Excursus E QUANTIFICATION TRANSFORMATIONS

Consider the situation expressed by the statement that a bird is a kind of animal. The logician may express this generalization by using the predicate calculus:
(x) BIRD(x) ANIMAL(x). “birds are animals”
It is also possible to express this without the quantifier as a configuration of two sets:
{ x | BIRD(x) } { y | ANIMAL(y) } “birds are animals”
Now with situational logic the analyst has further tools to allow him to express this relationship as the simultaneous validity of four propositions about these two predicates each expressed using one of the characterizing formal predicates:
Q-predicateAlternative Q-predicate
NOT(EXCLUSIVE(BIRD,ANIMAL))NOT(EXCLUSIVE(ANIMAL,BIRD))
IN(BIRD,ANIMAL)CONTAIN(ANIMAL,BIRD)
NOT(CONTAIN(BIRD,ANIMAL))NOT(IN(ANIMAL,BIRD))
NOT(EXHAUSTIVE(BIRD,ANIMAL))NOT(EXHAUSTIVE(ANIMAL,BIRD))
So instead of a single particular statement about propositions involving some generalized individual the logician makes four general (primitive) statements about two particular predicates or sets of individuals. Here predicate expressions stand for the sets they define. In the same way it is possible to let the symbol for a set configuration stand for the respective relationship between predicates, at least such predicates as have individual objects or sets as arguments.
inclusion(BIRD,ANIMAL) = +
BIRD ANIMAL “birds are animals”

Constraints on quantification.  
Logically the expression of this generalization brings with it certain constraints. The logician may always substitute a designation for a constant in the place of x. In other words if one has the proposition (x) BIRD(x), one may derive the proposition BIRD(a). Now, if the linguist uses “Pred” to stand for any predicate symbol and uses “x” for any variable, one may state the replacement transformation as:
UNIVERSAL INSTANTIATION (UI):
base:(x) Pred(x)
transform:Pred(a)
condition:a is any individual symbol
By “a” in this transformation the linguist means any individual symbol at all. Under the condition that it is known that a particular individual symbol refers to any arbitrarily selected individual (UI introduced it), the linguist may also write the following rule:
UNIVERSAL GENERALIZATION (UG):
base:Pred(a)
transform:(x) Pred(x)
condition:a has been arbitrarily selected
Provided one heeds appropriate constraints, one may also transform from and to an existential statement:
EXISTENTIAL INSTANTIATION (EI):
base:(x) Pred(x)
transform:Pred(a)
condition:a has no previous occurrence in derivation
EXISTENTIAL GENERALIZATION (EG):
base:Pred(a)
transform:(x) Pred(x)
condition:a is any individual symbol

Quantification reformulated.  
There is, however, a serious problem with this formulation. The base is much too restrictive. Consider the situation where the quantifier applies to an argument that appears in more than one predicate. The linguist will have to modify these transformations. In particular it will be advisable to formulate UI so that from
(x) BIRD(x) ANIMAL(x), “birds are animals,”
where the x appears twice, one may derive
BIRD(a) ANIMAL(a) “this bird is an (this) animal.”
The term "Pred" in the base will have to apply to all the predicates in an expression which have “x” as an argument. This is what logicians call the scope of the quantifier. What they are saying is that for the sake of quantification, logic requires that the analyst conceptualize as a single predicate all the expressions that share the same variable as an argument:
(x)IMPLIES-BIRD-ANIMAL(x).
At the very least one must interpret quantification so as to limit its argument to a particular scope. The logician must use such generalized quantification transformations with extreme caution. The use of an additional calculus, the lambda-calculus, makes this possible. This extension was developed by Alonzo Church before Stephen Cole Kleene brought it to light in 1935. Unfortunately the lambda-calculus is presently beyond the scope of this work.

Universal quantifier transformations revised.  
There are only sixteen possible ways for the logician to relate two propositions as to their logical truth values. There are sixteen homologous ways for one to relate two predicates as to their logical situations. One may view the process described by the four quantification transformations from the same perspective. The linguist may match the sixteen symbols for the logical operators (LOPi) with the sixteen symbols for the possible set configurations (SCi). For combinations of two predicates one may therefore formulate the universal quantification transformations as:
UNIVERSAL INSTANTIATION (UI2):
base:Pred1 + SCi + Pred2
transform:Pred1(a) + LOPi + Pred2(a)
UNIVERSAL GENERALIZATION (UG2):
base:Pred1(a) + LOPi + Pred2(a)
transform:Pred1 + SCi + Pred2
Existential quantifier transformations revised.  
There are four characterizing properties for the sixteen possible set configurations. These four Q-predicates serve to assert the existence or non-existence of elements relating to two sets. The linguist may express this equivalence in describing the reasoning process from a general statement involving these properties to the specific and vice versa.
EXISTENTIAL INSTANTIATION (EI2):
condition:a has no previous occurrence in derivation
1) base:NOT(EXCLUSIVE(Pred1,Pred2))
transform:Pred1(a) Pred2(a)
2) base:NOT(IN(Pred1,Pred2))
transform:Pred1(a) ~Pred2(a)
3) base:NOT(CONTAIN(Pred1,Pred2))
transform:~Pred1(a) Pred2(a)
4) base:NOT(EXHAUSTIVE(Pred1,Pred2))
transform:~Pred1(a) ~Pred2(a)
EXISTENTIAL GENERALIZATION (EG2):
condition:a is any individual symbol
1) base:Pred1(a) Pred2(a)
transform:NOT(EXCLUSIVE(Pred1,Pred2))
2) base:Pred1(a) ~Pred2(a)
transform:NOT(IN(Pred1,Pred2))
3) base:~Pred1(a) Pred2(a)
transform:NOT(CONTAIN(Pred1,Pred2))
4) base:~Pred1(a) ~Pred2(a)
transform:NOT(EXHAUSTIVE(Pred1,Pred2))
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