Non-Classical Logics

Review for the Midterm Exam
Spring, 2007

The midterm will be an open-book exam: you can bring Priest, An Introduction to Nonclassical Logics, and use it however you wish during the exam. I'll try to mostly write questions for which you can't simply look up the answers, though! But making the exam open-book will allow you to look up tableau rules, specific facts about the individual systems, etc. (Warning: the index of the book isn't particularly good!)

Important Systems:

• Classical propositional logic
• basic modal logic (no restrictions on the accessibility relation)
• normal modal logics (restrictions: rho, sigma, tau, eta, and combinations thereof)
• non-normal modal logics
• conditional logics: C (no restrictions), C⁺, S, C₁, C₂
• intuitionist(ic) logic
• many-valued logics: K₃, LP (don't worry about L, RM₃)
• FDE
• K₄, N₄, K_*, N_*
• Fuzzy Logic

Sample questions:

These are only samples, but hopefully they give the general idea of the sorts of questions I will ask. I want (mostly) questions that will require you to think about and apply the ideas we have discussed.

1. What does it mean to say that one logic is an extension of another logic? What does it mean to say that one logic is a sub-logic of another? If logic L₁ is an extension of logic L₂, which one is a sub-logic of the other? If L₂ is an extension of L₁, and T is a theorem of L₂, must it also be a theorem of L₁? If T is a theorem of L₁, must it also be a theorem of L₂? Explain your answer.
2. Consider the following relation R between logics: R(L₁, L₂) iff L₂ is an extension of L₁. Is this relation reflexive? symmetric? transitive? extendable? Explain.
3. Suppose we wanted to a modal logic for reasoning about space. What would "worlds" correspond to? If we use K_upsilon (no accessibility relation: every "world" is accessible from every "world"), what would the box mean? What would the diamond mean?
4. Present and evaluate a philosophical motivation for logic X (where X is a variable ranging over K, C, K₄, N₄, fuzzy logic, etc.).
5. proofs in the various logics using the tableau method and/or finding countermodels.
6. Stalnaker advocated condition 6, p. 88; Lewis recommended replacing it with condition 7, p. 90. What do these conditions mean, in ordinary English? What do they have in common? How are they different? Give an example that shows that counterfactual conditionals in English do not seem to be compatible with condition 6.
7. Invent a tableau method for fuzzy logic, and prove that it is sound and complete.
8. What is the definition of a relevant logic?
9. Prove that K₄ is not a relevant logic. (You can do this by proving, in K₄, a theorem that is incompatible with the definition of a relevant logic.)
10. As you know, in K3, p horseshoe p is not a logical truth. (Countermodel: v(p) = i.) Now, keep the same truth tables as K3, but revise the definition of logical truth as follows: p is logically true iff p is true in every supervaluation of every interpretation. Show that p horseshow p is a logical truth in the new system.
11. A different question with a content similar to the previous one. Define logical truth as in the previous question, and consider the truth tables of K3. Consider the sentence: p horseshoe (q v ~q). List every distinct interpretation of this sentence. (That is, give all the possible distinct functions v from the atomic sentences p and q to truth values.) (There should be nine.) For each of these functions, list every possible supervaluation of that function. (There should be a minimum of one and a maximum of four.) Is the sentence true in every supervaluation of every interpretation?
12. Suppose that A ⊨ B (in a given logic). What property of the tableau method for that logic guarantees that every branch of a tableau that begins with A and ~B will close?
13. Suppose that, in some logic, I build a tableau whose first node contains the sentences A and ~A, and I find that every branch of the tableau closes. What property of the tableau method for that logic guarantees that  A ⊨ B ?
14. The characteristic axiom of K_eta is: □p ⊃ ◊p. (a) Explain why, if this axiom is a logical truth, the accessibility relation R must be extendable. (You can do this by proving the contrapositive, i.e. by showing that if R is not extendable, there is a countermodel.) (b) Explain why, if R is extendable, the axiom must be true in every world of every interpretation, i.e. must be a logical truth.
15. I could ask the same question as 14 for the axioms related to other restrictions on R. For example, reflexivity is related to the logical truth of □p ⊃ p; symmetry is related to the logical truth of p ⊃ □◊p; and transitivity is related to the logical truth of □p ⊃ □□p.