OTHERWISE,’ ‘ELSE’

Another couple of English operators that seem to be construable either as disjunctions or conditionals are the words ‘otherwise’ and ‘else’ (or ‘or else’). That these are equivalent is suggested by the following two translations of Heraclitus’ fragment 40,^[28] an enthymeme:

(6) Much learning does not teach anyone to have intelligence; for else it would have taught Hesiod and Pythagoras, and again, Xenophanes and Hecataeus.

(7) Much learning does not teach insight.

If we symbolize “Much learning teaches insight” by M, and the compound conjunction “it would have taught Hesiod and Pythagoras, and moreover Xenophanes and Hecataeus” by T, we have: Not M. For else [or, otherwise] T. Implicitly, we also have “But not T.” That is,

Not M. For if M then T; but not T.

i. e.,

This is just an instance of Modus Tollens. Alternatively, just as with unless, we could have construed else as an ‘or,’ but this time with the first disjunct implicit, i.e., we could take “Not M. For else T” as “Not M. For either not M or T.” Then the argument would be symbolized

This is just an instance of Disjunctive Argument. Comparing this with the previous sym­bolization shows the close relationship between DS and Modus Tollens. In sum, the enthymemic argument

EXERCISES 7.1

1. Symbolize the following statements using the first letter of each capitalized word for the components:

(a) The Brooklyn Museum of Art will LOSE its $7.3m subsidy unless it SCRAPS the Sensation exhibition.—New York Mayor Rudy Giuliani

(b) If they think it is important to throw feces at NATIONAL or RELIGIOUS sym­bols, they can pay for it THEMSELVES.—New York Mayor Rudy Giuliani [N := they think it is important to throw feces at NATIONAL symbols, R := they think it is important...

(c) The mayor says Ofilfs work is OFFENSIVE to Catholics, and that he will also END the museum’s lease and take over its BOARD unless it toes his LINE.—“Hil­lary Backs Britart Show that New York Seeks to Ban,” Manchester Guardian Weekly 161, #14

(d) Unless I SEE it with my own eyes, and HEAR it with my own ears, I will never BELIEVE it.—Charles Dickens

(e) If the CAREER-ending or LIFE-threatening violence does not stop, [Dallas Stars centre Mike] Modano says he will get OFF the ice for good.—Associated Press report, October 8,1999 [C := the career-ending violence stops, L := the life-threat­ening violence stops.]

(f) Nothing can BE or be CONCEIVED without GOD.—Spinoza, Ethics II, P15 [construe “without God” as “unless GOD exists”]

(g) If one body PUSHES another, it cannot GIVE the other any motion unless it LOSES as much of its own motion at the same time; nor can it TAKE away any of the other’s motion unless its own is INCREASED by as much.—Rene Descartes, The World, second law of nature

2. “q unless p” can be symbolized either b>)r by p v q. Show that whichever

way it is done, (i) taken together with -∣p it entails q, and (ii) taken together with -∣ q it entailsp.

3. Symbolize and prove the validity of the following argument from the Science Times.^[29] (It is an enthymeme: you will need to supply the implicit premise.)

“People were presumably FLOATING some kind of boats as early as 50,000 years ago, or how else could humans have first SETTLED Australia.”

4. (CHALLENGE) (a) Symbolize and (b) prove the validity of this paraphrase of Des­cartes’ argument for the non-existence of atoms:

It is impossible that there should exist ATOMS, that is, pieces of matter that are by their very nature indivisible. Otherwise, however small we imagined them to be, they would still have to be EXTENDED. But either they are not extended, or we can divide them in our THOUGHT; and anything we can divide in our thought, must, for that very reason be DIVISIBLE. But by hypothesis, they cannot be divis­ible.

7.2