Theme One Program • Logical Cacti

Author: Jon Awbrey

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• Overview • Blog • Exposition • Logical Cacti • Appendices • Document History •

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Introduction

Up till now we've been working to hammer out a two-edged sword of syntax, honing the syntax of cactus graphs and cactus expressions and turning it to use in taming the syntax of two-level formal languages.  But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words interpretation, meaning, semantics, and their ilk take on so many different meanings from one moment to the next of their use.  A dedicated neologician might be able to think up distinctive names for all the aspects of meaning and all the approaches to them that concern us, but I will do the best I can with the common lot of ambiguous terms, leaving it to context and intelligent interpreters to sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that its graphs bear at least two distinct interpretations as logical propositions.  The two interpretations concerning us here are descended from the ones C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics.

Existential Interpretation

Table 1 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

${\text{Table 1. Existential Interpretation}}$

Entitative Interpretation

Table 2 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

${\text{Table 2. Entitative Interpretation}}$

Mathematical Structure and Logical Interpretation

The main things to take away from Tables 1 and 2 are the following two ideas, one syntactic and one semantic.

• The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.

• There are two ways of mapping these compositional structures into the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows.

• The node connective joins a number of component cacti $C_{1},\ldots ,C_{k}$ to a node:

• The lobe connective joins a number of component cacti $C_{1},\ldots ,C_{k}$ to a lobe:

The two ways of mapping cactus structures to logical meanings are summarized in Table 3, which compares the entitative and existential interpretations of the basic cactus structures, in effect, the graphical constants and connectives.

${\text{Table 3. Logical Interpretations of Cactus Structures}}$

Transformation Rules and Equivalence Classes

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming cacti among themselves and partitioning the space of cacti into formal equivalence classes.  The transformation rules and equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

• A reduction is an equivalence transformation which applies in the direction of decreasing graphical complexity.

• A basic reduction is a reduction which applies to a basic connective, either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

• A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.

• A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.

That is roughly the gist of the rules.  More formal definitions can wait for the day when we need to explain their use to a computer.

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• Overview • Blog • Exposition • Logical Cacti • Appendices • Document History •

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