How come these instances of modus tollens seem so weird?
January 23, 2013 8:23 AM Subscribe
Basic logic question. The validity of modus tollens is intuitive when the antecedent and consequent in the first conditional premise aren't negative statements. But whenever the argument begins with something like "If not-p, then q", the whole thing seems less intuitive to me. Can someone explain to my very rusty brain why those instances of modus tollens are still valid?
I don't have any trouble with arguments that assume the generic form of modus tollens:
(1)
If you’re murdered, you’re dead. (If p, then q.)
You’re not dead. (Not q.)
So you’re not murdered. (So not p.)
But this seems weirder, even though it's still modus tollens (correct?):
(2)
If you’re not murdered, then you’re alive. (If not-p, then q.)
But you’re not alive. (Not q.)
So you’re murdered. (So p.)
Or this:
(3)
If you’re not a racist, you like Jim. (If not-p, then q.)
But you don’t like Jim. (Not q.)
So you’re a racist. (So p.)
Any thoughts on why the validity of 2 and 3 isn't as apparent? (In each of them, my brain starts tossing out other ways you could have died, other reasons you might not like Jim...) Would translating them into sentences about necessary and sufficient conditions help?
I don't have any trouble with arguments that assume the generic form of modus tollens:
(1)
If you’re murdered, you’re dead. (If p, then q.)
You’re not dead. (Not q.)
So you’re not murdered. (So not p.)
But this seems weirder, even though it's still modus tollens (correct?):
(2)
If you’re not murdered, then you’re alive. (If not-p, then q.)
But you’re not alive. (Not q.)
So you’re murdered. (So p.)
Or this:
(3)
If you’re not a racist, you like Jim. (If not-p, then q.)
But you don’t like Jim. (Not q.)
So you’re a racist. (So p.)
Any thoughts on why the validity of 2 and 3 isn't as apparent? (In each of them, my brain starts tossing out other ways you could have died, other reasons you might not like Jim...) Would translating them into sentences about necessary and sufficient conditions help?
The problem is that that the initial statements made in #2 and #3 are not statements most of us would see as true.
For instance:
#1 says - If you're not murdered, you're alive.
This is simply not true. Lots of people are not murdered, but are also not alive. (eg- anyone who died of natural causes).
The statement "If you not alive, then you are murdered" *DOES* follow logically from the statement in #1. But since #1 isn't true, the statement that follows from it ends up not being true either.
posted by ManInSuit at 8:31 AM on January 23, 2013 [3 favorites]
For instance:
#1 says - If you're not murdered, you're alive.
This is simply not true. Lots of people are not murdered, but are also not alive. (eg- anyone who died of natural causes).
The statement "If you not alive, then you are murdered" *DOES* follow logically from the statement in #1. But since #1 isn't true, the statement that follows from it ends up not being true either.
posted by ManInSuit at 8:31 AM on January 23, 2013 [3 favorites]
I think you're confusing the English "if" with the logical "if", also denoted "iff". That is to say, "if X then Y" in English often does not mean "if and only if X then Y". As I recall, the "if" used in formal logic is "if and only if", which allows the contrapositive to hold as well.
posted by daveliepmann at 8:34 AM on January 23, 2013
posted by daveliepmann at 8:34 AM on January 23, 2013
As others have pointed out, the premises of 2 and 3 are incorrect, which is why you end up with incorrect conclusions.
posted by dfan at 8:38 AM on January 23, 2013 [1 favorite]
I think you're confusing the English "if" with the logical "if", also denoted "iff". That is to say, "if X then Y" in English often does not mean "if and only if X then Y". As I recall, the "if" used in formal logic is "if and only if", which allows the contrapositive to hold as well.If in formal logic means if, not if and only if. To be precise, "if X then Y" means "in all cases when X is true, Y is true as well", and says nothing about the cases in which X is false.
posted by dfan at 8:38 AM on January 23, 2013 [1 favorite]
I taught this sort of basic formal logic in LSAT preparation classes for a few years; this is a very common difficulty. In addition to the problem noted above that intuitively false statements are much harder to play around with, there is also the problem that negative statements are, all things being equal, harder to play around with than positive statements. My advice to students was to always try to think of hypotheticals that make intuitive sense in the real world. So an easier example of the pattern in your second formula might be:
If you're not moral, then you are evil.
You are not evil.
Therefore, you are moral.
Logically, of course, there's no difference between my example and yours, but our brains don't work in a strictly logical fashion, which is the reason we have to study logic to get good at it.
posted by skewed at 8:46 AM on January 23, 2013
If you're not moral, then you are evil.
You are not evil.
Therefore, you are moral.
Logically, of course, there's no difference between my example and yours, but our brains don't work in a strictly logical fashion, which is the reason we have to study logic to get good at it.
posted by skewed at 8:46 AM on January 23, 2013
No, p iff q would be "if p then q and if q then p". It says that p and q always have the same truth value.
The Boolean conditional operator ("if p, then q") asserts a very weak connection between p and q. ALL it says is that it is not possible for p to be true and q to be false, at the same time. In natural language, it normally suggests a much stronger association, such as causality, between p and q. The Boolean connectives, however, are only concerned with truth functions.
The example (2) is a valid argument. That is: IF the premises are true, then so must be the conclusion. However, the premises are not true, since there are ways to die other than being murdered. There is nothing wrong with the way modus tollens functions within it, however, and you could see that by looking at a similar argument:
If you’re not murdered, then you are named Alice (If not-p, then q.)
But you’re not named Alice. (Not q.)
So you’re murdered. (So p.)
Again, this is a valid argument, but here it is more obvious that the first premise is wrong..
posted by thelonius at 8:47 AM on January 23, 2013
The Boolean conditional operator ("if p, then q") asserts a very weak connection between p and q. ALL it says is that it is not possible for p to be true and q to be false, at the same time. In natural language, it normally suggests a much stronger association, such as causality, between p and q. The Boolean connectives, however, are only concerned with truth functions.
The example (2) is a valid argument. That is: IF the premises are true, then so must be the conclusion. However, the premises are not true, since there are ways to die other than being murdered. There is nothing wrong with the way modus tollens functions within it, however, and you could see that by looking at a similar argument:
If you’re not murdered, then you are named Alice (If not-p, then q.)
But you’re not named Alice. (Not q.)
So you’re murdered. (So p.)
Again, this is a valid argument, but here it is more obvious that the first premise is wrong..
posted by thelonius at 8:47 AM on January 23, 2013
I think you're confusing the English "if" with the logical "if", also denoted "iff". That is to say, "if X then Y" in English often does not mean "if and only if X then Y". As I recall, the "if" used in formal logic is "if and only if", which allows the contrapositive to hold as well.
No, there are two logical ifs: if (->) and iff (<>, if and only if).
The truth values are:
English if is more like logical if but depends greatly on contextual knowledge.
posted by jeather at 8:53 AM on January 23, 2013
No, there are two logical ifs: if (->) and iff (<>, if and only if).
The truth values are:
P Q P->Q P<>Q
T T T T
T F F F
F T T F
F F T TEnglish if is more like logical if but depends greatly on contextual knowledge.
posted by jeather at 8:53 AM on January 23, 2013
Another difficulty is that the propositional variables of Boolean logic have no internal structure at all - they are just atoms of truth or falsity. You need something more, like first order logic, to do formal translations of all the sentences about individual things and their properties that you might wish to do reasoning about.
There are other kinds of things that don't go over into Boolean logic very nicely at all, like subjunctives:
If it rains, I will bring my umbrella.
I will not bring my umbrella.
Therefore, it will not rain.
It's important to be clear that this kind of formal logic is an algebra of truth-bearing atoms and their connectives, and not a complete organon of reasoning.
posted by thelonius at 8:53 AM on January 23, 2013
There are other kinds of things that don't go over into Boolean logic very nicely at all, like subjunctives:
If it rains, I will bring my umbrella.
I will not bring my umbrella.
Therefore, it will not rain.
It's important to be clear that this kind of formal logic is an algebra of truth-bearing atoms and their connectives, and not a complete organon of reasoning.
posted by thelonius at 8:53 AM on January 23, 2013
If you don't have a valid ticket, then you pay extra fee. (If not-p, then q.)
You didn't pay extra fee. (not-q)
So you have a valid ticket. (p)
The causal intuition for me is that 'if' sets an iron tight condition, where event 'not-p' always triggers event q. If event q has not been triggered (not-q), then we can deduce that the trigger has not been activated: 'not-p' has not occurred, so 'p'.
If you don't avoid those landmines you will be missing a leg.
You are not missing a leg.
So you avoided those landmines.
If it helps, the negated statement can temporary be replaced with a positive statement in a way that is easier for your brain, and then turned back.
r = not-p
If r, then q
q
r
hit means don't avoid (r = not-p)
If you hit those landmines you will be missing a leg. (If r, then q)
You are not missing a leg. (not-q)
So you didn't hit those landmines. (not-r)
(So you avoided those landmines) (p)
posted by Free word order! at 8:56 AM on January 23, 2013 [1 favorite]
You didn't pay extra fee. (not-q)
So you have a valid ticket. (p)
The causal intuition for me is that 'if' sets an iron tight condition, where event 'not-p' always triggers event q. If event q has not been triggered (not-q), then we can deduce that the trigger has not been activated: 'not-p' has not occurred, so 'p'.
If you don't avoid those landmines you will be missing a leg.
You are not missing a leg.
So you avoided those landmines.
If it helps, the negated statement can temporary be replaced with a positive statement in a way that is easier for your brain, and then turned back.
r = not-p
If r, then q
q
r
hit means don't avoid (r = not-p)
If you hit those landmines you will be missing a leg. (If r, then q)
You are not missing a leg. (not-q)
So you didn't hit those landmines. (not-r)
(So you avoided those landmines) (p)
posted by Free word order! at 8:56 AM on January 23, 2013 [1 favorite]
Modus tollens is not generally valid in natural language (for example [pdf]), which as a formal system behaves radically different from classical propositional logic in many ways. I wouldn't personally invest much effort into trying to make MT valid for particular examples of natural language conditionals, given this.
posted by advil at 10:39 AM on January 23, 2013 [5 favorites]
posted by advil at 10:39 AM on January 23, 2013 [5 favorites]
I agree with others that you're having difficulty reasoning about examples 2 and 3 because the first premise in each of those examples is false.
It's also difficult to reason about statements of the form "If not X, then Y" because most "not" conditions in real life either have "not" results ("If you do not have a ticket, you will not be admitted to the concert") or they have exceptions ("If you do not pay taxes, you will go to jail." Well, maybe you're not required to pay taxes, or maybe you don't get caught, or maybe you only get a fine, etc.)
Other statements of the form "If not X, then Y," where X and Y describe two mutually exclusive and collectively exhaustive conditions ("If you're not a prokaryote then you're a eukaryote," at least for all cellular organisms) which, while both true and the result valid, may be less than helpful in thinking about this, as they are logically equivalent statements ("If and only if you're not a prokaryote, then you're a eukaryote"), so may not help in thinking about situations where "not X" and "Y" are not equivalent.
Here's the example I came up with:
If you don't have power to your house, your television is silent.
Your television is not silent.
Therefore, you have power to your house.
[If you did have power to your house, that wouldn't be enough information to say whether your television was silent. If your television was silent, that wouldn't be enough information to say whether you had power to your house.]
It might also be helpful to note that "If not X, then Y" is equivalent to "X or Y" (where "or" is taken to be inclusive or, as it is in formal logic, including the possibility that X and Y are both true). This gets rid of the "not" which makes intuitive reasoning more difficult.
E.g., my example above may be rephrased as "You have power to your house or your television is silent (possibly both)."
posted by DevilsAdvocate at 11:27 AM on January 23, 2013
It's also difficult to reason about statements of the form "If not X, then Y" because most "not" conditions in real life either have "not" results ("If you do not have a ticket, you will not be admitted to the concert") or they have exceptions ("If you do not pay taxes, you will go to jail." Well, maybe you're not required to pay taxes, or maybe you don't get caught, or maybe you only get a fine, etc.)
Other statements of the form "If not X, then Y," where X and Y describe two mutually exclusive and collectively exhaustive conditions ("If you're not a prokaryote then you're a eukaryote," at least for all cellular organisms) which, while both true and the result valid, may be less than helpful in thinking about this, as they are logically equivalent statements ("If and only if you're not a prokaryote, then you're a eukaryote"), so may not help in thinking about situations where "not X" and "Y" are not equivalent.
Here's the example I came up with:
If you don't have power to your house, your television is silent.
Your television is not silent.
Therefore, you have power to your house.
[If you did have power to your house, that wouldn't be enough information to say whether your television was silent. If your television was silent, that wouldn't be enough information to say whether you had power to your house.]
It might also be helpful to note that "If not X, then Y" is equivalent to "X or Y" (where "or" is taken to be inclusive or, as it is in formal logic, including the possibility that X and Y are both true). This gets rid of the "not" which makes intuitive reasoning more difficult.
E.g., my example above may be rephrased as "You have power to your house or your television is silent (possibly both)."
posted by DevilsAdvocate at 11:27 AM on January 23, 2013
Any thoughts on why the validity of 2 and 3 isn't as apparent? ... my brain starts tossing out other ways you could have died
One thing you need to have clear is the difference between validity and soundness. An argument is valid just in case its conclusion logically follows from its premises. An argument is sound if it is valid and its premises are true. Your numbers 2 and 3 are valid but, as your brain is pointing out to you, they contain false premises, so they are not sound.
posted by tractorfeed at 11:57 AM on January 23, 2013 [1 favorite]
One thing you need to have clear is the difference between validity and soundness. An argument is valid just in case its conclusion logically follows from its premises. An argument is sound if it is valid and its premises are true. Your numbers 2 and 3 are valid but, as your brain is pointing out to you, they contain false premises, so they are not sound.
posted by tractorfeed at 11:57 AM on January 23, 2013 [1 favorite]
As others are saying, the difficulty is getting a plausible If not-p then q. You need a really clean exclusive disjunction and those are rare without building in a lot of domain-narrowing qualifications.
If a person isn't a daughter, then they're a son.
This person is not a son.
Therefore this person is a daughter.
But basically, you can just build in the domain-narrowing qualifications. Even then, notice this works best with examples in geometry or laws/regulations/rules - because these are sort of artificial languages that are formalizable in the kind of way demanded by deductive logic.
If a [checkers piece in a standard set] isn't red, it's black.
This [checkers piece in a standard set] is not black.
Therefore it's red.
If a [regular polygon] is not a triangle, then it has a number of sides other than 3.
This [regular polygon] does not have a number of sides other than 3.
So it's a triangle.
[For all people in this swimming pool], if a person isn't a member, then they must get a guest pass.
This person [is in this swimming pool and] didn't get a guest pass.
Therefore they're a member.
posted by LobsterMitten at 12:39 PM on January 23, 2013 [1 favorite]
If a person isn't a daughter, then they're a son.
This person is not a son.
Therefore this person is a daughter.
But basically, you can just build in the domain-narrowing qualifications. Even then, notice this works best with examples in geometry or laws/regulations/rules - because these are sort of artificial languages that are formalizable in the kind of way demanded by deductive logic.
If a [checkers piece in a standard set] isn't red, it's black.
This [checkers piece in a standard set] is not black.
Therefore it's red.
If a [regular polygon] is not a triangle, then it has a number of sides other than 3.
This [regular polygon] does not have a number of sides other than 3.
So it's a triangle.
[For all people in this swimming pool], if a person isn't a member, then they must get a guest pass.
This person [is in this swimming pool and] didn't get a guest pass.
Therefore they're a member.
posted by LobsterMitten at 12:39 PM on January 23, 2013 [1 favorite]
Incidentally, this is why natural language tests of inferential ability are so horribly wordy and awkward - the need to build in all those domain-narrowing qualifications.
posted by LobsterMitten at 12:43 PM on January 23, 2013 [1 favorite]
posted by LobsterMitten at 12:43 PM on January 23, 2013 [1 favorite]
Formally, the inferences you are describing require an extra step that you are leaving out.
Modus Tollens has the form:
p -> q
~q
// ~p
So, if you have premisses:
~p -> q
~q
Then you really should conclude ~~p by MT. After that, you can conclude p by double negation elimination (in classical logic).
I don't know if the extra implicit step is enough to throw off your brain. I suppose the extra negations and the extra logical step could be adding enough cognitive load to be noticeable ... maybe?
Or, more humorously, maybe you are implicitly a big fan of intuitionistic logic, in which double negation elimination is invalid. Hence, as an intuitionist, you are balking at the fact that the inferences you describe are actually invalid. ;)
posted by Jonathan Livengood at 6:28 PM on January 24, 2013 [1 favorite]
Modus Tollens has the form:
p -> q
~q
// ~p
So, if you have premisses:
~p -> q
~q
Then you really should conclude ~~p by MT. After that, you can conclude p by double negation elimination (in classical logic).
I don't know if the extra implicit step is enough to throw off your brain. I suppose the extra negations and the extra logical step could be adding enough cognitive load to be noticeable ... maybe?
Or, more humorously, maybe you are implicitly a big fan of intuitionistic logic, in which double negation elimination is invalid. Hence, as an intuitionist, you are balking at the fact that the inferences you describe are actually invalid. ;)
posted by Jonathan Livengood at 6:28 PM on January 24, 2013 [1 favorite]
On review, I neglected to explain why in classical logic the inferences you mention are valid, which is what you initially asked. The usual way to prove validity is with truth tables. An argument is valid just in case there is no row of the truth table where all the premisses are true and the conclusion is false. The argument we are looking at has the form:
~p -> q
~q
// p
So, we construct the truth table below, where 0 = False and 1 = True:
We then note that there is only one row in the table (one "world") in which both premisses ~p -> q and ~q are true: the second row. In that row, the conclusion is also true. Hence, the argument is (classically) valid.
Another way to see that the argument works "intuitively" is to use contraposition. The conditional p -> q is truth-functionally equivalent to its contrapositive, ~q -> ~p. So, take the premisses of your initial argument:
~p -> q
~q
Replace the first premiss with its contrapositive to get the premisses:
~q -> ~~p
~q
By Modus Ponens, we get ~~p. And then by double negation elimination, we get p.
posted by Jonathan Livengood at 2:47 PM on January 25, 2013
~p -> q
~q
// p
So, we construct the truth table below, where 0 = False and 1 = True:
p q || ~p -> q ~q p 1 1 || 1 0 1 1 0 || 1 1 1 0 1 || 1 0 0 0 0 || 0 1 0The values for p and q on the left of the || are specifying what the atomic sentences look like at all the "possible worlds." The values on the right of the || are the values that the (possibly compound) sentences have at each possible world, e.g. in the world where p is false and q is true, the compound sentence ~p -> q is true.
We then note that there is only one row in the table (one "world") in which both premisses ~p -> q and ~q are true: the second row. In that row, the conclusion is also true. Hence, the argument is (classically) valid.
Another way to see that the argument works "intuitively" is to use contraposition. The conditional p -> q is truth-functionally equivalent to its contrapositive, ~q -> ~p. So, take the premisses of your initial argument:
~p -> q
~q
Replace the first premiss with its contrapositive to get the premisses:
~q -> ~~p
~q
By Modus Ponens, we get ~~p. And then by double negation elimination, we get p.
posted by Jonathan Livengood at 2:47 PM on January 25, 2013
This thread is closed to new comments.
Try something that's true.
If you're not a prokaryote then you're a eukaryote.
You're not a eukaryote.
So you're a prokaryote.
posted by jeather at 8:30 AM on January 23, 2013 [3 favorites]