Simple statements in logic? ( How do we know that " A implies B" is the same as such and such?)

Question

When do we know $A\implies B$ is same as $A\wedge B$ or $\neg B$??

When is it true that $A\implies B$ is same as $A\wedge B\implies 1==1$?

Answer 1

Well, $A\Rightarrow B$ is equivalent to $\neg A \vee B$ and to $\neg(\neg(\neg A\vee B))$ which is $\neg(A\wedge\neg B)$.

$A\Rightarrow B$ is the same as $A\wedge B$ if $A=B=1$.

Answer 2

Be careful by what you mean by 'same as'.

When it comes to logic statements, there are at least three different ways to interpret that:

1. We can say that statements $\varphi$ and $\psi$ are the 'same' if they are syntactically identical. So, for example, $\neg \neg A$ is the 'same' as $\neg \neg A$, and $\neg \neg A$ is 'different' from $A$

2. We can say that statements $\varphi$ and $\psi$ are the 'same' if they have the same truth-conditions. We could say that they 'say' or 'claim' the 'same thing'. For this, we use the term logical equivalence. So, for example, $\neg \neg A$ is logically equivalent to $A$, but $A \to B$ and $A \land B$ are not.

3. We can say that two statements are the 'same' in that they have the same truth-value .. when evaluated according to some evaluation function. So, for example, if $A=1$ and $B=1$, then $A \to B$ and $A \land B$ both evaluate to $1$.

I think you are using 'the same' in the third sense. That is, I think you are asking: under what evlautations do $A \to B$ and $A \land B$ have the same truth-value? Well, that would be if $A=B=1$, or if $A=1$ and $B=0$

Answer 3

In the same way, how do we know that : a - b is the same as a+(-b)?

We know this simply because we define the difference of two numbers as the sum of the first number and of the opposite ( i.e. additive inverse) of the second.

In other words, we know this because we decide it to mean this.

We can tell the " story of logic" in the following way.

The " inventors" or logic first had basic operators, say : NOT, AND, OR.

They decided to define more complex operators with these basics.

With AND and NOT, one can define the NAND operator, symbol : " | ".

So we will say that the " NAND " relation holds between 2 sentences just in case : it is not the cas ethat these 2 sentences are both true.

So we will dedice the following : "A NAND B" is ( by definition) "NOT ( A AND B)".

The inventors of logic also wanted an operator that could express the idea of " implication".

Saying that a sentence A implies a sentence B means that you do not have A true and B false.

A implies B iff : (A NAND ~B) is true. Or if you prefer, iff : (A AND NOT-B) is false.

But this is a bit long to write and they said to themselves that having a new symbol would be fine.

So they decided that a new operator ( able to capture the idea of implication) , namely " -->" would be defined as follows:

"(A-->B) is by definition an abbreviation for : ~ (A AND NOT-B)."

Now look at the truth table of ~ (A AND NOT B), in other words to the truth table of its abbreviation (A --> B).

A_B____ ~ (A AND NOT-B)

T_T_________T

T_F_________F

F_T_________T

F_F_________T

You see that the formula is true in only 3 cases , which can be reduced to only 2 :

either when A and B are both true ( line 1) OR when A is false ( lines 3 and 4) .

By this we know that

" A-->B is true" is equivalent to " we habe (A&B) OR ~A".

Using some logic laws ( DeMorgan and X & True = X ) this can be shown equivalent to :

B OR ~A .

And, to tell the truth, it is this last equivalence that was chosen as the official definition of the " if then" operator :

By definition : (A --> B) is an abbreviation for (~A OR B)

Answer 4

By drawing a truth table of (A--> B) and of ( A&B) OR ~B , one sees that the first " means" the second , that is, has the same truth truth value as the second, just in case A and B have the same truth value, that is just in case :

• A and B are both true

OR

• A and B are both false.

Which means that " A<->B" is true.

Indeed, a truth table shows that :

((A-->B) <-> ((A&B)OR ~B)) <-> (A <->B)

is a tautology.