A way to think about the disjunction elimination

Question

In his book Compactness and contradiction, Terence Tao writes:

The material implication “If A, then B” (or “A implies B”) can be thought of as the assertion “B is at least as true as A” (or equivalently, “A is at most as true as B”). This perspective sheds light on several facts about the material implication:

(1) [...]

(6) Disjunction elimination. Given “If A, then C” and “If B, then C”, we can deduce “If (A or B), then C”, since if C is at least as true as A, and at least as true as B, then it is at least as true as either A or B.

(7) [..]

What’s the point of thinking of an implication $A \implies B$ as asserting that $B$ is at least as true as $A$ in order to understand the disjunction elimination? In know that “$A$ or $B$” has the biggest truth value of the two truth values of $A$ and $B$. Thus if $C$ is at least as true as $A$ and also at least as true as $B$, then “$A$ or $B$” is at least as true as $C$ (since “$A$ or $B$” has the same truth value as $A$ or as $B$). But why does this shed light on the disjunction elimination? How does the viewpoint of Terence Tao help me understand the disjunction elimination?

Answer 1

Pretend $A,B,C$ are numbers and $A\Rightarrow B$ means $A\leq B$ (this is not some arbitrary idea; there is a good reason for this analogy).

Then:

“If A, then C” and “If B, then C”, we can deduce “If (A or B), then C”

translates to:

$$A\leq C \text{ and } B\leq C \Rightarrow \max(A,B) \leq C$$

which is true by the definition of $\max$.

Answer 2

Suppose that in C$\alpha$$\beta$ (this is Polish notation), you assert that '$\beta$' can have less truth than '$\alpha$'. Then if '$\alpha$' is true, and 'C$\alpha$$\beta$' is true also, then by a use of the rule of detachment, we could infer from a truth '$\alpha$' to a weaker conclusion '$\beta$'. In other words, we could start with a true premiss, a sound rule of inference, and end up inferring to a weaker truth, if not an outright falsity. Thus, as a matter of hope at least, I will hope that in C$\alpha$$\beta$, '$\beta$' will have at least as much truth as '$\alpha$' does.

So let's talk about slightly different rule of inference than disjunction elimination for a moment:

1) {Cac, Cbc} => CAabc

So, since we have 'Cac' as a premiss, 'c' is at least as true as 'a'. Since we also have 'Cbc' as a premiss also, 'c' is at least as true as 'b'. Thus, 'c' is at least as true as 'Aab' ("a or b"). Consequently, the above rule works out as sound (or at least that I hope the above convinces you of that). Now, since the above rule works out as sound, we'll consider disjunction elimination:

2) {Cac, Cbc, Aab} => c

By the above rule, since 'Cac' and 'Cbc' are premisses and rule 1) it follows that CAabc holds. But, since we have Aab as a premiss also, and we've thought of 'c' as at least as true as 'Aab', it follows that c holds also. Consequently, thinking of 'C$\alpha$$\beta$' as asserting that $\beta$ has at least as much truth as $\alpha$ has helped us to make sure that disjunction elimination makes for a sound rule of inference.

Notice that we didn't make any reference to truth tables or assume two logical values. And in intuitionistic logic which model-theoretically is infinite-valued, as well as Lukasiweicz infinite-valued logic (I would guess Goedelian infinite-valued logic also), disjunction elimination makes for a sound rule of inference.