1. Formal Proofs and Quantifiers
Language, Proof and
Logic
Formal Proofs and Quantifiers
Chapter 13

2. Universal quantifier rules
13.1
 Elim:
x variable
t constant term (variable-free term)
c constant which does not occur outside
the subproof where it is introduced
P(x), Q(x) --- any wffs only containing x free
P(c), Q(c) --- the result of replacing in
P(x), Q(x) all free occurrences
of x by c
P(t) the result of replacing in P(x) all
free occurrences of x by t
xP(x) …
P(t)
 Intro (General Cond. Proof):
 Intro (Universal introduct.):
c P(c) …
Q(c)
x[P(x)Q(x)] c …
Q(c)
xQ(x)
You try it, pp. 353, 354 [Universal 1-2]

3. Existential quantifier rules
13.2
 Intro:
 Elim:
P(t) …
xP(x)
xP(x) …
c P(c) Q
x variable
t constant term (variable-free term)
c constant which does not occur outside the subproof where it is introduced
P(x) --- any wff only containing x free
P(c) --- the result of replacing in P(x) all free occurrences of x by c
P(t) --- the result of replacing in P(x) all free occurrences of x by t
You try it, pp. 358 [Existential 1]

4. 1. Always be clear about the meaning of the sentences you are using.
Strategy and tactics
13.3.a
General tips:
1. Always be clear about the meaning of the sentences you are using.
Practically zero chance to succeed without that!
2. A good strategy is to find an informal proof and then try to formalize
it.
3. Working backwards can be very useful in proving universal claims.
You typically use  Intro in these cases.
4. Working backwards ( Intro) is not useful in proving an existential
claim xS(x) unless you can think of a particular instance S(c) of the
claim that follows from the premises.
5. If you get stuck, consider using proof by contradiction.

5. x[Small(x)LeftOf(x,b)] xLeftOf(x,b)
Strategy and tactics
13.3.b
x[Tet(x)Small(x)]
x[Small(x)LeftOf(x,b)]
xLeftOf(x,b)
Informal proof:
Look, Bozo, we are told that
there is a small tetrahedron.
So we know that it is small,
right? But we’re also told that
anything that’s small is left of
b. So if it’s small, it’s got to
be left of b, too. So, something
is left of b, namely, the small
tetrahedron.
Formal proof:
1. x[Tet(x)  Small(x)]
2. x[Small(x)  LeftOf(x,b)]
c Tet(c)  Small(c)
Small(c)  Elim: 3
Small(c)  LeftOf(c,b)]  Elim: 2
LeftOf(c,b)  Elim: 4,5
xLeftOf(x,b)  Inro: 6
8. xLeftOf(x,b)  Elim: 1, 3-7
You try it, p.366 [Quantifier Strategy 1]

6. Soundness and completeness
13.4
As in the propositional case, we have:
Q is provable in Fitch from premises P1,…, Pn
if (completeness) and only if (soundness)
Q is a FO consequence of P1,…, Pn