valid form of inference- and to show this by help of a counterexample and not by envisaging an evil demon confusing us - is proof of the ingenuity of a philosopher's ability to doubt. Other philosophers might be less impressed. They criticize McGee's counterexample, thinking it either rests on confusions or can, in some other way, easily be circumvented. I argue in this paper that such a reaction is unjustified. McGee's counterexample withstands the criticisms raised against it. Should we thus abolish modus ponens? It depends, I think, on what the right theory of conditionals is, and though I will provide some material for deciding this question, in the end, this material will be indecisive.

McGEE'S CLAIM ABOUT MODUS PONENS​

It is sufficient to focus our discussion on one of McGee's counterexamples - others follow the same recipe.

Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.​

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.​

This example shows that modus ponens is not an entirely reliable rule of inference. Sometimes the conclusion of an application of modus ponens is something we do not believe and should not believe, even though the premises are propositions we believe very properly. (McGee 1985, pp. 462f.)

Anyone understands what's going on here?
EB

What's going on is the  Principle_of_explosion. He's trying to sneak in a change of premises halfway through the argument. The conviction that "A Republican will win the election." rests entirely on the premise that the polls are accurate. The conditional "If it's not Reagan who wins" requires the polls to be inaccurate. So he ends up saying "if the polls are accurate and the polls are inaccurate", which is a contradiction.

 

[Jokodo]

Here is an interesting example to try your wits...

This is one philosopher, Vann McGee, who in 1986 proposed a counterexample to the Modus Ponens, no less!

Here is the thing:

McGee's counterexample
https://link.springer.com/article/10.1007/BF00355293

Almost 10 years ago, Vann McGee pushed philosophical doubt beyond another frontier. His attempt to show that modus ponens is not a valid form of inference- and to show this by help of a counterexample and not by envisaging an evil demon confusing us - is proof of the ingenuity of a philosopher's ability to doubt. Other philosophers might be less impressed. They criticize McGee's counterexample, thinking it either rests on confusions or can, in some other way, easily be circumvented. I argue in this paper that such a reaction is unjustified. McGee's counterexample withstands the criticisms raised against it. Should we thus abolish modus ponens? It depends, I think, on what the right theory of conditionals is, and though I will provide some material for deciding this question, in the end, this material will be indecisive.

McGEE'S CLAIM ABOUT MODUS PONENS​

It is sufficient to focus our discussion on one of McGee's counterexamples - others follow the same recipe.

Opinion polls taken just before the 1980 election showed the Republican Ronald Reagan decisively ahead of the democrat Jimmy Carter, with the other Republican in the race, John Anderson, a distant third. Those apprised of the poll results believed, with good reason:

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.​

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.​

This example shows that modus ponens is not an entirely reliable rule of inference. Sometimes the conclusion of an application of modus ponens is something we do not believe and should not believe, even though the premises are propositions we believe very properly. (McGee 1985, pp. 462f.)

Anyone understands what's going on here?
EB

What's going on is the  Principle_of_explosion. He's trying to sneak in a change of premises halfway through the argument. The conviction that "A Republican will win the election." rests entirely on the premise that the polls are accurate. The conditional "If it's not Reagan who wins" requires the polls to be inaccurate. So he ends up saying "if the polls are accurate and the polls are inaccurate", which is a contradiction.

From a more linguistic perspective, you could say that he's confusing two readings if the English indefinite article "a". It has an unspecific reading which can be paraphrased as "whoever wins the election is going to be a Republican", and a specific one roughly meaning "there is a certain Republican [in our context: namely Ronald Reagan] who will win the election". The conclusion only follows from the unspecific reading, which the context doesn't support. The specific reading is again in contradiction to the antecedent if the conditional.

 

[PyramidHead]

What's going on is the  Principle_of_explosion. He's trying to sneak in a change of premises halfway through the argument. The conviction that "A Republican will win the election." rests entirely on the premise that the polls are accurate. The conditional "If it's not Reagan who wins" requires the polls to be inaccurate. So he ends up saying "if the polls are accurate and the polls are inaccurate", which is a contradiction.

From a more linguistic perspective, you could say that he's confusing two readings if the English indefinite article "a". It has an unspecific reading which can be paraphrased as "whoever wins the election is going to be a Republican", and a specific one roughly meaning "there is a certain Republican [in our context: namely Ronald Reagan] who will win the election". The conclusion only follows from the unspecific reading, which the context doesn't support. The specific reading is again in contradiction to the antecedent if the conditional.

This is closer to the real problem, I think. In my view, the ambiguity is actually around "it" in "If it isn't Reagan...". Unpacked, it means "the winning Republican, in the event that a Republican wins." If you plug this into the conclusion, it follows from the premises without contradicting anything.

 

[Treedbear]

What's going on is the  Principle_of_explosion. He's trying to sneak in a change of premises halfway through the argument. The conviction that "A Republican will win the election." rests entirely on the premise that the polls are accurate. The conditional "If it's not Reagan who wins" requires the polls to be inaccurate. So he ends up saying "if the polls are accurate and the polls are inaccurate", which is a contradiction.

From a more linguistic perspective, you could say that he's confusing two readings if the English indefinite article "a". It has an unspecific reading which can be paraphrased as "whoever wins the election is going to be a Republican", and a specific one roughly meaning "there is a certain Republican [in our context: namely Ronald Reagan] who will win the election". The conclusion only follows from the unspecific reading, which the context doesn't support. The specific reading is again in contradiction to the antecedent if the conditional.

This is closer to the real problem, I think. In my view, the ambiguity is actually around "it" in "If it isn't Reagan...". Unpacked, it means "the winning Republican, in the event that a Republican wins." If you plug this into the conclusion, it follows from the premises without contradicting anything.

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.

But they obviously did have a reason to believe that, because when given the hypothetical "a Republican wins the election" and there are only two Republicans, if one was not the winner then it had to be the other. If p then q. If a Republican wins the election, then it will be either Reagan or Anderson.

 

[PyramidHead]

This is closer to the real problem, I think. In my view, the ambiguity is actually around "it" in "If it isn't Reagan...". Unpacked, it means "the winning Republican, in the event that a Republican wins." If you plug this into the conclusion, it follows from the premises without contradicting anything.

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.

But they obviously did have a reason to believe that, because when given the hypothetical "a Republican wins the election" and there are only two Republicans, if one was not the winner then it had to be the other. If p then q. If a Republican wins the election, then it will be either Reagan or Anderson.

That was my point. Properly interpreted, there is every reason to believe the conclusion given the premises, so this is not a counterexample to MP.

 

[Jokodo]

Put simpler, he's substituting a phrase in one meaning ("a Republican" under its specific reading) for the same phrase in another meaning (the unspecific one). So the argument is not entirely unlike saying "elephants are unique in having large trunks. My car has a large trunk. Therefore, my car is an elephant."

 

[Jokodo]

This is closer to the real problem, I think. In my view, the ambiguity is actually around "it" in "If it isn't Reagan...". Unpacked, it means "the winning Republican, in the event that a Republican wins." If you plug this into the conclusion, it follows from the premises without contradicting anything.

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.

Yet they did not have reason to believe

If it's not Reagan who wins, it will be Anderson.

But they obviously did have a reason to believe that, because when given the hypothetical "a Republican wins the election" and there are only two Republicans, if one was not the winner then it had to be the other. If p then q. If a Republican wins the election, then it will be either Reagan or Anderson.

One of the premises, restated in an unambiguous form, reads "if whoever wins the election is a Republican..." The other, "a certain Republican will win the election." Those are two different meanings, and you can't freely substitute one for the other and expect to arrive at a valid argument any more than by substituting the two meanings of "trunk",

 

[Speakpigeon]

OK, I broadly agree with everything which has been said. I would express the problem differently, though. To my mind, the problem is that our belief in this case is not grounded in the argument.

This argument is typical of the kind of arguments where we find it difficult to assess validity because it requires that we abstract the argument from the context. If you submit an argument involving Trump or Boris Johnson, people will forget the question about the validity of the argument and focus instead on the question of the truth of the conclusion.

Our argument here is indeed about a real situation about Ronald Reagan and the Republicans, and we are duly reminded of the opinion polls before the election. This is all an effective massaging of our belief gland and we look at the conclusion of the argument not as part of the argument but as if it was standing on its own and consequently, we all disbelieve the implication because, clearly, if it wouldn't have been Reagan, it would have been Carter.

Here we assess the conclusion in light of the general context of the actual situation referred to by the argument, including, crucially the opinion poll, and including our own memory and understanding of American politics.

There is a similar problem in particular with all arguments about God, which have the ability to upset people a lot and throw off their logic.

The problem, however, is more general. Whenever the argument is realistic and we have some expertise on the subject matter, our own beliefs will come in to derail our logical intuition. This is actualy quite potent. I can look at the Donald Reagan argument here and feel at the same time that the argument is valid and that somehow it doesn't work.

I agree with what has been said. However, the wording of the argument seems good enough and wouldn't in itself cause the problem. The problem arise because we read what are essentially the same sentences differently depending on whether they are part of an implication or the conclusion. This only happens because we are biased by being primed by our own expertise on the subject of American elections.

I will also add to this the interesting comment made by serious people...
EB

Three philosophers from Dartmouth defended modus ponens against McGee's example. They claim to find three confusions in McGee's argument, confusions that, if removed, leave modus ponens unchallenged.

First, modus ponens preserves truth, not grounds for believing or probabilities. A real counterexample would have to use modus ponens to go from true premises to a false conclusion. Second, an analogue of modus ponens for grounds or probabilities must not confuse good grounds or high probabilities for the premises separately with good grounds or high probability for the conjunction of the premises. Finally, the probability of a conditional must not be confused with the conditional probability. (Sinnott-Armstrong, Moor, Fogelin 1986, p. 300)
https://link.springer.com/article/10.1007/BF00355293

 

[Jokodo]

OK, I broadly agree with everything which has been said. I would express the problem differently, though. To my mind, the problem is that our belief in this case is not grounded in the argument.

This argument is typical of the kind of arguments where we find it difficult to assess validity because it requires that we abstract the argument from the context. If you submit an argument involving Trump or Boris Johnson, people will forget the question about the validity of the argument and focus instead on the question of the truth of the conclusion.

Our argument here is indeed about a real situation about Ronald Reagan and the Republicans, and we are duly reminded of the opinion polls before the election. This is all an effective massaging of our belief gland and we look at the conclusion of the argument not as part of the argument but as if it was standing on its own and consequently, we all disbelieve the implication because, clearly, if it wouldn't have been Reagan, it would have been Carter.

Here we assess the conclusion in light of the general context of the actual situation referred to by the argument, including, crucially the opinion poll, and including our own memory and understanding of American politics.

There is a similar problem in particular with all arguments about God, which have the ability to upset people a lot and throw off their logic.

The problem, however, is more general. Whenever the argument is realistic and we have some expertise on the subject matter, our own beliefs will come in to derail our logical intuition. This is actualy quite potent. I can look at the Donald Reagan argument here and feel at the same time that the argument is valid and that somehow it doesn't work.

I agree with what has been said. However, the wording of the argument seems good enough and wouldn't in itself cause the problem. The problem arise because we read what are essentially the same sentences differently depending on whether they are part of an implication or the conclusion. This only happens because we are biased by being primed by our own expertise on the subject of American elections.

I will also add to this the interesting comment made by serious people...
EB

Three philosophers from Dartmouth defended modus ponens against McGee's example. They claim to find three confusions in McGee's argument, confusions that, if removed, leave modus ponens unchallenged.

First, modus ponens preserves truth, not grounds for believing or probabilities. A real counterexample would have to use modus ponens to go from true premises to a false conclusion. Second, an analogue of modus ponens for grounds or probabilities must not confuse good grounds or high probabilities for the premises separately with good grounds or high probability for the conjunction of the premises. Finally, the probability of a conditional must not be confused with the conditional probability. (Sinnott-Armstrong, Moor, Fogelin 1986, p. 300)
https://link.springer.com/article/10.1007/BF00355293

It's true that a confusion of absolute truth it's partly to blame, but even without that the apparent paradox disappears once you substitute all occurrences of "a" and "it" for unambiguous paraphrases. The argument then becomes either obviously invalid because it rests on substituting a term for one if a different meaning, they become contradictory, or the conclusion actually does follow, or at the very least the premises are no longer supported by the context and therefore or intuitions about the conclusion become irrelevant.

For example, a. Specific reading of "a Republican" in the first premise implies a shape shifter who will take in either Reagan's or Anderson's form. Disambiguating "it" as "the Republican who wins" produces a valid conclusion with the second premise as a presupposition, etc.

I challenge you to find an unambiguous paraphrase that doesn't suffer from one if these problems!

 

[Speakpigeon]

It's true that a confusion of absolute truth it's partly to blame, but even without that the apparent paradox disappears once you substitute all occurrences of "a" and "it" for unambiguous paraphrases. The argument then becomes either obviously invalid because it rests on substituting a term for one if a different meaning, they become contradictory, or the conclusion actually does follow, or at the very least the premises are no longer supported by the context and therefore or intuitions about the conclusion become irrelevant.

For example, a. Specific reading of "a Republican" in the first premise implies a shape shifter who will take in either Reagan's or Anderson's form. Disambiguating "it" as "the Republican who wins" produces a valid conclusion with the second premise as a presupposition, etc.

I challenge you to find an unambiguous paraphrase that doesn't suffer from one if these problems!

We can formalise the argument without any ambiguity.

We start with:

A = Reagan wins

B = Anderson wins

A ⊻ B = Either Reagan or Anderson wins

Obviously, we have:

If either Reagan or Anderson wins (either A or B), then if Reagan does not win (¬A), then Anderson wins (B)

Given this, the argument is formalised as follows:

(A ⊻ B) ∧ ((A ⊻ B) → (¬A → B)) ⊢ ¬A → B

Which is valid, of course.

And then, we go back to good English:

p1 If either Reagan wins or Anderson wins, then if Reagan doesn't win, then Anderson wins;
p2 Either Reagan wins or Anderson wins;
C If Reagan doesn't win, then Anderson wins.

Which, I think may be accepted as the essence of McGee's idea, only properly phrased.

However, looking at my reformulation, I think the impression that something is amiss is lost. The argument seems now perfectly good. And it is. So, what happened? I think what happened is that the argument now is disconnected from the context of the actual election with opinions polls saying Reagan will win and if not Reagan then Carter.
EB

 

[Jokodo]

We can formalise the argument without any ambiguity.

We start with:



Obviously, we have:

If either Reagan or Anderson wins (either A or B), then if Reagan does not win (¬A), then Anderson wins (B)

Given this, the argument is formalised as follows:

(A ⊻ B) ∧ ((A ⊻ B) → (¬A → B)) ⊢ ¬A → B

Which is valid, of course.

And then, we go back to good English:

p1 If either Reagan wins or Anderson wins, then if Reagan doesn't win, then Anderson wins;
p2 Either Reagan wins or Anderson wins;
C If Reagan doesn't win, then Anderson wins.

Which, I think may be accepted as the essence of McGee's idea, only properly phrased.

However, looking at my reformulation, I think the impression that something is amiss is lost. The argument seems now perfectly good. And it is. So, what happened? I think what happened is that the argument now is disconnected from the context of the actual election with opinions polls saying Reagan will win and if not Reagan then Carter.
EB

As I said, as soon as we loose ambiguity, we loose the perception of a paradox. The "paradox" is grounded in us automatically assigning incompatible meanings to ambiguous English phrases. It rests in equivocation.

 

[Speakpigeon]

As I said, as soon as we loose ambiguity, we loose the perception of a paradox. The "paradox" is grounded in us automatically assigning incompatible meanings to ambiguous English phrases. It rests in equivocation.

If so then you will have no difficulty producing a formalisation complete with the equivocation you think there in McGee's argument. Your formalisation will need to be convincing as a model of McGee's argument and, crucially, not valid.
EB

 

[Jokodo]

As I said, as soon as we loose ambiguity, we loose the perception of a paradox. The "paradox" is grounded in us automatically assigning incompatible meanings to ambiguous English phrases. It rests in equivocation.

If so then you will have no difficulty producing a formalisation complete with the equivocation you think there in McGee's argument. Your formalisation will need to be convincing as a model of McGee's argument and, crucially, not valid.
EB

Sure.

(A -> (B -> C)). D. Therefore (E -> C).

Where A and D are two different readings of "a Republican", while B and E are "it" once referring to "the winner" and the other time to "the Republican who wins".

Your go.

 

[fromderinside]

Na, na na na, na ....naught na Say it isn't so.

 

[Speakpigeon]

As I said, as soon as we loose ambiguity, we loose the perception of a paradox. The "paradox" is grounded in us automatically assigning incompatible meanings to ambiguous English phrases. It rests in equivocation.

If so then you will have no difficulty producing a formalisation complete with the equivocation you think there in McGee's argument. Your formalisation will need to be convincing as a model of McGee's argument and, crucially, not valid.
EB

Sure.

(A -> (B -> C)). D. Therefore (E -> C).

Where A and D are two different readings of "a Republican", while B and E are "it" once referring to "the winner" and the other time to "the Republican who wins".

Your go.

Sorry, I don't get it. What are your two different readings of "a Republican"?

Further, in the original argument, there is no possibility of two different readings for the two occurrences of "it's not Reagan who wins".

You would need to explain why we make different readings.

The argument itself, taken literally, is valid. No one disputes that and it is in fact obvious. The issue, as McGee says, is that we believe the premises but not the conclusion. Why is that?

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.
If it's not Reagan who wins, it will be Anderson.

EB

 

[Jokodo]

Sure.

(A -> (B -> C)). D. Therefore (E -> C).

Where A and D are two different readings of "a Republican", while B and E are "it" once referring to "the winner" and the other time to "the Republican who wins".

Your go.

Sorry, I don't get it. What are your two different readings of "a Republican"?

"I don't know who will win, but whoever wins the election will be a Republican" vs "I know who is going to win, and he happens to be a Republican, but I won't mention his name", very roughly. In linguistics it's called the specific and unspecific reading.

Further, in the original argument, there is no possibility of two different readings for the two occurrences of "it's not Reagan who wins".

Sure there is. "It" is a pronoun, pronouns don't have intrinsic reference but are assigned a referent from context. The context offers "the Republican who wins" in the premise, but in the conclusion, ether thay context is lacking, we're tricked to interpret it as an unrestricted "the winner".

You would need to explain why we make different readings.

The argument itself, taken literally, is valid. No one disputes that and it is in fact obvious. The issue, as McGee says, is that we believe the premises but not the conclusion. Why is that?

If a Republican wins the election, then if it's not Reagan who wins it will be Anderson.
A Republican will win the election.
If it's not Reagan who wins, it will be Anderson.

EB

Where's your unambiguous paraphrase?

 

[Speakpigeon]

"I don't know who will win, but whoever wins the election will be a Republican" vs "I know who is going to win, and he happens to be a Republican, but I won't mention his name", very roughly. In linguistics it's called the specific and unspecific reading.

Further, in the original argument, there is no possibility of two different readings for the two occurrences of "it's not Reagan who wins".

Sure there is. "It" is a pronoun, pronouns don't have intrinsic reference but are assigned a referent from context. The context offers "the Republican who wins" in the premise, but in the conclusion, ether thay context is lacking, we're tricked to interpret it as an unrestricted "the winner".

Tricked? How so? What is there in this argument that would trick us that wouldn't trick us in another argument?

In both the first premise and the conclusion, the winner is obviously the winner of the election, not the Republican who wins.

Your comments apply to all arguments and therefore explain nothing specific to this argument.

There is nothing equivocal or ambiguous in the argument in itself. Any differences in the way we read identical sentences at different places in the argument comes from us, not from the argument.

We can make similar arguments without being "tricked". For example:

p1 If we consider the ordered set of even numbers, then if 0 is excluded, then 2 comes first;
p2 We consider the ordered set of even numbers;
C If 0 is excluded, then 2 comes first.

But we can still misread the conclusion as saying 2 comes before 1. If so, our fault.

Where's your unambiguous paraphrase?

Not anything difficult, yes?

p1 If a Republican wins, then if Reagan doesn’t win, then Anderson wins;
p2 A Republican wins;
C If Reagan doesn’t win, then Anderson wins.

Put into the context of the election poll and our memories and beliefs about Reagan, Anderson and Carter, we can still be "tricked". Tricked by what? There is no "it".
EB

 

[fromderinside]

There is no "it".
EB

As paraphrased from Reagan's successor's successor, Clinton, famously sayeth: "The problems is what is the meaning of what 'is' is re 'it'."

 

[Jokodo]

Tricked? How so? What is there in this argument that would trick us that wouldn't trick us in another argument?

In both the first premise and the conclusion, the winner is obviously the winner of the election, not the Republican who wins.

Your comments apply to all arguments and therefore explain nothing specific to this argument.

There is nothing equivocal or ambiguous in the argument in itself. Any differences in the way we read identical sentences at different places in the argument comes from us, not from the argument.

We can make similar arguments without being "tricked". For example:

p1 If we consider the ordered set of even numbers, then if 0 is excluded, then 2 comes first;
p2 We consider the ordered set of even numbers;
C If 0 is excluded, then 2 comes first.

But we can still misread the conclusion as saying 2 comes before 1. If so, our fault.

Where's your unambiguous paraphrase?

Not anything difficult, yes?

p1 If a Republican wins, then if Reagan doesn’t win, then Anderson wins;
p2 A Republican wins;
C If Reagan doesn’t win, then Anderson wins.

Put into the context of the election poll and our memories and beliefs about Reagan, Anderson and Carter, we can still be "tricked". Tricked by what? There is no "it".
EB

Are we though? Compared to the original example, I find it much more natural to read this formulation as "given a Republican winner, if Reagan doesn't win, Anderson has to", thus a valid conclusion.