3/4/06
You should now Launch Deriver and do the 3 exercises of Predicate Exercise 8 (Predex8).
Identify the scopes of the following formulas
a) (∀x)(Fx)
b) (∀x)(Fx⊃Gx)
c) (∀x)(Fx)⊃Gx
d) (∃x)(Fx)∧Gx
e) (∃x)(Fx∧Gx)
f) (∀y)(Hy⊃(∃x)(Fx∧Gx))
g) (∀y)((∀x)((∀z)(Hy∧(Fz∧Gx))))
ANSWERS
a) (Fx)
b) (Fx⊃Gx)
c) This is a bit of a trick question, sorry. (∀x)(Fx)⊃Gx has as its main connective the ⊃ so the formula as a whole is not a quantified formula and does not have a scope. However, if our interest is with the quantified formula (∀x)(Fx) which is a subformula of (∀x)(Fx)⊃Gx then it has scope (Fx).
d) This also is a trick. (∃x)(Fx)∧Gxhas as its main connective the ∧ so the formula as a whole is not a quantified formula and does not have a scope. However, if our interest is with the quantified formula (∃x)(Fx) which is a subformula of (∃x)(Fx)∧Gx then it has scope (Fx).
e) (Fx∧Gx)
f) (Hy⊃(∃x)(Fx∧Gx))
g) ((∀x)((∀z)(Hy∧(Fz∧Gx))))
Identify the free and bound occurrences of the variable x in following formulas
a) (∀x)(Fx)
b) (Fx⊃Gx)
c) Fx⊃ (∀x)(Gx)
d) (∀x)(Fx)⊃Gx
e) (∃x)(Fx)∧Gx
f) (∃x)(Fx∧Gx)
ANSWERS
a) (∀x)(Fxbound)
b) (Fxfree⊃Gxfree)
c) Fxfree⊃ (∀x)(Gxbound)
d) (∀x)(Fxbound)⊃Gxfree
e) (∃x)(Fxbound)∧Gxfree
f) (∃x)(Fxbound∧Gxbound)
Derive the following valid arguments
(a) (∀x) (Fx) ∴ Fa
(b) (∀x) (Fx) ∴ Fa∧Fb
(c) Fc,(∀x) (Fx⊃Gx) ∴ Gc
(d) Fa,(∀x) (Fx⊃Gx), (∀x) (Gx⊃Hx) ∴Ha
(e) (∀x) (Fx⊃Gx), (∀x) (Gx⊃Hx) ∴(Fa⊃Ha)