Logic 3003:
Notes for January 11.
1. Remarks about connectives (recommended extra reading: Appendix B: The
Connectives of Normal Language). Also, some remarks on studying language—
the distinction between syntax, semantics, and pragmatics.
Syntax: Language considered as an isolated set of ‘symbols’(vocabulary) of L,
with rules for generating the complex symbols (grammar) of L, and rules for
manipulating those symbols (proof theory for L).
Semantics: Language treated as having (a fairly abstract kind of) meaning—
interpretation, in particular, of the connectives of L as truth functions.
Pragmatics: Language in the real world, where we actually use it. Here is where
we can study how both the previous levels are abstracted from the real thing, how
our use of words/symbols (and ability to recognize them) grounds both the
syntactic relations/rules and the interpretations applied to them.
2. The syntactic approach to good argument: Rules and forms. Main connectives,
connective structure, substitution and form (a formal account to come). Rules:
encoding intuitions about simple argument forms that strike us as good ones. We
simplify a lot here, to avoid the complexities of natural language mentioned
above. The result is that a lot of what we do here, we do by stipulation. We
automatically treat all the connectives uniformly, by means of our rules,
regardless of any contextual information about the sentences they are joining
together or the tone of voice or prosody used to read the wffs, etc.
3. Structure vs. Language: In a really pure presentation, we would have exactly one
intro and one elim rule for each connective, together with certain structural rules
reflecting what we understand a proof in general to be. Jennings identifies 3
structural rules/principles that are worth reflecting on:
Reflexivity (p. 14), Monotonicity (p. 22) (Note the relation between monotonicity,
the weak form of reflexivity, and the more general form of reflexivity.), and
Transitivity (, A  B ,   A, /   B.) (Think about how we could prove
transitivity holds for our system: 1. Note that if we can prove the sequent ,A 
B, we can prove the sequent   A  B. 2. But if we also have   A, we can
add this proof to the end of the proof of   A  B, and then apply MPP to get B.
Now by the definition of proof,  will include all the assumptions needed to get A
 B and all the assumptions needed to get B, so the assumptions needed to get B
in the final proof will also be included in . QED! Now think about
monotonicity…)
(Reflexivity results just from the proof-starting rule, Assumption. )
4. Intro and elimination rules.
These are designed to capture straightforward conditions for what we can
conclude from having wffs with certain connectives under certain conditions, and
when we can add certain wffs using certain connectives to our ‘conclusions’ at a
given point in a proof. Learning to use them with confidence and clarity will go a
long way toward making the metatheory proofs here easy to understand. So let’s
spend a little time here considering the rules and running a few examples.
(Afterword on this section: We got through a few of the rules, but not all of them.
We’ll continue this on Thursday.) The general idea is that these rules allow us to
unpack and repack information in different forms, so that we can extract our
conclusions from the premises we are given…
5. Re. Simon: I’m still on the learning curve, and will be modifying the
assignments, but assignment 1 is set (it’s trivial—just a matter of getting online
and submitting the thing correctly—if you have any problems, contact me and/or
Nicole). I’ll be settling the early ones (now that I’ve learned how to edit them) in
the next few days. I may use more than the few I first thought of using, but they
will be very brief…