Logic Tutorial, Part 10.
Translating Arguments into Logic

(c) 2017 by Barton Paul Levenson

To translate an argument in English into its equivalent in formal logic strips away any emotion or surprise values in the original wording. Take, for example, these two simple logical statements:

a: Yoda is stunningly attractive.
b: Leia is horribly ugly.

The logical conjunction of the two (a ∧ b) can be used to represent any of the following sentences in English:

• Yoda is stunningly attractive and Leia is horribly ugly.
• Yoda is stunningly attractive, but Leia is horribly ugly.
• Yoda is stunningly attractive, yet Leia is horribly ugly.
• Although Yoda is stunningly attractive, Leia is horribly ugly.
• Yoda is stunningly attractive even though Leia is horribly ugly.
• While Yoda is stunningly attractive, Leia is horribly ugly.

This simplification is a great boon to the logical analysis of arguments. Cut away the emotional appeals and the shocking phrasing and just concentrate on what the sentence means. Or take a simple conditional, p → q. Waner and Costenoble list twelve different ways you could say this in English:

1. if p, then q
2. not p unless q
3. p implies q
4. p is a sufficient condition for q
5. p is sufficient for q
6. p only if q
7. q follows from p
8. q if p
9. q is a necessary condition for p
10. q is necessary for p
11. q whenever p
12. whenever p, q

I noted in chapter 6 that there were at least three ways to phrase a biconditional, e.g. p ↔ q:

1. p if and only if q
2. p is necessary and sufficient for q
3. p is logically equivalent to q

But so far all I've done is to translate formal logic into English. The trick is to do it the other way around. If you can do that, you can simplify an opponent's argument--or an argument of your own--into its logical equivalent and analyze it more easily.

I'll give two real-world examples. Both are from a political blog. They concern an incident where US immigration officials detailed and interrogated (for 26 hours) relatives of a government minister from the United Arab Emirates. Each example is an excerpt from a larger post.

1. What kind of abject moron would give a third degree
interrogation to a relative of a government minister of a
non-hostile country? The INS or DHS people responsible
should be fired. -Anon E. Mouse.

Translation into a syllogism:

M: Government officials who act like abject morons should be fired.
m: The INS or DHS people acted like abject morons.
C: They should be fired.

Further translation, into logical calculus:

M = is a government official who acted like an abject moron
F = should be fired
i = is an INS or DHS official involved in the said incident

∀x[Mx → Fx]
Mi
———————
∴ Fi

This argument appears to be logically valid. Whether the conclusion is true depends on whether the premises are true or not. Should government officials who act like abject morons--in this case, harshly interrogating family members of a friendly government's minister's family--be fired? This brings up another question. Is a government official who does this acting like an abject moron? The major premise here is being assumed. The argument would be more believable, and the major premise more likely to be true, if the major premise were the conclusion to an earlier argument on what constitutes acting like an abject moron. We can all agree that the INS or DHS officials in question conducted the harsh interrogation. To evaluate whether what they did was wrong or not, we need to have some theory which covers such actions. So, in short, Anon E. Mouse's argument is logically valid but may or may not be true.

2. We should certainly apologize for the incident at the
airport, but I wouldn't feel too bad about it. That's what
immigration services do and they are bound to make a
mistake. -SteveinVT.

Translation into a syllogism:

M: Government officials who make unavoidable mistakes should not be fired.
m: The INS or DHS people made unavoidable mistakes.
C: They should not be fired.

And into logical calculus:

M = is a government official who made an unavoidable mistake
F = should be fired
i = is an INS or DHS official involved in the said incident

∀x[Mx → ¬Fx]
Mi
————————
∴ ¬Fi

This is nearly the same argument as the first, except that the second predicate in the major premise, and the predicate in the conclusion, are negative rather than positive. Again, the argument seems valid, although again the major premise went unstated. Neither poster indulged obvious logical fallacies, unless you count the first poster's "abject morons" as prejudicial language.

We have not been given enough information to see which poster, if either, is right. But the translation did manage to pinpoint exactly where the disagreement lay. The key point at issue is evaluating how the officials acted. Was what they did proper or improper? Given only the information contained in these distillations of the posts, there is not enough information to tell. A sound analysis would have to include some theory as to how government officials in that situation should and should not act, and why.