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2| The linguist may characterize the expressions of a calculus by applying the generative portion of the GT grammar — phrase structure rules. These rules describe the structure of a language by giving the constituency of its parts explicitly. The computer scientist calls these rules the Backus-Naur Form (BNF) and his object described the aggregate model. The linguist orders these rules strictly, each one describing the constituency of an element in a previous rule. It is useful to loosen this restriction in the case of the initial element, however, to open up the possibility of recursion. The initial element, at least, may be a constituent part of itself or of one of its parts. Another difference for the linguist is the order of the parts aggregated that is implicit in the form of the rule |
PS-rules for definitions and entailments.In §4-2 I introduced two types of expressions of the propositional calculus: the definition and the entailment. |
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2| These two forms are so similar that the linguist may describe them with a single phrase structure rule.
The PS-rule itself has a form similar to the form of an entailment.
In a GT grammar, however, this form means that 1,
the element to the left of the arrow, is composed of or consists of 2, the element to the right.
This interpretation means that typically a PS-rule has multiple distinct elements to the right of the arrow. |
| Now, to describe the two kinds of propositions in a PS-rule:
A proposition (Pi) is composed of three parts: a proposition,
one of the symbols (either a “” or a “”, i.e., a connective, and a second proposition.
To symbolize the combination of ordered elements (a sequential concatenation) at the right, simply use a plus (+) sign. |
| Notice that this rule is recursive. The element on the left is defined in terms that refer back to it. |
| This rule for the connective gives two possibilities for its realization — each one described autonomously, i.e., the symbol stands for its own form in the object language. In this case there will be no rules to develop these symbols further. These symbols, which I have put in light yellow boxes, are entries in the lexicon, where their semantics is described on an individual basis. Linguists call such elements as have lexical entries to the right terminal constituents. My convention is to place terminal constituents in light green boxes. The rule that defines the proposition has terminal constituents as well, viz., any lowercase letter. |
| It should be mentioned as well that the arrow of implication for terminal constituents is a bit of an overgeneralization. It turns out that the lexical items cannot be so simply related to the terminal constituents. The better formalism will be introduced below. |
Aggregate model for PS-rules.Computer scientists have developed the Unified Modeling Language (UML) to express these concepts diagrammatically. In this language the constituent classes are diagrammed using an arrow with a diamond shaped head. The alternative structures, where braces appear in a PS-rule (or in BNF), must be considered structural sub-classes. These are in tan boxes brought together with an n-arity branch on an arrow. Even though UML shows optional elements with the sign of cardinality as a possible zero, I will, when necessary, as a more graphic representation, draw optional boxes using dashed lines. |
Transformations for the propositional calculus.In constructing a proof logicians first take one or more propositions as assumptions (also called premises), or possibly none at all. They then apply a set of derivational rules to these assumptions. Some of these rules may not even require an assumption. Some logicians call these derivational rules their rules of inference. By such a rule they may draw some (intermediate) conclusion. This allows them to take this conclusion as an assumption in applying some other derivational rule. This finally allows the argument to come to a conclusion, which if done correctly is the proposition that they wanted to prove. In this way each stage of the proof is either an assumption or a conclusion. |
The structure of a proof.Consider now how a linguist can describe the structure of an argument or proof using a phrase structure rule. A proof consists in zero or more propositions (taken as assumptions) followed by a proposition (taken as a conclusion). | |||||||||
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| The UML uses what is called the Kleene star (*) to show an unlimited cardinality, i.e., the fact that there may be several instances of a class in a set. It is sometimes also possible to “normalize” the classes so that there is only one box for a particular class. |
| Consider, for example, the principle of reasoning or rule of inference called modus ponendo ponens.
Suppose the logician takes the propositions and as assumptions.
This rule allows one to conclude the proposition .
This formalization of an argument does not capture the fact that the truth of the conclusion depends on the truth of the assumptions.
The validity of the argument does not depend on this truth.
The truth of the assumptions effects only the usefulness of the argument.
The argument decides on the appropriateness of asserting a proposition.
The above rules are still not enough to describe this.
Somehow the linguist must capture the semantics to describe the contents of each P. |
Rule of inference as a transformation.Linguists formulate each of the transformations of their GT grammar to mean that given certain forms they may derive certain other forms by applying that transformation. It is possible to use this notation to symbolize a step in a proof that is related by some principle of reasoning or rule of inference. There are two parts to a transformation of the GT grammar. The linguist would translate each assumption as a base of the transformation. One would translate the conclusion as the transform of the transformation. The rule of transformation is the linguist’s way of stating that given the forms in the bases one may derive the transform from them by this particular rule. |
Replacement rule for propositional calculus.The propositional calculus introduced here allows replacement in three cases: |
| REPLACEMENT 1 (REP1,MPP) | |
|---|---|
| base1: | P1 |
| base2: | P1 P2 |
| transform: | P2 |
| REPLACEMENT 2 (REP2) | |
|---|---|
| base1: | P1 |
| base2: | P1 P2 |
| transform: | P2 |
| REPLACEMENT 3 (REP3) | |
|---|---|
| base1: | P1 |
| base2: | P2 P1 |
| transform: | P2 |
| Consider the proof where the logician assumes
and and ,
and from this concludes .
In this example the linguist would apply REP1 twice.
First take and
as the bases to conclude as the transform.
One may then take and to yield .
As a rule of inference REP1 is also called modus ponendo ponens (MPP). |
Alternative replacement rule.In devising this particular proof some logicians may prefer to use a different rule of inference, say the principle called the hypothetical syllogism. As with any rule of inference, the linguist would give this principle in the form of a transformation: |
| REPLACEMENT HS (REP-HS) | |
|---|---|
| base1: | P1 P2 |
| base2: | P2 P3 |
| transform: | P1 P3 |
| For this proof the linguist now uses one application of REP-HS, thus saving two applications of REP1. In the slightly more developed predicate calculus there is in practice no end to the number and variety of transformations that the linguist could devise to help in constructing useful proofs. |
