Implication
2024 Sep 18
See all posts
In mathematical logic, implication is a logical operation typically denoted as \(P \Rightarrow Q\), which can be read as "if \(P\), then \(Q\)". It's a fundamental concept that forms the basis of many logical arguments and proofs. However, its definition often leads to confusion, particularly when the hypothesis is false.
Why is that dreaded implication true when the hypothesis is false?
Common explanations include:
- "Just remember the truth table and use it"
- "It's just a definition"
- "It's a vacuous truth"
- "Because I haven't broken a promise"
Somehow, none of these answers are satisfactory. Whoever came up with implication must have had a good reason for it. What is it?
I went through a thought experiment to understand this. To my pleasant surprise, I have also stumbled upon an article that went through a very similar thought experiment.
Let's begin.
Imagine you are a logician coming up with connectives. You are devising a formal logic system that is consistent and complete (at least until Gödel drops a bombshell on you). We already have negation, disjunction and conjunction, but we keep exploring what else we can do. So we want to come up with a new and unique connective that deals with two statements that may be related (but not necessarily). We want to come up with a connective that might tell us something about the conclusion if we know some information about the hypothesis (and they may very well be unrelated).
Let hypothesis be denoted by \(P\) and conclusion be denoted by \(Q\). Let's also denote the connective by \(\implies\) and call it "implication". We start thinking about what the truth table of \(P \implies Q\) might look like.
The first two rows are easy to determine. When \(P\) is true and \(Q\) is true, the implication is clearly true - if the hypothesis is true and the conclusion is true, the implication holds. When \(P\) is true and \(Q\) is false, the implication must be false - if the hypothesis is true but the conclusion is false, the implication doesn't hold.
But what about the last two? In a binary (true or false) system we have to pick one of the two. false feels "intuitive". What happens if we pick false for both?
Well, this is just conjunction:
| T |
T |
T |
T |
| T |
F |
F |
F |
| F |
T |
F? |
F |
| F |
F |
F? |
F |
We haven't produced any new concept here, so those two rows certainly aren't both false. Therefore, at least one of the two rows must resolve to true. Which one? Let's start with the third row.
| T |
T |
T |
| T |
F |
F |
| F |
T |
T? |
| F |
F |
F? |
What does this mean? It means that holistically our connective says: "\(Q\) whatever is \(P\)" or equivalently, quite literally, just "\(Q\)", and that's hardly useful. Again, this isn't the right configuration. Alright, perhaps we should pick true for the last row.
| T |
T |
T |
| T |
F |
F |
| F |
T |
F? |
| F |
F |
T? |
Intuitively, the last row feels wrong, but we have arrived at exactly this configuration through a rigorous process of thought and after exhausting all other options, so we can't throw it away based on how it "feels". There is something odd however with when \(P\) is false, \(Q\) is true and implication is false. We have a hypothesis that is false, conclusion that is true, but somehow our implication is still false. If our hypothesis \(P\) is "I exercise", and conclusion \(Q\) is "I lose weight" and it so happens that I have lost weight but did not exercise (I stopped chugging soda every day) our implication "I exercise => I lose weight" is false, despite me losing weight. This is wrong. We arrive at the only final configuration that must be the one, and as we know: it is. The implication table you all know and love.
paradoxes
Implication isn't without some peculiarities. Two classes of paradoxes arise from the definition of implication. Note, however, that paradoxes do not imply inconsistency (when you can prove both the statement and its negation). They are just unintuitive.
- Paradoxes arising when the hypothesis is false.
- Whenever the hypothesis is false, the whole conditional is true, regardless of the conclusion.
- This leads to counterintuitive results like "If moon is made of cheese, then life exists on other planets" being considered true.
- Paradoxes arising when the conclusion is true.
- Whenever the conclusion is true, the conditional is true, regardless of the hypothesis.
- This leads to counterintuitive results like "If life exists on other planets, then life exists on earth" being considered true.
These paradoxes stem from the truth table definition of material implication, where the conditional is only false when the hypothesis is true and the conclusion is false. In all other cases, including when the hypothesis is false or the conclusion is true, the conditional is considered true.
These are called "paradoxes" not because they lead to logical contradictions, but because they violate our intuitive understanding of "if-then" statements in everyday language. In natural language, we often assume a causal or relevant connection between the "if" part and the "then" part, which isn't necessarily the case in mathematical logic. This disconnect between formal logic and intuitive reasoning is what makes these situations feel paradoxical, even though they're logically consistent within the framework of material implication.
These paradoxes arise from trying to use material implication to capture all uses of "if...then" statements in natural language, when it is just one specific type of implication that does not always align with colloquial usage (if you are interested, try researching "non-classical logic").
exercises
Have a go at each before unfolding the answer underneath it.
Naming the rejects:
We arrived at implication by eliminating the three other ways to fill in the last two rows. Each of those rejects is a connective we already had. Name all three.
show answer
- F, F gives \(P \land Q\) – conjunction, as the post shows.
- T, F gives a table that is true exactly when \(Q\) is true, so the connective is just \(Q\); it ignores \(P\) entirely.
- F, T gives T, F, F, T – true exactly when \(P\) and \(Q\) agree. That's the biconditional \(P \Leftrightarrow Q\).
That last one is worth dwelling on, because it is the reject that comes closest to being useful. The biconditional is a perfectly good connective, but it is symmetric: swapping \(P\) and \(Q\) leaves it unchanged. We were looking for something that says one thing follows from another, and "follows from" has a direction. A symmetric connective can never express it, so the biconditional disqualifies itself for a structural reason, not merely an intuitive one.
The only way to fail:
Show that \(\lnot(P \Rightarrow Q)\) is equivalent to \(P \land \lnot Q\).
show answer
\(P \Rightarrow Q\) is false in exactly one row of its table – the one where \(P\) is true and \(Q\) is false. Its negation is therefore true in exactly that row and false in the other three, which is precisely the table of \(P \land \lnot Q\).
This is the most practical fact about implication: to refute "if \(P\) then \(Q\)" you must produce a case where \(P\) holds and \(Q\) fails. Nothing else will do. It also explains the false-hypothesis rows without any hand-waving about vacuous truth – when \(P\) is false you simply cannot build the one counterexample that would make the implication false, so it stands.
Implication out of older parts:
Show that \(P \Rightarrow Q\) is equivalent to \(\lnot P \lor Q\).
show answer
| T |
T |
F |
T |
T |
| T |
F |
F |
F |
F |
| F |
T |
T |
T |
T |
| F |
F |
T |
T |
T |
The last two columns match, so the two are the same connective.
There is a sting in this. The post set out to invent a new connective, and it turns out implication was expressible in negation and disjunction the whole time – so strictly speaking we added no new expressive power. What we gained was a notation matching how we actually reason, which is why implication earns its own symbol despite being redundant. (Read \(\lnot P \lor Q\) aloud as "either \(P\) fails, or \(Q\) holds" and the false-hypothesis rows stop feeling strange.)
Vacuous truth, on purpose:
I claim: "every unicorn in my garden is purple." I have no unicorns. Is my claim true, false, or meaningless? What would you have to show me to refute it?
show answer
True. The claim says: for every \(x\), if \(x\) is a unicorn in my garden, then \(x\) is purple. Each instance has a false hypothesis, so each is true, so the whole thing is true.
To refute it you would need exactly what exercise 2 identified – a case where the hypothesis holds and the conclusion fails, i.e. a unicorn in my garden that isn't purple. You cannot produce one, so the claim stands unrefuted.
Note that "every unicorn in my garden is green" is true for the same reason. Both a statement and its apparent opposite come out true, which feels alarming but isn't a contradiction: the negation of "all are purple" is "some unicorn of mine is not purple", and that one is false. Nothing inconsistent – the claims just say less than they appear to.
Converse, inverse, contrapositive:
Take "if it rains, the ground gets wet", i.e. \(R \Rightarrow W\). Write down the converse, the inverse and the contrapositive, and say which are equivalent to the original.
show answer
- Converse: \(W \Rightarrow R\) – "if the ground is wet, it rained." Not equivalent.
- Inverse: \(\lnot R \Rightarrow \lnot W\) – "if it doesn't rain, the ground doesn't get wet." Not equivalent.
- Contrapositive: \(\lnot W \Rightarrow \lnot R\) – "if the ground isn't wet, it didn't rain." Equivalent.
The contrapositive is equivalent because both statements are false in exactly the same single case, rain with dry ground. The example makes the failures concrete: a sprinkler leaves the ground wet without rain, which refutes both the converse and the inverse while leaving the original untouched.
Two things worth keeping. The converse and the inverse are contrapositives of each other, so they are equivalent to each other while both differ from the original. And quietly swapping a statement for its converse is the single most common error in informal argument – it is the same move as reading "sufficient" and concluding "necessary", which is the subject of the companion post.
conclusion
Understanding implication is crucial in logic and mathematics. It forms the basis of many logical arguments, proofs, and deductions. While its definition might seem counterintuitive at first, especially when the hypothesis is false, we've seen that it arises naturally when we try to create a logical operator that relates two statements in a meaningful way.
The seeming paradoxes of implication highlight the difference between formal logical systems and our intuitive reasoning. By understanding these nuances, we can use implication more effectively in formal logic while being aware of its limitations when applied to natural language.
Remember, in the realm of mathematical logic, implication is a precisely defined concept that behaves in ways that might not always align with our everyday understanding of "if-then" statements. This precision is what makes it so powerful in formal reasoning, even if it sometimes leads to results that feel paradoxical.
references
implication wikipedia article
paradoxes of material implication
material conditional is permissible
an extremely good resource on material conditional
Implication
2024 Sep 18 See all postsIn mathematical logic, implication is a logical operation typically denoted as \(P \Rightarrow Q\), which can be read as "if \(P\), then \(Q\)". It's a fundamental concept that forms the basis of many logical arguments and proofs. However, its definition often leads to confusion, particularly when the hypothesis is false.
Why is that dreaded implication true when the hypothesis is false?
Common explanations include:
Somehow, none of these answers are satisfactory. Whoever came up with implication must have had a good reason for it. What is it?
I went through a thought experiment to understand this. To my pleasant surprise, I have also stumbled upon an article that went through a very similar thought experiment.
Let's begin.
Imagine you are a logician coming up with connectives. You are devising a formal logic system that is consistent and complete (at least until Gödel drops a bombshell on you). We already have negation, disjunction and conjunction, but we keep exploring what else we can do. So we want to come up with a new and unique connective that deals with two statements that may be related (but not necessarily). We want to come up with a connective that might tell us something about the conclusion if we know some information about the hypothesis (and they may very well be unrelated).
Let hypothesis be denoted by \(P\) and conclusion be denoted by \(Q\). Let's also denote the connective by \(\implies\) and call it "implication". We start thinking about what the truth table of \(P \implies Q\) might look like.
The first two rows are easy to determine. When \(P\) is true and \(Q\) is true, the implication is clearly true - if the hypothesis is true and the conclusion is true, the implication holds. When \(P\) is true and \(Q\) is false, the implication must be false - if the hypothesis is true but the conclusion is false, the implication doesn't hold.
But what about the last two? In a binary (true or false) system we have to pick one of the two. false feels "intuitive". What happens if we pick false for both?
Well, this is just conjunction:
We haven't produced any new concept here, so those two rows certainly aren't both false. Therefore, at least one of the two rows must resolve to true. Which one? Let's start with the third row.
What does this mean? It means that holistically our connective says: "\(Q\) whatever is \(P\)" or equivalently, quite literally, just "\(Q\)", and that's hardly useful. Again, this isn't the right configuration. Alright, perhaps we should pick true for the last row.
Intuitively, the last row feels wrong, but we have arrived at exactly this configuration through a rigorous process of thought and after exhausting all other options, so we can't throw it away based on how it "feels". There is something odd however with when \(P\) is false, \(Q\) is true and implication is false. We have a hypothesis that is false, conclusion that is true, but somehow our implication is still false. If our hypothesis \(P\) is "I exercise", and conclusion \(Q\) is "I lose weight" and it so happens that I have lost weight but did not exercise (I stopped chugging soda every day) our implication "I exercise => I lose weight" is false, despite me losing weight. This is wrong. We arrive at the only final configuration that must be the one, and as we know: it is. The implication table you all know and love.
paradoxes
Implication isn't without some peculiarities. Two classes of paradoxes arise from the definition of implication. Note, however, that paradoxes do not imply inconsistency (when you can prove both the statement and its negation). They are just unintuitive.
These paradoxes stem from the truth table definition of material implication, where the conditional is only false when the hypothesis is true and the conclusion is false. In all other cases, including when the hypothesis is false or the conclusion is true, the conditional is considered true.
These are called "paradoxes" not because they lead to logical contradictions, but because they violate our intuitive understanding of "if-then" statements in everyday language. In natural language, we often assume a causal or relevant connection between the "if" part and the "then" part, which isn't necessarily the case in mathematical logic. This disconnect between formal logic and intuitive reasoning is what makes these situations feel paradoxical, even though they're logically consistent within the framework of material implication.
These paradoxes arise from trying to use material implication to capture all uses of "if...then" statements in natural language, when it is just one specific type of implication that does not always align with colloquial usage (if you are interested, try researching "non-classical logic").
exercises
Have a go at each before unfolding the answer underneath it.
Naming the rejects:
We arrived at implication by eliminating the three other ways to fill in the last two rows. Each of those rejects is a connective we already had. Name all three.
show answer
That last one is worth dwelling on, because it is the reject that comes closest to being useful. The biconditional is a perfectly good connective, but it is symmetric: swapping \(P\) and \(Q\) leaves it unchanged. We were looking for something that says one thing follows from another, and "follows from" has a direction. A symmetric connective can never express it, so the biconditional disqualifies itself for a structural reason, not merely an intuitive one.
The only way to fail:
Show that \(\lnot(P \Rightarrow Q)\) is equivalent to \(P \land \lnot Q\).
show answer
\(P \Rightarrow Q\) is false in exactly one row of its table – the one where \(P\) is true and \(Q\) is false. Its negation is therefore true in exactly that row and false in the other three, which is precisely the table of \(P \land \lnot Q\).
This is the most practical fact about implication: to refute "if \(P\) then \(Q\)" you must produce a case where \(P\) holds and \(Q\) fails. Nothing else will do. It also explains the false-hypothesis rows without any hand-waving about vacuous truth – when \(P\) is false you simply cannot build the one counterexample that would make the implication false, so it stands.
Implication out of older parts:
Show that \(P \Rightarrow Q\) is equivalent to \(\lnot P \lor Q\).
show answer
The last two columns match, so the two are the same connective.
There is a sting in this. The post set out to invent a new connective, and it turns out implication was expressible in negation and disjunction the whole time – so strictly speaking we added no new expressive power. What we gained was a notation matching how we actually reason, which is why implication earns its own symbol despite being redundant. (Read \(\lnot P \lor Q\) aloud as "either \(P\) fails, or \(Q\) holds" and the false-hypothesis rows stop feeling strange.)
Vacuous truth, on purpose:
I claim: "every unicorn in my garden is purple." I have no unicorns. Is my claim true, false, or meaningless? What would you have to show me to refute it?
show answer
True. The claim says: for every \(x\), if \(x\) is a unicorn in my garden, then \(x\) is purple. Each instance has a false hypothesis, so each is true, so the whole thing is true.
To refute it you would need exactly what exercise 2 identified – a case where the hypothesis holds and the conclusion fails, i.e. a unicorn in my garden that isn't purple. You cannot produce one, so the claim stands unrefuted.
Note that "every unicorn in my garden is green" is true for the same reason. Both a statement and its apparent opposite come out true, which feels alarming but isn't a contradiction: the negation of "all are purple" is "some unicorn of mine is not purple", and that one is false. Nothing inconsistent – the claims just say less than they appear to.
Converse, inverse, contrapositive:
Take "if it rains, the ground gets wet", i.e. \(R \Rightarrow W\). Write down the converse, the inverse and the contrapositive, and say which are equivalent to the original.
show answer
The contrapositive is equivalent because both statements are false in exactly the same single case, rain with dry ground. The example makes the failures concrete: a sprinkler leaves the ground wet without rain, which refutes both the converse and the inverse while leaving the original untouched.
Two things worth keeping. The converse and the inverse are contrapositives of each other, so they are equivalent to each other while both differ from the original. And quietly swapping a statement for its converse is the single most common error in informal argument – it is the same move as reading "sufficient" and concluding "necessary", which is the subject of the companion post.
conclusion
Understanding implication is crucial in logic and mathematics. It forms the basis of many logical arguments, proofs, and deductions. While its definition might seem counterintuitive at first, especially when the hypothesis is false, we've seen that it arises naturally when we try to create a logical operator that relates two statements in a meaningful way.
The seeming paradoxes of implication highlight the difference between formal logical systems and our intuitive reasoning. By understanding these nuances, we can use implication more effectively in formal logic while being aware of its limitations when applied to natural language.
Remember, in the realm of mathematical logic, implication is a precisely defined concept that behaves in ways that might not always align with our everyday understanding of "if-then" statements. This precision is what makes it so powerful in formal reasoning, even if it sometimes leads to results that feel paradoxical.
references
implication wikipedia article
paradoxes of material implication
material conditional is permissible
an extremely good resource on material conditional