6 Logical Metatheory for We shall focus on our metatheory on the so-called Propositional Modal Logic “soundness” and “completeness” results: i.e., that whenever a conclusion can be deduced from certain 6.1 Metatheory Generally premises in our deductive system, the corresponding argument is semantically valid, and vice-versa. Generally, a logical system or logical calculus (some- You might wonder what exactly the proper struc- times just a “logic”) is fully described by giving three ture and inference rules are for our metaproofs. things: In general, we shall not attempt to be systematic 1. The specification of a “syntax”, i.e., a descrip- here. While we shall sometimes identify and num- tion of the signs that make up the language in ber “steps” in a metaproof, and be clear what follows question and the rules governing which ways of from what, we shall not formalize these rules in any combining them yield well-formed sentences and detail. other expressions, and which do not. (Doing so with substantial rigor would begin an obvious infinite regress.) 2. The specification of a “semantics”, or theory de- In general, we shall simply allow ourselves the proof scribing the meaning or truth conditions of the techniques taken for granted in contemporary math- sentences of the language, usually couched in ematics. This includes metalanguage analogues of all terms of models, and thereby, definitions of such the standard rules for quantified and propositional logical notions as validity, satisfiability, and so logic, as well as some more sophisticated techniques on. such as mathematical induction and some rudiments 3. The characterization of a deductive system cap- of set theory. turing what is taken to be correct reasoning mak- Thus, for example, corresponding to the object lan- ing use of the logical system in question. guage rule of (MP), we will allow ourselves to reason like this. Generally speaking these things cannot be described in the language of the logical system itself, but must If yadda yadda, then blah blah blah. be discussed within another language. Yadda yadda. The object language is the language of the logical Therefore, blah blah blah. system being investigated. The object language for all our studies this semester In the metalanguage, however, we write “if . then → so far has been the language of propositional modal ...” rather than “ ” to avoid any possible confusion logic, i.e., the language of propositional logic supple- between meta- and object language. mented with the operators ‘2’ and ‘3’. Our present focus is simply to establish the following relationships (for various notions of validity The metalanguage is the language used to de- and deductive systems). scribe, discuss explain and prove things about the object langauge. A1, A2,...,An ⊢ B A1, A2,...,An B In our case, the metalanguage is English, or perhaps, (soundness) English supplemented with some technical jargon and symbolism (“model”, “operator”, “”, etc.) We have already described in some detail how to (tree conversion) (tree modeling) carry about deductions or proofs within various sys- tems for modal logic: K, T, B, S4, S5, etc. The tree for A1, A2,...,An, ∼B closes For various reasons, however, it may be worth- while to prove things about logical systems. These (The two relationships at the bottom, put together, proofs are carried out not in a given object language, yield completeness.) but within our metalanguage. They could be called We may not be able to give the proofs of all these in “metaproofs”. rigorous detail; it may suffice if we simply understand Taken together, the examination of what can be why they hold more informally. proven about a given logical system is called its We begin with the relationship on the left side of metatheory. this triangle. 26 6.2 Converting Trees to Proofs • A ∨ B is already in the proper form. Our first task is to show that if all branches for a • A → B yields ∼A ∨ B (by (DN) and (Def ∨)). given tree for checking the K-validity of an argument • ∼(A & B) yields ∼A ∨∼B (by (DM)). A1, A2,...,An /∴ B close, then a proof can be con- structed in system K from A1, A2,...,An to B. We • A ↔ B yields (A & B) ∨ (∼A & ∼B) and do this by showing that the tree can be converted into ∼(A ↔ B) yields (A & ∼B) ∨ (∼A & B). a proof by a mechanical procedure. This transition made, the two branches underneath Given the great similarity between tree rules and can be seen as the two subproofs making up a proof inference rules, the point may seem somewhat obvi- by cases (or as Garson calls, it (∨Out)). ous, but it is worth going over the details. Since all branches of a closed tree must close, the A tree for the argument is headed by the assump- goal then becomes to convert each branch into a proof tions A1, A2,...,An and ∼B in a given world (typ- of ’⊥’ for each case ically “w”), and if closed, ends in ‘⊥’. The cor- Branches within branches require proofs by cases responding proof takes the form of an (IP) with within proofs by cases. A1, A2,...,An as initial premises or hypotheses and Here’s an example from an earlier handout: ∼B an additional assumption for reductio. The goal is to convert the remainder of the tree p → ∼∼q w into a proof of ‘⊥’ within this subderivation, yielding ∼p ∨∼q an indirect proof of B: ∼∼p w A1 p A2 ∼p ∼∼q . ⊥ q . ∼ ∼ An p q ⊥ ⊥ ∼B . Skipping (Reit) steps, this converts to: . 1. w p → ∼∼q ⊥ 2. ∼p ∨∼q B (IP) 3. ∼∼p The non-branching rules for propositional operators 4. p 3 (DN) correspond rather obviously to inference rules: 5. ∼p ∨ ∼∼q 1 (Impl) • ∼∼ The tree rule for changing A to A 6. ∼p corresponds to (DN). 7. ⊥ 4,6 (⊥In) • The tree rule for going from A and ∼A to ⊥ ∼∼ corresponds to (⊥In). 8. q 9. q 8 (DN) • The tree rule for ∼(A → B) corresponds to (→F). 10. ∼p • The tree rule for A & B corresponds to (&Out). 11. ⊥ 4,10 (⊥In) ∼ • The tree rule for ∼(A ∨ B) corresponds to a 12. q combination of (DM) and (&Out). 13. ⊥ 9,12 (⊥In) More interesting is the conversion process for branch- 14. ⊥ 2,10–11,12–13 (∨Out) ing rules. The first thing to note is that each state- 15. ⊥ 5,6–7,8–14 (∨Out) ment form that leads to a branch is equivalent to, and yields in system PL, a disjunction: 16. ∼p 3–15 (IP) 27 This isn’t even close to the shortest or most eloquent This gives us all we need to convert this tree from an proof, which would be something more like: earlier handout: p → ∼∼q w 1. 2(p & q) 2. ∼p ∨∼q ∼(2p & 2q) 3. p ∼2p ∼2q ∼∼ 4. q 1,3 (MP) ∼p u ∼q v 5. ∼p 2,4 (DS) p & q p & q p p ⊥ ⊥ 6. 3,5 ( In) q q ⊥ ⊥ 7. ∼p 3–7 (IP) However, the procedure generating the longer proof w 1. 2(p & q) from the tree was entirely mechanical. Indeed, con- structing a tree and then converting it provides a 2. ∼(2p & 2q) recipe for generating a proof if there is one. 3. ∼2p ∨∼2q 2 (DM) To convert any K-trees, we need only add methods for converting the tree rules for statements that begin 4. ∼2p 2 3 with ‘ ’ or ‘ ’ and their negations. 5. 3∼p 4 (∼2) For their negations, it is simply worth noting that u 2 a statement of the form ∼3A is treated precisely like 6. , ∼p 2∼ A with regard to trees, and one of the form is 7. p & q 1 (2Out) ∼2A is treated precisely like 3∼A. Since we have the derived rules (∼2) and (∼3) for swapping them in 8. p 7 (&Out) proofs, we shall do so, and hence focus all our atten- 9. q 7 (&Out) tion on the rules for unnegated 2- and 3-statements. ⊥ ⊥ The tree rule for 3A creates a new world accessible 10. 6,8 ( In) from the given world in which A is true. In convert- 11. ⊥ 5, 6–10 (3⊥) ing, this corresponds to a new world-subproof (see ∼2 page 7 of the handouts) headed by 2, A. 12. q The tree rule for statements of the form 2A allows 13. 3∼q 12 (∼2) us to introduce A into an already present world box v 2, ∼q accessible from the world where we found 2A. The 14. proof conversion of this is simply an application of the 15. p & q 1 (2Out) (2Out) rule to bring A into a world-subproof where p we had 2A prior to that subproof. 16. 15 (&Out) Strictly speaking, if “⊥” is derived in a world- 17. q 15 (&Out) subproof rather than in our original subproof headed 18. ⊥ 14,17 (⊥In) by the reductio assumption, this only gives us “3⊥” outside of that world subproof by means of (3Out). 19. ⊥ 13, 14–18 (3⊥) However, this eventually yields “⊥” in the main sub- 20. ⊥ 3,4–11,12–19 (∨Out) proof owing to the fact that ⊢K ∼3⊥. We shall ab- breviate this as follows: 21. 2p & 2q 2–20 (IP) 3 A You may notice that here I have tagged the world 2, A subproofs with the corresponding world label (“u” . and “v”)from the tree (and the main subproof with . “w”). This is simply to make the conversion easier to ⊥ follow. Untagged subproofs are not world subproofs (or boxed subproofs of any sort), and hence the (Reit) ⊥ (3⊥) rule always traverses past them.