Mathematical Logic 13–14
Week 2
1. (a) Prove the following:
Proposition. Let α be a formula whose propositional variables
are among p0 , . . . , pn . Let v, w be valuations such that v(pi ) =
w(pi ) for all 0 ≤ i ≤ n. Then v(α) = w(α).
Hint. By induction on α.
(b) Derive that, in order to define the truth value of a formula α
in which at most propositional variables p1 , . . . , pn do occur, it
suffices to assign truth values just to p1 , . . . , pn .
2. So far we have seen examples of a 0-ary connective (⊥), a 1-ary connective (¬), and four binary connectives (∧, ∨, →, ↔), but there are n-ary
connectives, for each n ∈ N. What matters is not the symbol used to
denote the connective, but rather its semantics. For instance, the semantics of connective ∧ is given by the mapping f∧ : {0, 1}2 → {0, 1},
f∧ (x, y) = min(x, y).
So, a n-ary connective IS a mapping f : {0, 1}n → {0, 1}.
Let n ∈ N. How many n-ary connectives are there? (Pay special
attention to the case n = 0: you will realize why we said that ⊥ is not
only an atomic formula, but also a connective.)
3. (a) What are the valuations that satisfy the empty set of formulas?
(b) Prove that, for α ∈ PROP,
α is a tautology
⇔ |= α.
4. Exercises 3, 6 pag. 14 (the rank of a formula is defined on page 12); 9
pag. 15 in van Dalen, Logic and Structure, 4th ed.
5. Exercises 1 (a)(e)(g)(h) pag. 20; 2, 3, 14, 15 pagg. 27–29 in Logic and
Structure, 4th ed.
Hint for Exercise 14(i) : let p be a propositional variable not occurring
neither in ϕ nor in ψ. Try with σ = ϕ ∨ (ψ ∧ p). (Can you see why we
pick such a σ?)
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6. Before working out this exercise, read the statement of Theorem 1.3.6
in Logic and Structure (you don’t have to go through the proof !) and
the comments after the proof.
Assuming that {¬, ∨} is a functionally complete set of connectives,
prove that also {¬, ∧}is functionally complete.
7. Let | be the binary connective (called Sheffer’s stroke) whose valuation
function is 1 − min(x, y). Prove that {|} is functionally complete.
Hint: if you do not succeed, see Exercise 5, pag. 28 in Logic and
Structure, 4th ed.
8. Notice that the proof of functional completeness of {¬, ∨} proves also
the so called Disjunctive Normal Form Theorem, namely the statement
that every α ∈ PROP is logically equivalent to a formula of the form
_ ^
βij ,
i≤n j≤mi
where each βij is an atom or a negated atom. (See Definitions 1.3.7
and 1.3.8, pag. 25, in Logic and Structure.)
(a) Follow the hint given in Exercise 7, pag. 28, Logic and Structure
to state and prove a Conjunctive Normal Form Theorem.
(b) State and prove a criterion (i.e. a necessary and sufficient condition) for a formula in conjunctive normal form to be a tautology.
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