I just started studying logic (high school) anyway...for the truth table of logical implication
If sentence $A$ is true and $B$ is true then $A\implies B$ is true.
does that mean if $A$ and $B$ are both true then there is a way to prove $B$ is true from $A$, always?
the same for if $A$ is false can you get anything either True or false proved from this $A$?
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Look at $A$ and $B$ as something that is either false or true. For example let $A$ be the event that tomorrow is Tuesday and let $B$ be the event that the day after tomorrow is Wednesday.
Look at $$ A\implies B $$
as a promise - if $A$ is true then so is $B$.
In our example, if $A$ is true then indeed so is $B$ and so the implication $A\implies B$ is true.
However, now consider $C$ as the statement that tomorrow is Friday, and I state $$ C\implies B $$
that is - I promise you that if $C$ will happen so will $B$.
Tomorrow is not Friday (at the time of writing), and so $C$ is false, regardless of if $B$ is false or true - my promise was kept.
Now regarding the terminology of proofing $B$ is both $A,B$ are true. Note that statements like $$ \text{My cat walks on four}\implies1+1=2 $$
is true, since both are true, but what would it mean to prove $B$ from $A$ ?
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Maybe it's more clear if we separate the logical operator meaning of implication from its logical statement meaning.
When we use it as a logical operator, we conceive it simply as an entity, that given two logical values (thus true or false), produces a third logical value, using a common defined rule (its truth table).
So it makes perfect sense to say $A \implies B = true$ if $A = B = true$, and we don't concern about what actually proposition $A$ and $B$ means, we care exclusively about their logical values.
It's different when we use implication as a logical statement.
In this case we really say something about the meaning of the propositions involved in our statement. So, while proposition $A = My\;cat\;is\;black$ is true, and proposition $B = I\;am\;hungry$ is true as well, $A \implies B$ is not a valid statement.
Such meaning is linked to set theory and formal logic. Using implication in this context means that you can infer $B$ from $A$, in a way called modus ponens.
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As a logical proposition, the material conditional $A \implies B$ is a very weak one: as you've noticed, it's very easy to satisfy it just by accident. In fact, this happens whenever $A$ is false, or whenever $B$ is true. Thus, merely observing that $A \implies B$, for some specific $A$ and $B$, says very little.
Instead, the usefulness of implication lies in the fact that, precisely because of its weakness, it is often possible to assert $A \implies B$ as a universal statement (either an axiom or a provable theorem) that holds for any valuation of any free variables mentioned in the propositions $A$ and $B$.
For example, consider the statement: $$x > 2 \;\land\; x \text{ is prime} \implies x \text{ is odd}.$$ Merely observing that this statement holds for some $x$ says very little — there are plenty of numbers for which it is trivially true, either because they are odd, or because they are not primes greater than 2. What makes this statement useful is that we can prove that it holds for all $x$ — there isn't a single number which would be greater than 2 and prime, but not odd.
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One way to understand implication is to remember that $A\Rightarrow B$ is equivalent to $\neg A \lor B$. If you understand negation ($\neg$) and disjunction ($\lor$), then you understand implication.
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The problem is understanding that there's a difference between the everyday mathematical meaning of "implies" - "I can prove it (or someone can)" - and the formal logical meaning expressed by the truth table. This may help: In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?.
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If sentence $A$ is true and $B$ is true then $A\implies B$ is true.
Yes. In mathematics and logic as widely applied, $A\implies B$ is equivalent to $\neg[A\land\neg B]$.
So, if $A$ is true and $B$ is true, then $\neg[A\land\neg B]$ and hence $A\implies B$ will also be true.
Does that mean if $A$ and $B$ are both true then there is a way to prove $B$ is true from $A$, always?
Yes. Here is a formal proof using a form of natural deduction:
The same for if $A$ is false can you get anything either True or false proved from this $A$?
Yes. Here is a formal proof:
That the above may seem counter-intuitive may stem from the fact that implication is often confused with temporal notions of causality. The statement, "If it is raining, then it is cloudy" does not mean that rain always causes cloudiness, or that cloudiness always cause rain. It means only that, at a given instant in time (usually the present), its is not the case that it is both raining and not cloudy.
On
The sentence $A\implies B$ is not a sentence about proving anything. It is a compound sentence, but otherwise just like the sentences $A$ and $B$. For the truth table, the compound sentence got to have truth values like $A$ and $B$ has, and it has to depend on these sentences truth values. The values for this compound sentence is the same as for the compound statement $\neg A \vee B$.
On
Implication can be understood as thinking in NECESSARY and SUFFICIENT conditions. A good example is to think about someone who has born in Dallas, Texas.
Proposition P could stand for "Someone who was born in Dallas"
Proposition Q could stand for "Someone who is texan"
Truth table for implication is:
So, in this example is easy to check line by line the validity of truth table above:
Who was born in Dallas is therefore texan; So the first implication is TRUE.
Who was born in Dallas MUST be texan. So, saying that someone who was born in Dallas, in the state of Texas, and is not texan is FALSE;
Saying who wasn't born in Dallas but is texan anyway (take someone who has born in Houston, for example) is a TRUE statement;
At last, who wasn't born in Dallas AND also is NOT texan is also TRUE.
So, thinking in terms of NECESSARY (to be texan is a necessary condition for who was born in Dallas) and SUFFICIENT (to born in Dallas is a sufficient condition to be texan) conditions ease the process to translate implication to natural language.
Other examples of propositions containing necessary and sufficient conditions are:
On
This is the way I understand it (I make no guarantee this is considered a correct interpretation).
the truth value of an implies statement is basically 'innocent until proven guilty', or 'my statement is true so long as the proposition values don't disprove it'.
So with an example, propositions P and Q, with the statement P implies Q. Lets say proposition P is 'is a man' and proposition Q is 'lives in japan', so our implication statement is basically saying 'all men live in japan'.
The truth table:
is basically saying, for any example where P and Q are true, we haven't disproved our statement that P -> Q, and for any example where P is false we haven't disproved our statement, only if we have an example of P being true and Q being false is our statement shown to be false.
So imagine there are only 4 humans in existence, we look at each of them and categorise them in our truth table. By doing this we can work out if our statement that P -> Q is true. We find 2 of the people are men, and that they do live in Japan, OK good, our statement still holds true, the 3rd person is a woman, we don't care if she lives in Japan or not because in either case our statement about men is still holding true.
A woman living in or outside japan has no bearing on the truth of our statement about men living in japan.
The 4th person then will decide if our statement is true or not, if the 4th person is a man living in japan, or a woman, then is it logically true to say that being a man implies that you live in japan.
Why is this useful? well the truth table itself doesn't help us prove the truth of our statement by itself, what it does is clearly set out what conditions are relevant to the truth of our statement - i.e. that we don't care about women, and that only a man not living in japan will falsify our statement.
Of course, you could make a statement like 'all green men live on mars'. This is a true statement because we cannot verify who lives on mars, and we don't have any examples of green men. But this is whats called a "vacuous truth", it has no bearing on reality for us because the propositions are intangible.
I would say that $A$ being true and $B$ being true does not mean you can always prove (deduce) $B$ from $A$.
Here's an example. A: Alice lives in Atlanta. B: Bob lives in Boston. Even if these are both true, there is no (apparent) relationship. So you can't logically deduce $B$ from $A$ even though $A\Rightarrow B$ is true in this case.
I guess this means that if you can logically deduce statement $Q$ from statement $P$, then $P\Rightarrow Q$ is true; but knowing $P\Rightarrow Q$ is true does not guarantee the existence of a deduction of $Q$ from the assumption of $P$.
Logical implication is a defined logical connective, so as long as $P$ and $Q$ have truth values (true or false), so does $P\Rightarrow Q$.