Formal Proofs and Quantifiers

1
Formal Proofs and Quantifiers
Language, Proof and Logic
Chapter 13
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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 results
of replacing in P(x), Q(x)
all free occurrences of x
by c P(t) --- the results 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) ?xP(x)?Q(x)
c Q(c) ?xQ(x)
You try it, pp. 353, 354 Universal 1-2
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Existential quantifier rules
13.2
? Intro
? Elim
P(t) ?xP(x)
?xP(x) c P(c) Q 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 results of replacing in P(x)
all free occurrences of x by c P(t) --- the
results of replacing in P(x) all free occurrences
of x by t
You try it, pp. 358 Existential 1
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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.
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Strategy and tactics
13.3.b
?xTet(x)?Small(x) ?xSmall(x)?LeftOf(x,b) ?xLe
ftOf(x,b)
Informal proof Look, Bozo, we are told
that there is a small tetrahedron. So we know
that it is small, right? But were also told
that anything thats small is left of b. So if
its small, its got to be left of b, too. So,
something is left of b, namely, the
small tetrahedron.
You try it, p.366 Quantifier Strategy 1
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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