Preliminary Explanation of Modeling Logic in Patterns
Modeling Systems of Logic in Pattern takes a bit of Alchemy
A preliminary explanation as to the forthcoming approach to dissecting logical systems is in order. Pattern logic is one coherent method for modeling meaning in pattern. But what any given pattern comes to mean may show up in various places in other systems and therefore, will cross lines that experts in any one logical system will surely feel compelled to protest as being too “loose and fast” with the details. This is inevitably going to be the case in many instances. But readers are encourage to consider the benefits of a unified grounding of meaning across logical systems and models before reflexively rejecting these efforts. There is certainly more work to be done here and it requires expertise and sensitivity to nuanced meaning but it is likely to be well worth the effort. The goal is to push pattern logic as far as needed to express what is required while maintaining coherence of the grounding and fidelity across systems. This won’t be achieved on the first pass, but remains a worthy goal.
What this modeling effort requires is the combining of primitive patterns into more complex ones, just like atoms combine to form molecules in chemistry. Just as we required a nomenclature to distinguish the atomic occasions, we will require some additional nomenclature to categorize and distinguish the “molecular” components such as logical concepts and operators. This is like moving beyond the periodic table of elements to express chemistry equations describing the interactions of molecules. In order to explain this increase in complexity of pattern we will introduce terms that identify the count of occasions in the pattern (.ie monadic, dyadic…) and the nature of their inter-dependency (.ie internality, externality, relationality).
Generically, we will continue to refer to the combination of more than one primitive occasion to model something more “molecular” in the "real world" as a Pattern Set.
Logical Quantifiers
Logic meets Mereology through Quantification
One such crossing of the lines between logical systems as explained above happens very early on in the exploration of ADEPT LION patterns and it is between the concepts of logical quantification and those of mereology.
By logical quantification we mean specifically the Existential (ꓱ) and Universal Quantifiers (ꓯ) of First Order Logic which were given their modern symbolic representation by American logician C. S. Peirce.
These quantifiers are actually extended in pattern logic because of their association with a related set of patterns of "dyadic externality" and what comes into view are both the components of the Aristotelian syllogism and the building blocks of Mereology, which is the philosophical study of parts and wholes, long a topic of many philosophers, but formulated in the modern era through the work (among others) of Alfred North Whitehead.
The newly introduced quantifiers expressed in pattern logic are called "Non quantification (∄)" and "Partial Quantification (∂)". Their derivation from the patterns of dyadic externality follows.
American Logician Charles Saunders Peirce
Patterns of Dyadic Externality
Predication by Superior is Universal Quantification
Any concept is predicated by its superior
Pattern Logic
Any concept alpha has a superior gamma that predicates it (occasion beta means gamma said-of alpha).
Mereology
Gamma is the "whole" of a given concept alpha.
Syllogism
Gamma is "All" of a syllogism term alpha occurring in a premise.
First Order Logic
Universal Quantification gamma means "for every alpha".
Counter-Predication by Same is Existential Quantification
Any concept is counter-predicated by its same
Pattern Logic
Any concept alpha has a same gamma that counter-predicates it (occasion beta means gamma present-in alpha).
Mereology
Gamma is the "improper part" of a given concept alpha.
Syllogism
Gamma is "A/An (existent)" of a syllogism term alpha occurring in a premise.
First Order Logic
Existential Quantification gamma means "there exists an alpha".
Counter-Predication by Non-predication is Partial Quantification
Any concept is counter-predicated by its non-predication
Pattern Logic
Any concept alpha has a non-predication gamma that counter-predicates it (occasion beta means gamma present-in alpha).
Mereology
Gamma is the "part" of a given concept alpha.
Syllogism
Gamma is "Some" of a syllogism term alpha occurring in a premise.
Predication by Disjoint is Non Quantification
Any concept is predicated by its disjoint
Pattern Logic
Any concept alpha has a disjoint gamma that predicates it (occasion beta means gamma said-of alpha).
Mereology
Gamma is the "excluded whole" of a given concept alpha: "non-alpha".
Syllogism
Gamma is "No/Not alpha" for the syllogism term alpha occurring in a premise.
Quantification Operations
Classifying Quantification Patterns of Dyadic Externality
There are ten patterns of dyadic externality, of which we are only considering four in our discussion of quantification here. Incidentally, the concepts of complement and negation from first order logic and elsewhere are also found in the dyadic externality family of patterns.
Mereology is usually axiomatized upon the initial concept of either Parthood or Proper Parthood. We see above the association in pattern between proper parthood in Partial Quantification. Rather than follow the known axiomatic approaches however, pattern logic suggests an alternative starting point for mereology in which the concepts of wholehood, excluded wholehood and improper parthood are also given as defined concepts from the outset and unified by their representation as related patterns.
What are these "relations"? First, it is that these are all found within the family of patterns that we have called dyadic externality. Second, we can understand what alterations in pattern can transform one quantification pattern into another. These transformations will have greatest significance in syllogistic logic where we consider truth-preserving rules such as the "conversion of an I-premise".
There are two alterations of pattern (regardless of the matter of truth-preservation), and they are called aversion and inversion (see diagram below). At a data-model level of ADEPT LION these involve alterations to the placement of values for the thing and use channels. We will not dive into the technical aspects of this here. But as mentioned already within syllogistic logic, these alterations form the basis of understanding truth-preserving rules of transformation like premise conversion.
Within the logical systems of mereology and first order logic, these alterations simply provide a coherence to a proposed unification method: what we might call an onto-mereo-logical system.
In the diagram above we see that the "ontological" quantifiers of first order logic, universal and existential quantification, are related by the pattern logic alteration of inversion and so are the "mereological" quantifiers or mereology, excluded wholehood and proper parthood.
Furthermore, to move between the "distributed" quantifiers of syllogistic logic, universal (ALL) and non quantification (No/Not), the pattern logic alteration of aversion is involved. Similarly, aversion relates the "undistributed " quantifiers of syllogistic logic, existential (A/An existent) and partial (Some). The terms distributed/undistributed are drawn from their usage in analyzing syllogisms.
With this brief introduction of quantification, we can proceed to an exploration of syllogistic reasoning which makes the broadest use of the patterns introduced here.
Aversion and Inversion of Pattern
Aversion (^) and Inversion (#) are transformations of primitive patterns that are truth-preserving in some cases and not in others. The self-returning dashed arrows reflect cases in which the transformation results in no effective change.
For instance, across the bottom of the diagram we see that Aversion of a None (OEH*-) results in a Disjoint (OEH-\) and subsequently, aversion of a Disjoint results in a Superior (OEH\*, and finally, aversion of a Superior leads back to a Disjoint. Inversion of a None (OEH*-) results in a Some (OEH-*) and so on...
