Tjyylli

There are multiple good answers here, including references to modus tollens and the contrapositive (both of which are correct).

What helps me understand this concept is a more intuitive/layman's perspective.

Say that if it's raining, you ALWAYS bring an umbrella outside. (Assume that you have a perfect memory). Therefore we have:

Raining -> Umbrella

However, if you have an umbrella, does it mean that it's raining? Well, no. You might be bringing an umbrella for a trip, using it as a parasol in the scorching summer sun, or even simply returning it to a friend.

But if you don't have an umbrella, is it raining? No, and this is supported by modus tollens/contrapositive. That is to say:

Not(Umbrella) -> Not(Raining)

And that makes sense to us. If it were raining, you would've brought your umbrella. We said that you ALWAYS bring an umbrella if it's raining. So if you don't have it, it can't possibly be raining.

Let's apply it to your example:

"In order to have internet (B), the router must be on (A). So, if
there is no internet (not-B), that means the router is not on (not-A)"

This is actually not formulated properly; A and B have been reversed. This is probably why it's confusing. (Intuitively, it's in line with what you've been saying: if we don't have Internet, it's not necessarily because of the router.)

So let's word it differently:

If we have an Internet connection (A), the router must be on (B).

InternetConnected -> RouterOn

A -> B

But if the router isn't on, then we can't possibly connect to the Internet. (How would we even do that?) We know this is the case, because we said that if we're connected to the Internet, the router MUST be on.

Not(RouterOn) -> Not(InternetConnected)

Not(B) -> Not(A)

Hope this helps! :)

‘If not B then not A’ is the contrapositive of ‘If A then B’ and is logically valid.

By saying ‘if A then B’, the author is saying that whenever A happens, B will definitely happen. Hence, if B does not happen, it is clear that A did not happen (if not B then not A).

You are correct: The statement does not show a sufficient condition!

Consider this statement: "If (A) it rains, then (B) the street will be wet."

Assuming the street is always wet after it rained (because it is not covered by a roof or something), the contraposition would hold true!

Contraposition: "If the street is not wet, it did not rain"

I think what is giving you troubles is that the street in this example can be wet without rain (someone sprayed water onto it), therefore the street is not always wet just because of rain.
That's called an inverse and differentiation can sometimes be hard.

No fallacy

Router necessary for internet

Can be restated

Internet sufficient for router

Let's restate the second more elaborately

Internet (found to be) working is sufficient (evidence)

that the router (has to be) working

In addition to contrapositive suggested by @virmalor you may like to see also modus tollens

"In order to have internet, the router must be on" corresponds to the implication "if there is internet, the router is on." The contrapositive is "if the router is not on, there is no internet."

The router being on is a necessary condition for internet, while having internet is a sufficient condition for the router being on. (It can be tricky translating statements to implications, especially when there are modals like "must"! The translation reduces the statement to its barest essence with a material implication, and the translation is usually lossy.)

As others have said, there is no fallacy in a contrapositive argument, i.e. "if A then B. Not B, therefore not A". But your example is faulty:

Here is an example I could think of, in natural language: "In order to have internet (B), the router must be on (A). So, if there is no internet (not-B), that means the router is not on (not-A)" (Which is not true because there may be a problem with the providers, for example.)

What you have written is "if B then A. Not B, therefore not A"--which is of course invalid reasoning. A better way of putting an example of the a contrapositive into natural language would be "If we have an internet [connection], then the router is on. The router is off [not on], therefore we do not have an internet [connection]." Which is a valid argument of the contrapositive form.

Your confusion might stem from the fact that logical implications don't always correspond intuitively to cause and effect in the real world. I'll get back to your example later; for now, let's focus on a more intuitive implication:

If [a: there is fire], then [b: there is smoke].

This is a natural process in which the cause (fire) always produces the effect (smoke), and that's a situation that can be described with an implication. (In the real world, there are substances that burn without smoke, so let's constrain ourselves to wood fires.) In this situation, you probably agree that the contrapositive is true as well:

If [not(b): there is no smoke], then [not(a): there is no fire].

This makes intuitive sense - we're not observing the effect that is always produced by the cause, so the cause can't be occurring.

We can formally prove why the contrapositive is equivalent to the original statement. Here is the truth table that defines implications (T for true and F for false):

| a | b | a -> b |

+---+---+--------+

| F | F | T      |

| F | T | T      |

| T | F | F      |

| T | T | T      |

So an implication is only false if the left hand side is true and the right hand side is false. Let's interpret this via the example. Our implication is intended to be a description of the real world, so all the true rows should correspond to situations that can happen in the real world, and all the false rows should correspond to situations that can not:

1. It can happen that there is no fire and no smoke.


2. It can happen that there is no fire, but there is smoke (which might be produced by something else than fire).


3. It can not happen that there is fire but no smoke.


4. It can happen that there is both fire and smoke.

Now, let's add the contrapositive to the truth table:

| a | b | a -> b | not(b) | not(a) | not(b) -> not(a) |

+---+---+--------+--------+--------+------------------+

| F | F | T      | T      | T      | T                |

| F | T | T      | F      | T      | T                |

| T | F | F      | T      | F      | F                |

| T | T | T      | F      | F      | T                |

The not(b) -> not(a) is identical to the a -> b column, so the two statements are equivalent.

In your example, you seem to think that the router is the cause that produces the effect of internet. However, unlike in the fire and smoke example, internet isn't a guaranteed effect of the router alone, and therefore there is not an implication from router to internet. Perhaps surprisingly, there is an implication, but it goes the other way. The router is just one of several necessary ingredients for having internet, and in an implication, the necessary part goes on the right hand side - so even though the router is one of the technical "causes" of the internet, the implication looks like this:

If [a: you have internet], then [b: the router is on].

This unintuitive order is easier to understand if you stop thinking about an implication as the left hand side being a physical cause of the right hand side, and just think of it as the statement "it is not the case that the left hand side is true while the right hand side is false". Let's confirm that this is a correct description of the real world (the only thing that shouldn't happen is that the left hand side is true and the right hand side is false):

1. It can happen that you don't have internet and the router is off.


2. It can happen that you don't have internet, but the router is on (so you've got a tech problem somewhere).


3. It can not happen that you've got internet but the router is off.


4. It can happen that you've got internet and the router is on.

In this example, the contrapositive is actually the more intuitive one:

If [not(b): the router is off], then [not(a): you don't have internet].

It's not a fallacy, but it's only guaranteed to hold in Aristotelian logic, where we have the law of the excluded middle, which states that every proposition is either true or false.

Example: Anyone who is a Christian believes that Jesus is their savior.

A is the proposition that my dog is a Christian.

B is the proposition that my dog has a belief in Jesus as her savior.

Now B is false, because my dog has no belief in Jesus as her savior. If you're working in Aristotelian logic, then it's a perfectly valid logical conclusion to say that my dog is not a Christian: if B is false, then A must be false.

But it's at least as reasonable to apply non-Aristotelian logic here. We can say that Christian and non-Christian are silly categories to apply to a dog. Then we say that neither A nor not-A is true. The truth-value of A is undefined. The fact that B is false tells us that A is not true, but it doesn't tell us that not-A is true.

Most mathematicians spend most or all of their time working within Aristotelian logic, but that doesn't mean that non-Aristotelian logic is invalid or not accepted by the mathematical community. Similarly, some people play basketball, some play tennis, and some play both. There is no controversy over which set of rules is the right set of rules, and everyone understands implicitly which set of rules is agreed upon when they're playing.

In most cases in everyday life, Aristotelian logic just isn't a good model of how we reason about the world. If a friend asserts that jazz is more fun than pop music, I would probably answer, "Well, yes and no..."

Your translation from natural English to formal logic is where you go wrong.

"If-A then B" precisely means that A is a sufficient condition for B. - Ie if we have case A we can know that we also have case B.

"In order to have internet (B), the router must be on (A)." describes a necessary condition. This can be written as "Only if A then B" or "If B then A". The contrapositive would be "If Not-A then Not-B"; Ie. "If the router is not on then we do not have internet."