An Explanation of the Rules of Quantification

from Boardman's Supplement for Symbolic Logic (c)

UNIVERSAL INSTANTIATION

To use the inference rule of UI, you must take as your premise a schema which has a universal quantifier as its major operator; now delete that prefix, and substitute--for every occurrence of the previously bound variable--some one variable (or, instead, some one constant). For the letter which may be used for this substitution: you may retain the letter which previously was there as a bound variable; you may use as your letter one which occurs elsewhere in the premise as a free variable (or as a constant); or you may use a different letter so long as it will permit the following condition to be satisfied. Wherever the variable--which was formerly bound by the major operator--occurred in the premise, now a single free variable (or, a single constant) must occur in its place in the conclusion. Notice that the free variable (or the constant) in the conclusion may appear at more places than the original bound variable appeared in the premise; but the free variable must appear at least in all the places where the original bound variable did.
It may help to remember the point of UI: when something is true of everything, then it must be true of any particular thing you wish to consider—either a named individual, or an unspecified individual.

EXISTENTIAL GENERALIZATION

To use the inference rule of EG, you must take as your premise a schema which has at least one occurrence of a free variable or one occurrence of a constant; it will make no difference what the major operator of the premise is. In that premise, you must choose some one free variable, or some one constant, to generalize over (but not both ). In addition, you have a further choice: you may selectively generalize over some occurrences of the chosen free variable or constant, leaving other occurrences as they were before. [Such choices are exercised in accordance with strategic considerations: you decide what you need to infer given the over-all problem.] Now take those occurrences of the free variable or the constant which you have chosen, and replace them with some one variable; you may use for this replacement the variable which you are generalizing over, or you may use a different one so long as it will satisfy the conditions in the next sentence. Finally, affix to this new schema as its major operator an existential prefix which binds every occurrence of the replacing variable and which binds no other variable.
You will have noticed that the substitution specified in the schematic statement of the rule of EG runs "backwards," running from the conclusion to the premise: its purpose is to require that the new bound variable will occur in the conclusion only at places where the original free variable, or the original constant, occurred in the premise; the new bound variable need not appear at every place where the original appeared, but it must not appear in the conclusion at any place where the original had not occurred. In addition, the "backwards statement" of substitution is what permits you to replace selectively all or merely some of the occurrences of the original variable, or the constant, with the new bound variable. Finally, remember the point of EG: if some property or relation is true of a named individual, or of an unspecified individual, it must then be true of something.

UNIVERSAL GENERALIZATION

To use the inference rule of UG/, you must take as your premise a schema which has at least one occurrence of a free variable; it will make no difference what the major operator of the premise is. If there are several free variables in your premise, then you will have a choice of which of them to generalize over. But notice that unlike the first two rules, there is no second pattern in which constants are featured: you may not universally generalize over a constant. Now substitute for every occurrence of the chosen free variable, some single variable--which may be the same letter as before, or may be a different letter so long as its use will satisfy the conditions in the next sentence; finally, attach to the schema a universal prefix as the new major operator binding every occurrence of the substituted variable. Your choice of letter for the substitution must allow these conditions to be satisfied: the newly bound variable must occur in the conclusion everywhere the original free variable occurred in the premise; and the newly bound variable must occur in the conclusion only where the original free variable occurred in the premise.
There is a further, very important restriction on your choice of free variables for UG/: if you are at a place in a proof where some assumption is still open, you may not choose a particular free variable (i.e., as your candidate for “”) to generalize over if that letter happens to occurs free in (the first line of) any still-open assumption.¹ (An assumption is identified by an arrow's pointing to it, and the assumption is still open if the end of the arrow has not yet been bent across the proof and succeeded by the use of Conditional Proof.) Whenever you use the inference rule of UG/, you must, in your citation of the rule, specify the free variable (i.e., your “”) in the premise that you have chosen to generalize over--i.e., you must identify your candidate for "square" (“”) in the line you cite as your premise: then you must routinely check to be sure that you are not violating this last restriction; compare the letter you propose to generalize over with the first lines of all open assumptions to be sure that the letter is not free at any of them. If that letter is free in any still-unclosed assumption, then you may not use UG/ over that letter (over that candidate for “”) at this point in the problem: you will accordingly have to find some alternative strategy.
Underlying its complex restrictions, the point of UG/ is that if something has been proved of any arbitrarily chosen individual, then it must be true of everything.

Footnote to Explanation of Rules for UG/:
¹ Here is an example of the sort of INVALID argument which is FORESTALLED by the restriction on generalizing within an open assumption:

   1. (x)(Wx ⊃ Mx)

   2. Wy ⊃ My) [1 UI]
->3. Wy

| 4. My [2, 3 MP]

| 5. (x)Mx [4 UG/y
thus violating the restriction on UG over a variable (y in this example) which is free within an open assumption (at line 3)

----------------- --------------

   6. Wy ⊃ (x)Mx [3-5 CP]

   7. (z)[Wz ⊃ (x)Mx] [6 UG/y]

   8. (∃z)Wz ⊃ (x)Mx [7 Equiv E]

Here is the sort of invalid argument which is blocked by the restriction on generalizing over a variable which was free within an assumption which remains open: Suppose that we begin with the (true) premise, “Everything in Wisconsin is in the Midwest.” Were it not for the restriction on UG/—which is violated here in line5—we could invalidly "prove" the (false—and absurdly insular) conclusion, “If anything is in Wisconsin then everything is in the Midwest” (a false conclusion as well as a discouraging prospect).

The important EXISTENTIAL DRILL is our tool for using the quantification rules to work proofs containing existentially generalized premises.

TO USE AN EXISTENTIAL PREMISE, PROCEED THUS:

Equivalence E will routinely be used in "the existential drill" to work proofs having existentially quantified premises, since we have no rule by which an existential prefix may simply be removed. You can see why we cannot simply remove an existential prefix if you think about it in common sense terms for a moment: suppose that we know that someone is a spy (that, then, will be our premise); clearly this is not by itself sufficient to infer that the spy is Bob, or that the spy is a person with red hair. Thus, we cannot simply remove an existential prefix and infer that the premise is true of some specified individual. But because we can prove Equivalence E to be a logical truth, in certain standard circumstances we can use our existentially quantified premise in a Modus Ponens inference. [Note that what Copi calls "EI" is merely his abbreviation for the existential drill, and we will not use it since doing so would require us to memorize an altogether new set of restrictions.] To work a problem which contains an existentially quantified premise, you will typically have to resort to the standard pattern of proof displayed above. Because the sequence--following discharge of its characteristic assumption--of its last four steps is a routine which varies only slightly from proof to proof, we call the sequence "The Existential Drill." (See the discussion on using the existential drill, "An Explanation of the Existential Drill.".)

EQUIVALENCE E

Equivalence E, unlike the other three quantification rules, is a replacement rule: it may be used to rewrite a schema which is a premise, or a schema which is merely part of a premise; it may be used from right to left as well as from left to right. Since we have no rule by which an existential prefix may be removed, Equivalence E will routinely be used in "the existential drill" to work proofs having existentially quantified premises. [What Copi calls "EI" is merely an abbreviation for the existential drill, and we will not use it.]
To use Equivalence E from left to right, the schema to be rewritten must have a universal prefix as its major operator; in addition, the schema must have a horseshoe as its operator next in rank; and finally, the major operator must not bind any variable in the consequent which follows the horseshoe—the consequent being represented here by "P." If these three conditions are satisfied, you may then rewrite the schema as follows: delete the universal prefix; substitute (throughout) for the variable originally bound by that prefix any variable which will satisfy the conditions in the following clause; finally, attach to the antecedent, as its major operator, an existential prefix which binds every occurrence of the variable you have substituted and which binds nothing else. Notice that when you use Equivalence E from left to right, you take a schema whose major operator is a universal prefix, and rewrite it as a schema whose major operator is a horseshoe.
To use Equivalence E from right to left, the schema to be rewritten must have a horseshoe as its major operator, and its antecedent must have an existential prefix as its major operator. If these two conditions are satisfied, you may then rewrite the schema as follows: delete the existential prefix from the antecedent; substitute (throughout) for the previously bound variable, any variable so long as it does not occur free in the consequent—in what is represented here by "P"—and so long as it will satisfy the further conditions in the following clause; and finally, affix to the schema a universal prefix which binds every occurrence of the variable you have substituted and which binds nothing else, changing the punctuation to make the new prefix the major operator of the schema. Notice that when you use Equivalence E from right to left, you take a schema whose major operator is a horseshoe, and rewrite it as a schema whose major operator is a universal prefix.
Copi discusses Equivalence E on page 96 of our text, where he proves it to be a logically true equivalence. Although Equivalence E will at first strike you as arcane, magical, and counter-intuitive, you will later discover that it fits wonderfully well with our intuitive recognition of two sorts of English phrases as alternative paraphrases of each other. (Later, you might return to the matter when you look at Section 2 on L 38 on symbolizing in predicate logic.)

QUANTIFER NEGATION
QN

Note that this is a replacement rule: accordingly, one can use it to rewrite part of a schema, and can go from right to left or left to right.
~() ≡ (∃) ~
~(∃) ≡ () ~

The rationale behind Quantifier Negation involves the the Square of Opposition which recognizes that there is a logical connection between universally general statements and existentially general statements; for example, a universally general affirmative sentence is equivalent to the denial of an existentially general negative sentence. Using Quantifier Negation, together with the other replacement rules with which we are already familiar, we are able to rewrite a universally general affirmative sentence as the denial of an existentially general negative sentence (and vice versa):
"(x)(Hx ⊃ Mx) ≡ ~(∃x)(Hx · ~Mx);" in English, the sentence, "All humans are mortal," is equivalent to the sentence, "It is false that some humans are not mortal."
To express this relationship generally, “ (x)x ≡ ~(∃x)~x ” and “ ~(x)x ≡ (∃x)~x ”. (Refer to the discussion on the Square of Opposition, below, for further discussion.)

(The following is not itself a rule of quantification, but it is useful in understanding QN and why we symbolize sentences as we do.)

The Square of Opposition
The statements in English on the opposing ends of each arrow in the figure, below, contradict each other. Accompanying each English sentence is the schema which symbolizes it. Any statement must be equivalent to the negation of its contradictory (as an instance of double negation). As shown in the yellow boxes, when we take the negation of a schema's contradictory and rewrite it using Quantifier Negation together with other, familiar replacement rules, we will end with the schema itself.

In the symbolization of a typical English sentence (or sub-sentence)—one which does not sound artificially contrived² —when its prefix is universal, the major operator within the prefixed expression will be a horseshoe; when its prefix is existential, the major operator within the prefixed expression will be a dot. This is an essential feature to notice: it is required by the "Square of Opposition," which Copi discusses on pages 66-9. In order that the symbolization of "Some S is not P" will turn out to be the contradictory of the symbolization of "All S are P," the copula—the verb (usually "is" or "are") connecting the subject to the predicate—must be represented by a "· " in existential sentences and by a "⊃" in universal sentences.
As a result, symbolic logic construes universally general sentences such as "All humans are mortal" and "No humans are mortal" as not implying that humans exist. Thus, "All humans are mortal" and "No humans are mortal" would both be true were no humans to exist, so that the existentially general sentences which do imply existence, "Some humans are not mortal" and "Some humans are mortal," would both be false. The universally general sentence, "All S's are P," is represented by the schema, "(x)(Sx ⊃ Px)," which should be understood as saying, "For any individual, if it is S then it is P."
Consider the following typical examples:

(∃x)[Fx · (∃y){Ky · (z)[(∃v)Mxzv ⊃ Dxzy]} ]
_{(∃x)----> · (∃y)------> ·
(z)---------------> ⊃}

(x) [Fx ⊃ (y){Ky ⊃ (∃z)[(∃v)Mxzv · ∼Dxzy]} ]
_{(x)-----> ⊃ (y)-----> ⊃
(∃z)----------------> ·}

Notice how each existential prefix typically dictates a dot as the operator next in rank which it controls, and how each universal prefix typically dictates a horseshoe as the operator next in rank which it controls; this general rule provides an effective way to "proof-read" one's first-approximation of a symbolization. (Incidentally, the two sentence schemata used as examples contradict each other; if you prefix a tilde to the one schema and then bring it to the inside, you will derive the other.)

Footnote to discussion of Square of Opposition:

² Consider the peculiar-sounding sentence, "Something exists such that if it is S, then it is P;" this would be symbolized as "(∃x)(Sx ⊃ Px)." Notice that while the sentence does imply that something exists, it does not assert that this existing individual actually has the property, S; rather, it asserts that if the individual has the property, S, then it also has the property, P. On the other hand, the sentence schema, "(x)(Sx . Px)," asserts "For any individual, it is both S and P"—that is, "Everything is both S and P;" unlike the familiar sentence, "All S is P," this one does imply that something has the property, S.

PRINCIPLES OF IDENTITY

Where represents either some variable or some constant:

Id.

Note that this is an inference rule; the premises of an argument must be generated by one of these patterns.

=
———
∴
~
—————
∴ ~( = ) =
———
∴ = P
———
∴ =

The idea behind the Principles of Identity comes from Leibniz who reasoned that if two individual symbols refer to the same individual, then whatever property is true of the one must also be true of the other. Common sense tells us that if you are talking about an individual whom you call "Ralph" and someone else is talking about an individual he calls "the Midnight Flasher," then if the one has a characteristic which the other lacks (e.g., hair, or perhaps, modesty), these terms cannot refer to the same individual; police investigators use this principle when they rule out a suspect because he lacks some characteristic of the culprit (e.g., since they have different fingerprints, they cannot be the same individual). The principles add a rule of commutation for identity expressions, and mark as a tautology the claim that an individual is identical with itself, which accordingly will validly follow from any premise whatever.