Excursus D A GT GRAMMAR FOR QUANTIFICATION, LOGICAL OPERATORS & PRIMITIVE SENTENCES

PS-rules for the quantifiers.  
In the proposition containing operators, there are two lists of arguments which partially overlap. Some of the arguments — the bound variables — must appear in both lists. This restriction is particularly difficult to capture in PS-rules and I am beginning to wonder if maybe such rules are not up to the task. The present attempt is not fully successful in describing the quantifiers for the following reasons: Rule 1 limits the expression to a single predicate. Rules 2, 3 and 4 do not exclude a collision between i and j.

Transformations with variables.  
In transformations one may use a variable in much the same fashion that one might use any symbol of the metalanguage. For the replacement that may take place based on a definition there are these two transformations:
DEFINITION1 (DEF1)
base1:P1
base2:DEFINE(P1,P2)
transform:P2
DEFINITION2 (DEF2)
base1: P1
base2: DEFINE(P2,P1)
transform: P2
Using variables of the calculus these would read:
DEFINITION1 (DEF1)
base1:x
base2:DEFINE(x,y)
transform:y
DEFINITION2 (DEF2)
base1:x
base2:DEFINE(y,x)
transform:y
There is a convention to use Greek letters as symbols for a variable binary value: a, b, etc. For an integer value, when quantified “n” is available and for a real value, “r.” For additional number values these letters receive subscripts.

PS-rules for negation.  
To account for negation the GT grammar for propositional calculus would need another terminal constituent and an adjustment to the PS rule describing a proposition:
There is at least one transformation to interpret the negative symbol:
LAW OF DOUBLE NEGATION (DNN)
base: ~ ( ~ (P))
transform:P
The parentheses in the above expression are a part of the propositional calculus. They are to help delineate the structure of the expression; this is how one may indicate which parts of the string to take as propositions.

PS-rule for logical connectives.  
When the logician writes a logical operator as a symbol, one is thinking of it as a logical connective () of an extended propositional calculus. In this form one places it between the arguments:
(1, 2) = + (1, 2) 1 2.
In effect the propositional calculus takes on much of the power of the predicate calculus. This means that the linguist’s GT grammatical description needs one of its PS rules enlarged.

Transformations for primitive sentences.  
It is possible to give any primitive sentence, in particular those of the last section, in the form of a transformation rule. The null premise of a primitive sentence is a simple unanalyzed proposition. This becomes the base of the transformation. Different logicians usually select their primitive sentences differently. A given sentence some would designate as primitive and others would express in a rule of derivation. Such a rule in a GT grammar of the propositional calculus would appear as on the left. I translate this to a corresponding rule in the predicate calculus such as given on the right.
LAW OF NON-CONTRADICTION (NC)
base:Pbase:P
transform:~(P ~P) transform:NOT(AND(P,NOT(P)))
LAW OF EXCLUDED MIDDLE (EXC-MID)
base:Pbase:P
transform:PV ~P transform:OR(P, NOT(P))
LAW OF DOUBLE NEGATION (DNN)
base:Pbase:P
transform: ~(~P) P transform:ENTAIL(NOT(NOT(P)),P)
The linguist may also adapt this primitive sentence to the form of a rule of derivation. By MPP the law of double negation may become:
LAW OF DOUBLE NEGATION (DNN)
base:NOT(NOT(P))
transform:P
This means in fact that the linguist may translate any primitive sentence expressing an entailment by MPP so as to have a more complicated base.
DE MORGAN'S LAWS-1 (DML)
base1:P1
base2:P2
transform:ENTAIL(NOT(OR(P1,P2)), AND(NOT(P1), NOT(P2)))
By MPP this may become:
DE MORGAN'S LAWS-1 (DML)
base:NOT(OR(P1, P2))
transform:AND(NOT(P1), NOT(P2))
DE MORGAN'S LAWS-2 (DML)
base1:P1
base2:P2
transform:ENTAIL(NOT(AND(P1, P2)), OR(NOT(P1), NOT(P2)))
By MPP this may become:
DE MORGAN'S LAWS-2 (DML):
base:NOT(AND(P1, P2))
transform:OR(NOT(P1), NOT(P2))
With this we see how the linguist can express primitive sentences in terms of GT transformations of the predicate calculus. Some additional rules of derivation that logicians commonly use are:
ADDITION (ADD)
base:P1
transform:OR(P1, P2)
CONSTRUCTIVE DILEMMA (CD)
base1:AND(ENTAIL(P1, P2), ENTAIL(P3,P4))
base2:OR(P1, P3)
transform:OR(P2,P4)
SIMPLIFICATION (SIMP)
base:AND(P1, P2)
transform:P1
CONJUNCTION (CONJ):
base1:P1base1: P2
base2:P2base2: P1
transform:AND(P1,P2) transform:AND(P1,P2)
The linguist may also express primitive sentences as GT transformations of the functional calculus, for example:
LAW OF NON-CONTRADICTION (NC)
base: A
transform:negation(conjunction(A, negation(A) = +) = +) = +
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