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Pattern Logic Primer
Quantification
A logical term is a quantified modal concept. There are ten ways to quantify a logical term using pattern.
Composing Quantifiers in Pattern
What is a Quantification in Pattern Logic?
A logical term is composed of a monadic (purple), a dyadic (blue), and a triadic (green) occasion. The monadic occasion expresses the modality of the logical term. The dyadic and the triadic occasion determine the quantification of the logical term. The locus of quantification is the triadic occasion, but its meaning depends on both the monadic and dyadic occasions. There are four types for the monadic occasion, five types for the dyadic occasion and two directions for the triadic occasion, leading to 40 distinct patterns of quantified modal terms in Pattern Logic (the diagram displays only the quarter of necessary modality). The transitive reasoning function of these quantifiers is captured in two properties called continuity and directionality.
Continuity Property of Quantifiers
There are five pattern types for the continuity property:
- Reflexive: the thing and use channel inputs originate from the same occasion.
- Cyclic: the dyadic occasion has non-absent (may be a self-input) thing and use channels, and with the triadic occasion, it creates a loop-back to the monadic occasion.
- Acyclic: the dyadic occasion has non-absent thing and use channels, and with the triadic occasion, it does not create a loop-back to the monadic occasion.
- Dynamic: the dyadic occasion has an absent thing or use channel and the channel it has from the monadic occasion, it also supplies to the triadic occasion.
- Broken: the dyadic occasion has an absent thing or use channel and the channel it does not have from the monadic occasion, it supplies to the triadic occasion.
Directionality Property of Quantifiers
There are two pattern types for the directionality property:
- Distributed: in the triadic (quantification) occasion, the monadic (modal) occasion is limited by the dyadic (derivative) occasion.
- Undistributed: in the triadic occasion, the monadic occasion limits the dyadic occasion.
Quantification and Inference
An Example of Limitation's Transitivity
The relational properties of transitivity and asymmetry found in limitation allow us to relate the quantifiers to each other.
For example, we intuitively know that if I predicate something about "all blue", such as "all blue is relaxing to me", then I am also predicating something about "any blue", such as "any blue is relaxing to me".
Alternatively, if I assert that "some blue is relaxing to me", I do not imply anything about whether "all blue is relaxing to me".
These intuitions are captured in the asymmetric transitive relations between the pattern logic quantifiers.
Quantifier Inferences
From the transitivity types we can derive rules of quantifier inference. Testing the intuition of the relations between these quantifiers and understanding how these relations are captured in pattern is crucial to reasoning in pattern. This table summarizes the transitive relations between the ten quantifications. The columns indicate a given quantifier and the rows indicates a transitive target quantifier. The grid then represents whether a transitive inference from the given to the target quantification (of the same modal term) is implied (⊢), not implied (⊣) or potentially implied (?).
This grid is a restatement of the transitivity relations found in the graphical representation of patterns shown in the figure at the top of this page.
A “potential” transitivity relation is the summary result of the distribution pattern for quantifiers with dynamic or broken continuity. It is linked to the absence of existential import for these quantifier inferences and this will be one characteristic of abductive reasoning.
