Symbolic Logic II Philosophy 520 Prof. Brandon Look. Final Exam

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1 Symbolic Logic II Philosophy 520 Prof. Brandon Look Final Exam Due Wednesday, May 6, at 12:00pm. Put your completed exam in my mailbox in POT Please note: U means that a problem is required of undergraduates; G means that a problem is required of graduate students in Philosophy. Graduate students in Philosophy may do either (8) or (9). Each problem or subproblem is worth 10 points, unless marked otherwise. (1) (U) Produce a counterexample (that is, an interpretation of the predicate for some set of objects) to show that the following wff is not logically valid: [ x y z((rxy Ryz) Rxz) x Rxx] x y Rxy (2) (U/G) Show the following (see Sider 5.5): (a) F P C xf x F a (b) F P C xf x ( y y = a F a) (3) (U/G) For the following wff give a semantic validity proof if the wff is SQML-valid or a countermodel if it is invalid: x(nx y(ny y = x) Ox) x(nx y(ny y = x) Ox) (4) (U/G) Demonstrate the VDQML invalidity of the following wff: x y y = x (5) (U/G) Construct an axiomatic proof in SQML for the following wff: x y y = x (G) Translate this formula into English. Do you find this proposition problematic? If so, since we derived it from our axioms and rules, is there anything wrong with our axioms and rules? (Answer in no more than a paragraph.) 1

2 (6) (U/G) Demonstrate by means of a semantic validity proof the following fact: (7) (G) Derive the following: (a) Lemma LP C=S5 φ ψ just in case LP C=S5 φ ψ (b) BF α φ αφ For (b) you should use the Lemma and our rules and axioms. (8) (G) Some Philosophy: Suppose we deny transworld identity, that is, we deny the thesis that individuals/objects can exist in more than one world. So, let us say that a exists at a world, w. Then, at any other world, w, since a does not exist there, one could say that F a is false there. And it follows that F a is false at w, unless Rww, in which case F a is true at w iff F a is. If we take w to be the actual world and assume that it accesses itself, then we have fatalism: the only things that can be true are the things that are actually true. On the other hand, suppose we accept transworld identity. Then we are committed to the thesis that the very same object can exist in more than one possible world. But how can that be? According to Leibniz s Law, α = β (φ(α) φ(β)). But it would seem that a s properties in w should differ from its properties in w thus invalidating Leibniz s Law. Continue this discussion with special appeal to the logical, metaphysical, and epistemological concepts and principles we have discussed this semester. (No more than 500 words (2pp.). 20 points.) (9) (G) Some more Philosophy: In Plato s Sophist, we are confronted with the Paradox of Non-Being. There the Eleatic Stranger says that the term that which is not can t be applied to any of those [things] which are... So if you can t apply it to that which is, it wouldn t be right either to apply it to something. (237c) In other words, we cannot or ought not say of x that it does not exist. Compare this with Richard Cartwright, in his essay, Negative Existentials : To deny the existence of something of unicorns, for example we must indicate what it is the existence of which is denied; and this requires that unicorns be referred to, or 2

3 mentioned: the negative existential must be about them. But things which do not exist cannot be referred to or mentioned; no statement can be about them. So, given that we have denied their existence, unicorns must after all exist. The apparently true negative existential is thus either false or not really a statement at all; and, since the argument applies as well in any other case, we seem forced to conclude that there are no true negative existentials. (J. Phil. 1960: 629) So, consider the following argument: (a) If S denies the existence of a, then S refers to what S says does not exist. (b) Things which do not exist cannot be referred to or mentioned; no statement can be made about them. (c) Therefore, if S denies the existence of a, then what S says does not exist does exist. How can you respond to this argument and this problem in general? Answer by appealing to the logical, metaphysical, and epistemological concepts and principles that we have discussed this semester. In particular, consider: (a) what this problem means for what we are quantifying over ; (b) what this means in terms of actualist vs. possibilist quantification; (c) and whether this problem should move us towards or away from some of the tenets of Free Logic. (No more than 500 words (2pp.) 20 points.) 3

4 Tautologies of Propositional Logic 1 PC1 (φ ψ) φ PC2 (φ ψ) ψ PC3 (φ ψ) ((φ χ) (φ (ψ χ))) Law of Composition PC4 φ (ψ (φ ψ)) Law of Adjunction PC5 (φ ψ) ((ψ φ) (φ ψ)) PC6 (φ ψ) ((ψ χ) (φ χ)) Law of Syllogism (or HS) PC7 (φ (ψ χ)) ((φ ψ) χ) Law of Importation PC8 (φ ψ) ((ψ (χ ξ)) ((φ χ) ξ)) PC9 φ (φ ψ) PC10 ψ (φ ψ) PC11 (φ ψ) ((χ ψ) ((φ χ) ψ)) PC12 φ φ Law of Double Negation (DN) PC13 (φ ψ) ( φ ψ) De Morgan s Law I (DM) PC14 (φ ψ) ( φ ψ) De Morgan s Law II (DM) PC15 (φ ψ) ( ψ φ) Law of Transposition PC16 (φ ψ) (ψ φ) Commutativity PC17 (φ ψ) (ψ φ) Commutativity PC18 ((φ ψ) χ) (φ (ψ χ)) Associativity PC19 ((φ ψ) χ) (φ (ψ χ)) Associativity PC20 φ (φ φ) PC21 φ (φ φ)) Axioms of Propositional and Quantificational Logic 2 PL1 φ (ψ φ) PL2 (φ (ψ χ)) ((φ ψ) (φ χ)) PL3 ( ψ φ) (( ψ φ) ψ) 1 αφ φ(β/α) 2 α(φ ψ) (φ αψ) 2* If φ ψ and α is not free in φ, then φ αψ. 1 See G.E. Hughes & M.J. Cresswell, A New Introduction to Modal Logic, (London: Routledge, 1996), p For 2*, see Hughes & Cresswell, p

5 Axioms of Modal Logic K (φ ψ) ( φ ψ) K (φ ψ) ( φ ψ) D φ φ T φ φ T φ φ B φ φ B φ φ S4 φ φ S4 φ φ S5 φ φ S5 φ φ Rules of Inference MP If φ ψ and φ, then ψ UG If φ, then αφ NEC If φ, then φ Other DT If Γ, φ ψ, then Γ φ ψ RX α = α LL α = β (φ(α) φ(β)) (Leibniz s Law) Problematic Formulas LI x y(x = y (x = y)) Law of Identity (the necessity of identity) LNI x y(x y (x y)) Law of Non-Identity PII F x y((f x F y) x = y) Principle of the Identity of Indiscernibles BF x F x xf x Barcan Formula BFC xf x x F x Converse Barcan Formula 5