Motivating the Implication Operator

first-order-logiclogic

I am self-studying from Number Systems and the Foundations of Analysis By Eliott Mendelson.

The book has just defined the truth table for the implication operator and in the following problems seeks to motivate why everything other than T$\Rightarrow$F is evaluated as T. The question begins as follows:

"Verify the following justification for our truth table for A$\Rightarrow$B. It is obvious that $(C\land D)\Rightarrow C$ must always be true. Hence: (i) When $C$ and $D$ are both true, we have $(T\Rightarrow T)=T$. This gives the first row of the table for $\Rightarrow$. (ii) When $C$ is false and $D$ is true, we have $(F\Rightarrow T)=T$, which is the second line of the table. (iii) When $D$ is false, we have $(F\Rightarrow F)=T$, which is the fourth line of the table."

$$\begin{array}{ccc}
A & B & A\Rightarrow B \\ \hline
T & T & T\\
F & T & T\\
T & F & F\\
F & F & T
\end{array}$$

My question is what about the assertion $(C\land D)\Rightarrow C$ is obvious? You can of course use the truth table to see that this is true. But if the question is trying to motivate the truth table, I'm left to assume that I can't use the truth table. So it must be true on the basis of an intuitive understanding of implication, or something else.

Can anyone shed light on why this would be an "obvious" assertion?

Best Answer

We do not take implication as some form of causality, but as a form of logical consequence, i.e., that the truth of a first thing forces us to accept that a second thing is true. The essence of "and" is: that one thing and another are true necessitates that each of these two things is true. So with $C\land D$ representing the statement that $C$ and $D$ are true forces us to accept that $C$ is true (it also forces us to accept that $D$ is true). If you do not agree, you may have an unusual notion of "and" (or of implication). And this fact, i.e., that the truth of $C\land D$ forces us to accept that $C$ is true. The accepting-force behind this argument is always correct, no matter whether $C\land D$ itself is actually true. This now precisely what $(C\land D)\Rightarrow C$ symbolizes.

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