Allen/Hand, Section 3.3: Instances, Open Formulas and Quantifier Scopes

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Proofs in Predicate Logic

We want to extend our system for proofs to include predicate logic sequents. To do that, we will add two pairs of rules: Elimination and Introduction rules for each of the quantifiers. These require some explanation in advance. First, we need a definition of an open formula:

An OPEN FORMULA is the result of replacing at least one occurrence of a name in a wff by a new variable (one not already occurring in the wff)

Examples:

Note that an open formula is not a wff.

Now we need to define precisely the scope of an occurrence of a quantifier. Intuitively, the scope of a quantifier is the part of the sentence that the quantifier works on. Quantifiers always take the smallest possible scope. To put this formally:

The SCOPE of a quantifier in a given formula is the shortest open formula to the right of the quantifier.

Some examples:

Instances of Universal and Existential Quantifications

An INSTANCE of a universal or existential quantification is a wff that results by performing the following two steps on it:

1. Remove the initial quantifier
2. Replace all occurrences of the free variable in the resulting open formula with a name (using the same name for all instances)

Note these points:

• Only universal and existential quantifications have instances.
• You must substitute the same name for all the occurences of the free variable.
• You do not need to choose a name that does not already occur in the formula: any name can be used.

Universalizations and Existentializations of wffs

A UNIVERSALIZATION of a wff is the result of performing the following two steps on it:

1. Replace one or more occurrences of a name in it by a new variable (that is, a variable not already occurring in it).
2. Prefix a universal quantifier containing that variable to the resulting open formula.

An EXISTENTIALIZATION of a wff is the result of performing the following two steps on it:

1. Replace one or more occurrences of a name in it by a new variable (that is, a variable not already occurring in it).
2. Prefix an existential quantifier containing that variable to the resulting open formula.

Points to note:

Introduction and Elimination Rules for Quantifiers

Our system will have rules of Universal Elimination (∀'E), Universal Introduction (∀I), Existential Elmination (∃E), and Existential Introduction (∃I). As it happens,

Universal Elimination

UNIVERSAL ELIM (∀E): Given a universally quantified sentence (at line m), conclude any instance of it.
ASSUMPTION SET: the same as the line containing the universal quantification.

If ∀x;Φ is a universally quantified wff, then Φ is an open formula with x as an unbound variable. If α is a name, then let's use Φ(α) to mean "the result of replacing x with the name α in Φ". Then, we can summarize this as:

Examples:

In this example, Φ = (Fx→Gx) and α = b

The next example is a proof of the sequent Fc, ∀y(Fy→Gy) ⊦ Gc :

Note these points:

1. This rule can only be applied to universal quantifications, that is, wffs that begin with a universal quantifier having the entire remainder of the wff as its scope.
2. You can use any name whatever in producing the instance: it does not have to be a name that already occurs somewhere in the proof.
3. You can apply ∀E repeatedly to the same wff: universal quantifications do not get "used up" or discharged when you use them once.

THINGS NOT TO DO. The following are all mistaken applications of ∀E:

Existential Introduction

EXISTENTIAL INTRO (∃I): Given a sentence (at line m containing any occurrence of a name, conclude any existentialization of that sentence with respect to that name.
ASSUMPTION SET: the same as the original line.

We will use Φ(α) to mean 'a wff that contains the name α'. We can then use Φ(x) to mean 'the universalization of Φ(α) with respect to α'. Note that x has to be a variable that does not already occur in Φ(α).

Example: a proof of the sequent (Ha ∨ ~Ga) ⊦ ∃x(Hx ∨ ~Gx) :

Example: a proof of the sequent (Ha ∨ ~Ga) ⊦ ∃x(Hx ∨ ~Ga) :

These are both correct: nothing says that we have to replace every occurrence of the name with a variable in making an existentialization of it.

Example using both ∀E and ∃I: a proof of the sequent Fa, ∃xFx→∀yHy ⊦ Hb :

Universal Introduction

∀I is a simple rule, with a catch: it has a restriction on when it can be used. The restriction is absolutely critical:

UNIVERSAL INTRO (∀i): Given a sentence (at line m) containing at least one occurrence of a name, conclude a universalization of the with respect to the name
ASSUMPTION SET: the same as the original line.
RESTRICTION: the name α must not occur in any assumption in the assumption set for line m.

The crucial part of this rule is the restriction. We can explain it as follows. If we are able to get, as a line of a proof, a sentence containing the name α from premises that do not contain α, then it must not be anything pecular about α that made this possible: we could have done the same proof with any other name besides α. In that case, it's reasonable to say that we could have come to the same conclusion about any name whatsoever. And that, in effect, is what a universal quantifier means.

Existential Elimination

This rule, unfortunately, is considerably more complicated to state than the other quantifier rules. Like ∀I, it also has a restriction. However, in this case the restriction is rather complicated as well.

EXISTENTIAL ELIM (∃E)Given the following:

1. a sentence (at line m),
2. an existentially quantified sentence (at line k),
and
3. an assumption (at line i) that is an instance of the sentence at line k,

conclude the sentence at line m again.
ASSUMPTION SET: all the assumptions on line m and k, with the exception of i.
RESTRICTION: the instantial name at line i does not occur in any of the following:

1. the sentence at line m,
2. the sentence at line k, or
3. any sentence in the assumption sets of lines m or k except for line i.

This is much more complicated than any of the other rules. To explain it, we can first consider an argument in English:

Someone stole a huge bag of cash from the bank yesterday. We don't know who it was, so let's call that person "Rob". So, Rob stole a huge bag of cash from the bank yesterday. Now, unfortunately for Rob, this huge bag of cash was booby-trapped with a can of malodorous, fluorescent dye designed to explode when the bag is opened and cover whoever opened it with evil-smelling fluorescent dye. Rob is bound to have opened the bag, so Rob is now covered with stinky day-glow stuff. Therefore, someone is now covered with bad-smelling fluorescent dye.

In this argument, "Rob" functions as a made-up name with no other purpose than allowing me to reason about whoever it was that stole a huge bag of cash from the bank yesterday. The conclusion of this argument doesn't mention Rob and only talks about someone. Intuitively, this looks like a valid argument. We don't draw any conclusions about "Rob" except those that follow from the assumption that he stole a huge bag of cash, etc.

Here's another example:

There is no largest prime number. For suppose that there is a largest prime number. Call this number N. Now consider the number N!, which is the product of all the numbers up to N: N! = 2 × 3 × ... × (N - 1) × N. Add one to this number. Either N!+1 is prime or it is not. If it is prime, then it is a prime number larger than N, and so N is not the largest prime number. So, suppose that N is not a prime number. Then it must be divisible by some number K. Now, for any number M not larger than N, N!+1 is one greater than a multiple of M. But then M is not a divisor or N! + 1. So, K must be different from every number less than N. In that case, K is greater than N, and again N is not the greatest prime number. Therefore, there is no greatest prime number

In the second example, N is used as an arbitrary name for 'the largest prime number.' In fact, the entire argument shows that there is no such number. However, within the context of the argument, once we have assumed (for the sake of RAA, you could say) that there is one, we can then give it a name and draw some conclusions about it, using only the assumption that it is the largest prime number.

Now, here is how the rule of ∃E embodies this line of reasoning. Suppose that we have, as a line of a proof, an existential sentence ∃xΦ(x). This sentence says, in effect, that for some name α, the instance we get from ∃xΦ(x) by instantiating with α is true. So, we introduce a procedure that lets us do the following:

1. Assume an instance of an existential sentence that we already have as a line of the proof
2. Draw further inferences from that assumption
3. Conclude a sentence that does not contain the instantial name of the assumption any more (usually by using ∃I).
4. Add another line that contains the same sentence just concluded, but with the assumption set changed so that the number of the assumption we made is replaced with the assumption set of the existential sentence

Here is an example using both ∃I and ∃E: a proof of the sequent ∃xFx, ∀x(Fx→Gx) ⊦ ∃xGx:

Exercise 3.2.2

S87. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S88. ∃x(Gx & Fx), ∀x(Fx → ~Hx) ⊦ ∃x~Hx

S89. ∀x(Gx → ~Fx), ∀x(~Fx → Hx) ⊦ ∀x(Gx →~Hx

S90. ∃x(Fx & Ga), ∀x(Fx → Hx) ⊦ Ga & ∃x(Fx & Hx)

S91. ∀x(Gx → ∃y(Fy & Hy)) ⊦ ∀x~Fx → ~∃zGz

S92. ∀x(Gx → (Hx & Jx)), ∀x((Fx ∨ ~Jx) → Gx) ⊦ ∀x(Fx & Hx)

S93. ∀x((Gx & Kx) ↔ Hx), ~∃x(Fx & Gx) ⊦ ∀x~(Fx & Hx)

S94. ∀x(Gx → Hx), ∃x((Fx & Gx) & Mx) ⊦ ∃x(Fx & (Hx & Mx))

S95. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S96. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S97. ∀x~(Gx & Hx), ∃x(Fx & Gx) ⊦ ∃x(Fx & ~Hx)

S98. ∃x(Fx & ~Hx), ~∃x(Fx & ~Gx) ⊦ ~∀x(Gx → Hx)

S99. ∀x(Hx → (Hx & Gx)), ∃x(~Gx & Fx) ⊦ ∃x(Fx & ~Hx)

S100. ∀x(Hx → ~Gx), ~∃x(Fx & ~Gx) ⊦ ∀x~(Fx & Hx)

S101. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S102. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S103. ∃xFx ∨ ∃xGx, ∀x(Fx → Gx) ⊦ ∃xGx

S104. ∀x(Fx → ~Gx) ⊦ ~∃x(Fx & Gx)

S105. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S106. ∃x((Fx & Gx)→ Hx), Ga & ∀xFx ⊦ Fa & Ha

S107. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S108. ∃y(Fa → (∃xGx → Gy)), ∀x(Gx → Hx), ∀x(~Jx → ~Hx) ⊦ ∃x~Jx → (~Fa ∨ ∀x~Gx)

S109. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S110. ∃xFx ↔ ∀y((Fy ∨ Gy) → Hy), ∃xHx, ~∀z~Fz ⊦ ∃x(Fx & Hx)

S111. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S112. ∃x(Gx & ~Fx), ∀x(Gx → Hx) ⊦ ∃x(Hx & ~Fx)

S113. ∃x(Fx & ∀yGxy), ∀xy(Gxy → Gyx) ⊦ ∃x(Fx & ∀yGyx)

S114. ∃x~∀y(Gxy → Gyx) ⊦ ∃x∃y(Gxy & ~Gyx)

S115. ∀x(Gx → ∀y(Fy → Hxy)), ∃x(Fx & ∀z~Hxz) ⊦ ~∀xGx

S116. ∀xy(Fxy → Gxy) ⊦ ∀x(Fxx → ∃y(Gxy & Fyx))