Download IUMA Máster MTT, Métodos, 2015-2016 Examen 22 febrero 2016

IUMA Máster MTT, Métodos, 2015-2016 Examen 22 febrero 2016 Nombre:
1. Lógica deductiva formal de proposiciones. Comentar las siguientes reglas de deducción
natural ND 1, 2 y 3 con sus casos a, b, c, d: Given a set of premises D, and the goal you are
trying to prove, G , there are some simple rules that will be helpful in finding ND proofs for
problems. These rules are not complete, but will get you through many problem sets. Here it
goes. Given D, prove G:
1. Apply premises (atomic propositions and formulas) pi D, to prove G.
2. If you need to use a -premise, then apply -e and prove G for each disjunct (case).
3. Otherwise, work backwards from the type of goal you are proving:
(a) If the goal G is a conditional formula (A  B), then assume A and prove B; use -i rule.
(b) If the goal G is a negative formula (A), then assume A and prove contradiction PC; use
-i rule.
(c) If the goal G is a conjunction (A  B), then prove A and prove B; use -i rule.
(d) If the goal G is a disjunction (A  B), then prove one of A or B; use -i rule.
2. Comentar: A formula A is said to be consistent if ⊬¬A. Also: a formula is consistent if and
only if it is satisfiable.
IUMA Máster MTT, Métodos, 2015-2016 Examen 22 febrero 2016 Nombre:
3. Comentar: A1, . . . ,An ⊨ B means that formula B is valid: it is true in all situations in which
A1, . . . ,An are true. It implies the use of semantics. However a theorem is a formula that can
be established (‘proved’) by a given proof system. We write A is a theorem as ⊢ A, A is a
theorem. A proof system is sound (sólido) if every theorem is valid, and complete if every valid
formula is a theorem. Natural deduction in formal logic is both sound and complete. When
some premises are hypotheses or postulates (rather than axioms) then what is proved,
deduced, is called a thesis. We then say ⊢ A, A is a thesis. Or if A is not a thesis ⊬A.
4. Lógica deductiva formal de predicados de primer orden. Siguiendo los ejemplos vistos en
clase, decir en español:
a) ∀x∃y bought(x, y)
b) ∃y∀x bought(x, y)