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5AANA014
Section A:
1. (a) Identify the free and bound variable occurrences in the following logical formulas:
• ∀x∃y(Rxz ∧ ∃zQyxz),
• ∀x((∃yRxy→Ax)→Bxy),
• ∀x(Ax→∃yBy) ∧ ∃z(Cxz→∃xDxyz).
(b) Give the definition of a atomic formula of predicate logic and of a valuation of
terms s based on a variable assignment s.
(c) Prove the following by induction (on construction of formulas): if variable assignments s and s0 agree on all variables occurring in a formula φ, then |=U φ [s]
iff |=U φ [s0 ] for all formulas φ and interpretations U.
(You may assume that the language has only the connectives ¬ and → and that
∀ is the only quantifier symbol. Moreover, you may assume that for all terms t:
s(t) = s0 (t)).
2. (a) Give a definition of a valuation ν of formulas based on the truth assignment ν
in Sentential/Propositional Logic.
(b) Let Σ ∪ {φ} be a set of formulas of sentential logic. In terms of valuations
say what it means that (a) φ is a tautological consequence of Σ, that (b) φ is
tautology, and that (c) the set Σ is satisfiable.
(c) Show that if Σ is a satisfiable set, at least one of Σ ∪ {φ} and Σ ∪ {¬φ} is also
satisfiable.
3. (a) In the statement |=U φ [s], i.e, ‘interpretation U satisfies wff φ with variable
assignment s’, describe the components U, s and φ.
(b) Work out, to the atomic formulas, the truth conditions of
• |=U ∀x(Ax→¬Bxy) [s]
• |=U ∀x(Dx→∀y¬Cy) [s]
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(c) Prove the following:
If the variable x does not occur free in φ, then |=U φ [s] if and only if |=U ∀xφ [s].
(Hint: you may assume the Finiteness Lemma: whenever variables assignments
s and s0 agree on all free variables in φ, then |=U φ [s] if and only if |=U φ [s0 ]).
(d) Let ‘∀xφ’ stand for the wff ‘¬∃x¬φ’ and let
|=U ∃xφ [s] ⇐⇒ there is an x-variant s of s0 such that |=U φ [s0 ].
Show that then |=U ∀xφ [s] gives ‘∀’ the expected meaning.
Section B:
4. (a) Formulate the ‘Deduction Theorem’, and the principles of ‘Ex Falso’, ‘Reductio
ad absurdum’, and ‘Indirect proof’.
(b) Use these principles to show
i. ¬P →¬Q ` Q→P ,
ii. ` ¬P →(P →Q),
iii. ((P →Q)→P ) ` P .
(Hint: use (ii) to show that {(P →Q)→P, ¬P } is inconsistent).
(c) Prove (a) the ‘semantic deduction theorem’ : Σ ∪ {φ} |= ψ if and only if Σ |=
φ→ψ and (b) ‘semantic reductio’: if Σ ∪ {φ} is unsatisfiable, then Σ |= ¬φ.
5. (a) Give the definition of a Kripke Frame and a Kripke Model for Propositional
Modal Logic
(b) Let Kripke Frames (W1 , R1 ), (W2 , R2 ).(W3 , R3 ), be defined by
W1 = {a}, R1 = {ha, ai}
W2 = {a, b}, R2 = {ha, bi, hb, ai, hb, bi}
W3 = {a, b}, R3 = {ha, ai, ha, bi, hb, bi}
(hx, yi in Ri means xRi y).
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For each of the three frames determine which of the following formulas is/are
valid on the frame.
i. 2P ⊃ P
ii. P ⊃ 23P
iii. 2P ⊃ 3P
Moreover, if one of the formulas φ is not valid on frame (Wi , Ri ), give a world
w in Wi and a forcing relation ` between Wi and {P } such that w ` ¬φ.
(c) Show that P →23P is valid on a Kripke Frame if and only if the accessibility
relation is symmetric.
(Hint; for the ‘if’ direction, assume that 2P holds at a(n arbitrary) world, for
the ‘only if part assume the accessibility relation is not symmetric and use the
fact that you are free to assign ‘true’ or ‘false’ to the sentence letter P , for every
w ∈ W independently ).
6. (a) Define the notions Φ |= α and Φ ` α for Φ ∪ {α} a set of sentences of Predicate
Logic.
(b) Use the soundness theorem to show that 6` ∃x(Ax→∃yAy).
(c) Argue that Repetition, Extension and Cut hold for both |= and ` .
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