Exercise (3.1). (The Grand Leap) We translate the sentence The ...

Exercise (3.1). (The Grand Leap)
We translate the sentence
The grand leap of the whale up the Falls of Niagara is accounted by
all who have seen it, one of the finest spectacles of Nature.
into the language of L in stages, by rewriting it successively as follows:
Every person who has seen the grand leap of the whale up the Falls
of Niagara believes that it is one of the finest spectacles of Nature.
For every person x, if x has seen the grand leap of the whale up the
Falls of Niagara then x believes that this grand leap is one of the finest
spectacles of Nature.
Finally we get the following sentence of L. Let σ be the sentence:
˙∀x (P x −→ Qx)
We now see if |= A σ. The following statements are equivalent:
(1)
|= A σ
|= A ˙∀x (P x −→ Qx)[s] (for any s)
(∀a ∈ A) |= A (P x −→ Qx)[s x a]
(∀a ∈ A) |= A P x[s x a] =⇒ |= A Qx[s x a]
(∀a ∈ A) a ∈ P A =⇒ a ∈ Q A
P A = ∅, since there never have been whales in Niagara Falls, and so
the statement a ∈ P A is false for each a. The statement
a ∈ P A
=⇒ a ∈ Q A
is, therefore, true for each a. Hence the statement (0.1) is true, and so
Benjamin Franklin was telling the truth.
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Exercise (3.2). (Sarah) The language L has a binary relation symbol
G (Gxy means “x grades y”) and a constant symbol c (for Sarah). Let
σ be the sentence ˙∀x (Gcx ↔ ¬Gxx). We want to determine if
(2) {σ} |= Gcc ,
that is, we want to know if for every structure A for L, and s,
(3) |= A σ =⇒ |= A Gcc .
Let A = (|A|, G A , c A ) be a structure for L. The following statements
are equivalent:
|= A σ
|= A ˙∀x (Gcx ↔ ¬Gxx)[s] (for any s)
∀a ∈ |A| |= A (Gcx ↔ ¬Gxx)[s x a]
∀a ∈ |A| (c A , a) ∈ G A ⇐⇒ not (a, a) ∈ G A .
But the last line is false since it implies
(c A , c A ) ∈ G A ⇐⇒ not(c A , c A ) ∈ G A .
Since the antecedent of (3) is false, the statement (3) is true (for every
A), and hence (2) is true.
Exercise (Enderton p 79 1(d)). We translate (d) successively as:
Any uninteresting number with the property that all smaller numbers
are interesting, certainly is interesting.
For every x, ( if (x is a number and x is uninteresting and all
smaller numbers are interesting) then x is interesting).
For every x, (if (x is a number and x is uninteresting and (for every
y, if y is a number and y is smaller than x implies y is interesting))
then x is interesting).
˙∀x (Nx ∧ ¬Ix ∧ ˙∀y (Ny ∧ y ˙

Exercise (Enderton p79 4). We translate successively:
If (horses are animals) then (heads of horses are heads of animals).
If (horses are animals) then (every head of a horse is the head of an
animal).
If (for every x (if x is a horse then x is an animal)) then (for every
x (if x is the head of a horse then x is the head of an animal))
We translate
for every x if x is a horse then x is an animal
by ˙∀x (Ex → Ax).
We translate
x is the head of a horse by
for some y, y is a horse and x is the head of y.
which becomes
˙∃y(Ey ∧ x ˙=hy)
and similarly for: x is the head of an animal. Finally, we get:
˙∀x (Ex → Ax) −→ ˙∀x ( ˙∃y (Ey ∧ x ˙=hy) → ˙∃y(Ay ∧ x ˙=hy))
Exercise (Enderton 89,10). We want to write the sentence ˙∃v 1 P v 1 ∧P v 1
without abbreviations. We use the convention for abbreviations on
page 78.
Let α = ˙∃v 1 P v 1 (which is actually in turn an abbreviation for
(¬ ˙∀v 1 (¬P v 1 )). And let β = P v 1 . Then gradually eliminating abbreviations,
we get successively:
(1) ˙∃v 1 P v 1 ∧ P v 1
(2) α ∧ β
(3) (α ∧ β)
(4) (¬(α → (¬β)))
(5) (¬((¬ ˙∀v 1 (¬P v 1 )) → (¬P v 1 )))
(5) is in unabbreviated form.
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