Propositional Logic in Patterns

The governing principle of pattern logic is that unifying logic in pattern is about one general form for every logical statement. When we come to the idea of a proposition, it is important to realize that even the “atomic proposition” is not exempt from this principle. In other words, even an atomic proposition must be understood as a logical statement in pattern logic. In propositional logic, the atomic proposition is a given, it is the starting point. And if the proposition has some meaning, this is a “non-logical” concern in propositional (and predicate) logic.

The requirement that every element of pattern logic should conform to the general form of a logical statement is quite pragmatic from a computational perspective. Indexing all logical statements effectively identifies all components of any logical system within an ADEPT LION instance, and thereby distinguishes logical elements from every other expression found in that instance. This effectively isolates the workings of logic, and subsequently the workings of reasoning within the ADEPT LION instance.

Simple Logical Connectives

We may start with these elements of a propositional logic:

• A proposition
• The complement of a proposition
• TRUE
• FALSE

Then we add these logical connectives:

• conjunction
• nonconjunction
• disjunction
• nondisjunction
• implication
• nonimplication
• converse nonimplication
• converse implication

The most fundamental and unifying requirement of each of these listings is the relations of opposition that govern them. A proposition’s opposite is the complement of that proposition. The opposite of TRUE is FALSE. The opposite of conjunction is nonconjunction … and so on. Pattern Logic understands each of these relations of opposition as an expression of one pattern transformation known as involution.

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If we were to map the idea of an atomic concept to a pattern in ADEPT LION, it would be of the type “shadow entity of aether (SEH)” and of one of the four subtypes for a “monadic occasion” which will be one of the four alethic modalities. The modality that appears to fit the most common understandings of atomic propositions will be the necessary occasion (SEH*-) but pattern logic would also accommodate “possible propositions (SEH**)”, “impossible propositions (SEH — )”, and “contingent propositions (SEH-*)”. But since there is not a well-established understanding of a modal propositional logic, we can leave further discussion of this aside for now.

The patterns for a proposition and the complement of that proposition will have this necessary monadic occasion in common. The proposition and the complement of that proposition each have a pair of two patterns available for their expression.

Figure 1: Single Concept Junctive Simple Connectives and their Opposites. Proposition A and Proposition not-A (larger shape).

The larger rounded squares in figure 1 correspond to the two pairs of logical statements that express a proposition and its complement. These patterns are categorized in the following manner:

1. Single Concept — the initial and subsequent terms of the statement are derived from the same concept (the necessary occasion “A” at the center of the diagram).
2. Junctive — the quantification of the initial and subsequent terms of the four copulas represented by the larger rounded squares found in each quadrant are the same in each case (both existential(ꓱA), or both non-existential (∄A)) The alternative in which the quantification is different is an “implicative” type of statement.
3. Simple — the terms of each statement are not themselves statements (this would make the pattern “complex”)
4. Connectives — each connective statement is composed using only existential or non-existential quantification of its terms and has either an overlap or union copula.
5. Opposites — the dashed lines on the diagram represent an axis of transformation. A shape (representing an occasion) on the left side is positioned opposite a corresponding involution of its pattern on the right side (.ie the occasion (and its supporting occasions) expressing nonconjunction is the involution of the occasion (and its supporting occasions) expressing conjunction).

The existence of any of these patterns (the diagram demonstrates four) establishes either the (necessary) proposition A, or the (necessary) proposition not-A. For the two expressions of proposition A, we can distinguish these as a statement of conjunction and a statement of disjunction. For the two expressions of the proposition not-A, we can distinguish these as a statement of nonconjunction and a statement of nondisjunction.

These “atomic propositions” as developed so far, have no truth value associated with them. The next patterns we will introduce are for the opposing concepts of TRUE and FALSE. Similarly, we find two statements that can express each of these concepts.

Figure 2: Simple Concept Implicative Simple Connectives and their Opposites. True and False (larger shapes).

Now we see that introducing the TRUE value involves taking a concept and making an “implicative” type of statement about itself. If the statement is an implication or a converse implication, it is the expression of TRUE. If the statement is oppositely, a nonimplication or a converse nonimplication, it is the expression of FALSE. The reason for duplicate expressions for each pair of opposition can now be verified to actually be an arbitrary decision regarding whether these statements, having the same logical terms, choose to arrange those terms within the statement. The implication statement is a non-significant rearrangement of the converse implication statement and the nonimplication statement is a non-significant rearrangement of the converse nonimplication statement.

Aren’t these means of establishing the elements of propositional logic unnecessarily complex? Observe first that the use of pattern involution in this development provides a coherence between the elements of propositional logic. If there is an axiomatic requirement at play here, it is not specific to the needs of propositional logic, but rather of the broader requirements of pattern logic, specifically the requirement that every element of a logical system conform to the general form of a logical statement. So while it is reasonable to acknowledge criticism about introducing this “unnecessary” degree of complexity, it will also be shown to be quite understandable why we should quickly dismiss this critique.

The reason for this dismissal becomes more apparent as we move to a very similar set of patterns which conform to the general form of a logical statement in pattern logic, but involve double concepts rather than a single concept. We start with the junctive statements.

Figure 3: Dual Concept Junctive Simple Connectives and their Opposites. Propositional connectives of conjunction/nonconjunction and disjunction/nondisjunction (larger shapes).

As figure 3 demonstrates, these represent the first half of the eight basic logical connectives. As anticipated, nonconjunction is in opposition to conjunction and nondisjunction is in opposition to disjunction. We will observe that as they are introduced in figure 3, there is no representation of validity or truth values for these simple connective statements. Rather, they have necessary modality.

The remaining four basic logical connectives are found in figure 4.

Figure 4: Dual Concept Implicative Simple Connectives and their Opposites. Propositional connectives of implication/nonimplication, converse nonimplication/ converse implication (larger shapes).

Again, as anticipated, implication opposes nonimplication and converse nonimplication opposes converse implication. As above, there is no assertion of validity (TRUE/FALSE) in these patterns.