meeting -*- Outline -*-

* Deductive Systems (3.3)

  We now want to reason about the denotations of terms.

  Fix AT and SG.

** defintions
   Q: What's a formula over SG?
      a term of type Bool

   Q: So is a formula either true or false?
      Yes, in any given environment

   Q: What's a sequent, and what does it have to do with deduction?
      It's a pair (Hyp, t),
            where Hyp is a finite set of formulas over SG (the hypotheses),
            and t is a formula over SG (the consequent).

   Q: When is a sequent satisfied in a model?
      when it is assigned true in all environments over that model
           which make the hypotheses true.

   Q: When is a sequent valid?
      When it is satisfied in all standard models.

   Q: What's an inference?
      A list of sequents (Hyp-seqs, conc-seq)

   Q: When is an inference valid?
      When it its conclusion is satisifed in all standard models in
      which the hypotheses are satisfied.

   Q: What's the difference between an inference and a deductive
   system?

   Q: What's a side condition?  What role does it play in a deductive
   system?

   Q: What's the difference between an axiom and an axiom scheme?
      an axiom scheme has metavariables, an axiom is closed

*** deductions
   Q: What's the difference between a deduction and a proof?
      a proof has no hypotheses

   Q: What's the difference between a metavariable and a variable?

** theorems (3.4)

   Q: What's a theory?
      A set of theorems (that are valid).
   Q: How is a theory specified?
      Using a type structure, a signature, and a set of inference
      rules.

   Often we refer to the presentation of a theory as the theory itself.

   Q: How can a theory be inconsistent?
      if it contains all sequents

   Q: Is an inconsistent theory sound?  complete?

   Q: If we extend a theory's presentation, can we still prove
   everything we could prove before?

   Q: What's a conservative extension?  Why is it important?
      it's important because it preserves consistency