Implication
2024 Sep 18
See all posts
Implication
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.
| \(P\) |
\(Q\) |
\(P \Rightarrow Q\) |
| T |
T |
T |
| T |
F |
F |
| F |
T |
? |
| F |
F |
? |
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:
| \(P\) |
\(Q\) |
\(P \Rightarrow Q\) |
\(P \land Q\) |
| 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.
| \(P\) |
\(Q\) |
\(P \Rightarrow Q\) |
| 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.
| \(P\) |
\(Q\) |
\(P \Rightarrow Q\) |
| 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.
| \(P\) |
\(Q\) |
\(P \Rightarrow Q\) |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
F |
T |
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").
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").
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