truth 1, falsity 0
Sat 27 Dec 2014 07:45 PM
I just posted version 1.30 of forall x. As usual, the update corrects a number of typos. It also makes some changes. I taught logic this Fall for the first time in several years, and the time away from the book made me realize that some things weren't working as well as they should.
It is standard in philosophy to do truth tables with Ts and Fs, but this makes the shift from sentential logic to quantified logic awkward. The sentential connectives operate in pretty much the same way in a quantified formula as they do in a sentential formula, except that the operators are truth functions in SL and satisfaction functions in QL. I want students to understand the difference between truth and satisfaction, but I also want them to apply what they know about the truth-functional connectives. So I end up saying things like, "Both conjuncts are satisfied, so the conjunction is satisfied. This part's just like in a truth table. Both parts are true, sort of... true-ish... something that works a lot like truth."
In the previous edition of the book, I tried to smooth this over in the chapter on formal semantics by giving the function which defines 'truth in SL' in terms of 1 and 0 rather than in terms of T and F. The definition of 'satisfaction in QL' is also in terms of 1 and 0, and the clauses which define satisfaction for sentential connectives look exactly the same as they do in the definition of truth. So I can say instead, "Both conjuncts are 1, so the conjunction is 1."
The problem is that, by that point, students have acquired habits in terms of T and F from doing truth tables. So I decided to start with 1 and 0 earlier, doing truth tables entirely using 1s and 0s.
This is common in computer science and electronics, even though it's not common in philosophy. My motivation is philosophical, though. Doing truth tables in terms of 1 and 0 underscores the step of abstraction, that these are formal, mathematical values rather than metaphysical truth and falsity. And because they are formal values rather than rich concepts, they can be interpreted differently (as satisfied/not, rather than as true/false).
I think I made this change consistently everywhere, but there are probably still some lingering mention of T and F. New content means new typos.
Proofs in QL
The chapter on proofs is the barest part of the book. It would be the hardest part to learn from directly, if someone were just reading the book rather than taking a course.
In this update, I just made some changes to the presentation of the quantifier rules.
I changed the typographic mark for a substitution instance, and I think it's clearer now. (I won't try to produce it here on the blog.)
I rewrote the Existential Elimination rule so that the proxy constant cannot occur anywhere else in the proof. You have 'Exists x Px' and assume 'Pc' for some entirely new constant c. This is stronger than what's strictly required, but it underscores the conceptual point that c is only functioning as a placeholder name for whatever thing it is that's P. The subproof is the only place where c occurs, because the subproof is the moment in the argument when you say "Something is P. We don't know what, but let's call it c."
I am considering splitting the chapter on proofs into two chapters: One on proofs in SL and another on proofs in QL. This would allow me to add material to both discussions. It would also allow instructors who want to do proofs in SL immediately after doing truth tables to do so more easily. That's not a change I made in this revision, though, and I'm still mulling it over.
I archived earlier versions of the book at a SUNY digital repository. Recently, the library here at UAlbany has set up a local digital repository which should offer more features and more visibility. I think that the submission needs to be approved by a librarian, but version 1.30 will appear there soon enough. Until then, it's available directly from my website.