Are Piecewise Definitions Conjunctions or Disjunctions?

Question

I'm wondering how to look at piecewise definitions from a logical point of view. For example, should $\vert a \vert =$ \begin{cases} a & \text{ If $a>0$} \\ 0 & \text{ If $a=0$} \\ -a & \text{ If $a<0$} \end{cases}

be viewed as a conjunction like$$\,$$ $$(a>0 \implies \vert a \vert = a)\,\land (a=0 \implies \vert a \vert = 0)\,\land(a<0 \implies \vert a \vert = -a)$$$$\,$$ or a disjunction like$$\,$$ $$(a>0 \implies \vert a \vert = a)\,\lor (a=0 \implies \vert a \vert = 0)\,\lor(a<0 \implies \vert a \vert = -a)$$$$\,$$

I'm not sure how to think about this in the context of definitions, or in general now for that matter. I appreciate any help. Thanks.

Answer 1

You have to view it as a conjunction, since all three claims ("the absolute value of a positive is itself", "the absolute value of 0 is 0", "the absolute value of a negative is its negation") are being forced to be true.

Using disjunctions would mean that either the absolute value of a positive is itself or the absolute value of 0 is 0 or the absolute value of a negative is its negation. This is kind of silly because it wouldn't actually define the function on any particular range. It would seem like $f(x)=x^3$ is a perfectly good absolute value function, just because it satisfies the $f(0)=0$ part.

Answer 2

It should be a conjunction. You are probably getting confused because there is a disjunction in terms of conditions. That is, we know that $a>0 \lor a=0 \lor a <0$. However, in terms of the conditionals, you need to have a conjunction.

Relatedly, note the following equivalence:

$$(A \lor B) \rightarrow C \Leftrightarrow (A \rightarrow C) \land (B \rightarrow C)$$

See how the disjunction of conditions becomes a conjunction of conditionals?

Indeed, a classic sentence students are asked to symbolize is 'fruits and vegetables are wholesome', for which they often use $(F \land V) \rightarrow W$, but it should of course be $(F \rightarrow W) \land (A \rightarrow W)$ ... so you see that our language really isn't helping us to think clearly about these matters!

Another example is when cognitive psychologist Philip Johnson-Laird gave his subjects the following task: 'If there is a king in the hand, then there is an ace in the hand, or else if there is not a king in the hand, then there is an ace in the hand. What can be inferred?'. Most subjects answered that it can be inferred that there is an ace in the hand, but Johnsson-Laird said the were wrong, as the 'or else' is between two conditionals, and you need both conditionals inorder to infer that there is an ace. In fact, he said, the 'or else' is meant as an exclusive disjunction, so wat you get is $(K \rightarrow A) \oplus (\neg K \rightarrow A)$, meaning that one of the conditionals is false, and since $A$ would be false either way, we can in fact that there is no ace in the hand!

Now, I have always found Johnsson-Laird's analysis a bit suspect (for one, while the falsity of the mathematically defined material conditional does indeed imply that the consequent is false, it is far from clear that humans should be said to make a logical reasoning mistake when they claim that 'Alice does not live in Germany' does not follow from the falsity of the English conditional 'If Alice lives in London, then Alice lives in Germany'), because I think quite a few subjects quite naturally interpreted the 'or else' as between the conditions $K$ and $\neg K$, rather than between the conditionals $K \rightarrow A$ and $\neg K \rightarrow A$.