Tuesday, September 14, 2010

Writing about propositions

After reading up on the truth literature last fall, I've discovered some embarrassing problems in my past writing, which I've also seen in student writing, when propositions are used. Now when I see these, I cringe. It's mostly just a matter of grammar. Here are some cases of the sort of error I mean:
  1. "If p is a true proposition, then someone could know that p."
  2. "Given the truth that p, we can ask whether p is contingent or necessary."
  3. "If p explains q, and p explains r, then p explains (q and r)."
  4. "If someone knows that p, then p is true."
The easy way to see that these are ungrammatical is to substitute "that snow is white" and the like for variable letters that range over propositions and to substitute "snow is white" for variable letter that abbreviate sentences or that should be understood via substitutional quantification. If one does that, one gets ungrammaticalities:
  1. "If that snow is white is a true proposition, then someone could know that that snow is white."
(The antecedent is fine, but the consequent is ungrammatical.) And sometimes one realizes that one doesn't know how to interpret a variable letter. For instance, in (4) we could take p to be a substitutionary sentence variable, in which case the antecedent would be fine ("If someone knows that snow is white") but the consequent would be ungrammatical ("then snow is white is true"), or we could take p to be a proposition, in which case the antecedent is ungrammatical.
Many cases of this grammatical error are easy to fix. For instance, in (1) and (4), one should simply change "knows that p" to "knows p". (This introduces an ambiguity between knowing p in the "to be the case" sense, and being acquainted with the abstract proposition p, but context should take care of that.) In (2), one changes "the truth that p" to "the truth p". Alternately, in (1), (2) and (4), one can use sentential variables instead, perhaps changing p to s to mark this, and making the requisite changes ("If that s is a true proposition..."; "... whether it is contingent or necessary that s"; "then it is true that s").
However, (3) is trickier to fix up. The problem is that "and" is a sentential operator, while q and r are propositions, so we get "(that snow is white) and (that grass is green)", say. An easy thing is to use sentential variables, and then say a little more verbosely:
  1. If that s explains that u and that s explains that v, then that s explains that s&v.
However, this limits the generality of (3) to those propositions that can be expressed by a sentence. Maybe that's all propositions, but this is not obvious. (This is a general problem with using sentential variables instead of propositional ones.) Another move is to replace "p and q" with "the conjunction of p and q". This fine in this case, and may be stylistically the best solution, but it won't work well with more complex logical forms.
I think the right move to take is to make "&" not be a connective, but a function that takes a pair of propositions and returns their conjunction. If one uses this convention, then one can replace "p and q" with "p & q", and all is well. One may not even need to be explicit about using this convention.
Poor writing as in (1)-(4) may lead to philosophical problems, though hopefully usually it doesn't. Here is a somewhat more serious issue. One way that some authors—and I am pretty sure I've done this myself—handle the problems in (1)-(4) is by using truth. Thus, they might replace (4) by:
  1. "If someone knows that p is true, then p is true."
This neatly avoids the ambiguity of "knows p". However, (6) does not say that if someone knows a proposition p, then p is true, which is what (4) was intended to say. Rather, (6) says that if someone knows the second-order proposition that p is true, then the first-order proposition p is true. This is, of course, true, but does not capture (4). For instance, from (4) one should be able to derive:
  1. If someone knows that snow is white, then it is true that snow is white.
But this does not follow from (6), since someone who knows that snow is white might not know that the proposition that snow is white is true (e.g., a small child or a philosopher). In general, one cannot in a non-extensional context replace a first order proposition p with the second order claim that p is true.
This is all obvious, but somehow it wasn't taught to me when I was in grad school.


Mike Almeida said...


This is a strange worry. In propositional logic, you don't substitute names of propositions in the varibles p, q, r. Indeed, you don't even substitute English propositions. You substitute constants A, B, C, etc., so that the schema p -> q takes as substitution instances things like (A & B) -> C and D -> B and (A v C) -> (G & F). The upper case letters are atomic propositions. They are not names of propositions. Otherwise, truth-functional formulations such as A v B would be completely senseless (e.g., that it is snowing or that it is raining??). So, the analogue in quasi-propositional logic, where English is our language, would be to substitute English propositions, not names of propositions in English, for the propositional varibles.

Alexander R Pruss said...

What is an "English proposition"?