Revising details in forall x

I’m teaching Introduction to Logic for the first time in several years. The course text is my own forall x. It’s always been an open textbook, even back before I had good vocabulary for explaining what that means. But now it’s available from SUNY OER Services, and they’ve partnered with SUNY Press so that my students are able to buy a hardcopy from the campus bookstore for just $8.50.

Working through it this time, I’ve hit a couple of things which I am considering changing.

First, the translation of “unless”.

It seems to be that the following sentences make sense in English.

Ingmar will go running tomorrow unless it rains. And let’s be honest— if it rains he will go running anyway.

Suppose we let I mean Ingmar will go running and R mean It will rain. Then we can translate the first sentence as (¬R→I) and the second as (R→I).1

Since (¬R→I) is logically equivalent to  (I∨R), I have always just recommended translating “unless” using “∨” (inclusive disjunction).

Some students always want to translate “unless” as an exclusive disjunction; for example, translating the first sentences as (I↔︎¬R). This has always seemed wrong to me, because it treats the two sentences in the passage above as inconsistent. Yet the passage seems to me like something that one could coherently say.

Nevertheless, one might see the passage as following a train of thought in which the speaker says something and then takes it back. They first say (I↔︎¬R) but then withdraw that and instead say something different.

The majority of my students this semester preferred the latter reading. Competent English speakers can disagree about how to read it, and the different readings make for different translations into sentential logic.

My current inclination is to say that “unless” is like “or”. Depending on context, it can be read as inclusive or exclusive. Reading inclusively, the first sentence is (I∨R). Read exclusively, it is (I↔︎¬R).

This is messy, because it means that there are no unambiguously right translations of the first sentence. But I think this just reflects the messiness of language.

Second, the strictures about parentheses.

Like many logic books, mine defines a well-formed formula (WFF) so that parentheses are only added along with two-place connectives. For example, one gets from (I∨R) and (I↔︎¬R) to ((I∨R)&(I↔︎¬R)).

One may leave out parentheses when they make things harder to read rather than easier, as a matter of notational convention. So one may write (I∨R)&(I↔︎¬R) without the outside parentheses and (A&B&C&D) without the pairwise parentheses.

One may not add parentheses in any other way, even though they don’t decrease readability. So ¬(¬I) is not a WFF or sentence.

I’ve notice this time, though, that some students are tempted to insert parentheses where they aren’t allowed. For example, they write ∀xMx as ∀x(Mx).

Some systems in some books do allow this, so curing them of it isn’t especially important. It doesn’t make the expression any less readable. Arguably it makes it clearer.

So I am considering adding an additional notational convention: For any WFF A, you can write (A).

P.D. holds a copy of forall x.
  1. I am pasting in logical symbols, which is something I usually don’t do in my blog. It’s Unicode, so it should display just fine. It may just look like bullets or boxes, though.

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