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A logical term is a *quantified modal concept*. There are * ten* ways to

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

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.

There are two pattern types for the * directionality *property:

**Distributed**: in the triadic (quantification) occasion, the monadic (modal) occasionthe dyadic (derivative) occasion.**is limited by****Undistributed**: in the triadic occasion, the monadic occasionthe dyadic occasion.**limits**

**Individual (@)** – an exemplar of the concept (* “each blue*”)

**Total (Σ )** – a complete collection of the concept (“* every blue*”)

**Partial (∂)**– part* of the concept (“* some/some of blue*”)

* * In the terminology of mereology, the pattern of interpretation for partial quantification is of the ambiguous general parthood***,*** rather than proper parthood. This means that a partial quantification of a concept may be equal to the concept itself, or it may be something less.*

**Non-existential (∄)** – non-existence of the concept (“* not any blue*”)

**Existential (ꓱ)** – existence of the concept (“* any blue*”)

**Universal (Ɐ)** – a complete set** of the concept (“* all blue*”)

* ** The "every" of Total Quantification is a more expansive concept than the "all" of Universal Quantification. It may accommodate concepts that go beyond the mere concept being quantified but will at least accommodates everything that is associated with the concept. As indicated by the renderings, this may also mean that the universal is a "complete set" of unique elements falling under the concept, while the total is a "complete collection" with potential duplication of elements falling under the concept. The distinction is found in their dyadic occasions. The extent of total quantification is an assertion, which is an approximation of the concept; and the extent of the universal concept is (counter-intuitively) a subordinate of the concept (making it equivalent with the concept).*

**Strong greater inequality (>)** – more than the concept (“* more than blue*”)

**Strong lesser inequality (<)** – less than the concept (“* less than blue*”)

**Weak greater inequality (≥)** - at least the concept (“* not less than blue*”)

**Weak lesser inequality (≤)** – at most the concept (“* not more than blue*”)

**e ≡ t ≈ e **

*"entity e means translation t approximates entity e"*

**e ≡ t ≈ [e1] **

*"entity e means translation t approximates (this entity e1)"*

**e ≡ t ≈ ¬e1 **

*"entity e means translation t approximates not entity e1"*

**e ≡ t ≈ e ⊇ e1 **

*"entity e means translation t approximates entity e limits entity e1"*

**e ≡ t ≈ e1 ⊇ e **

*"entity e means translation t approximates entity e1 limits entity e"*

The * Generic Quantifiers* are a transformational grouping that consists of:

**individual**(each) and**total**(every) quantification.

These quantifiers correspond to a limitation between a derivative **assertion** occasion and a modal monadic occasion.

The * General Quantifiers* are a transformational grouping that consists of:

**partial**(some),**non-existential**(not any),**existential**(any), and**universal**(all) quantification.

These quantifiers correspond to a limitation between a derivative **particular**, **complement**, **superordinate**, or **subordinate **occasion and a modal monadic occasion.

The * Relational Quantifiers* are a transformational grouping that consists of:

**strong greater inequality**(more than),**strong lesser inequality**(less than),**weak greater inequality**(not less than), and**weak lesser inequality**(not more than).

These quantifiers (also) correspond to a limitation between a derivative **particular**, **complement**, **superordinate**, or **subordinate **occasion and a modal monadic occasion.

The quantification diagram above is slightly misleading when it comes to modality. These diagrams more accurately reflect the reality of modal quantification transformations. Taking just general quantifications (green) of universal, existential, particular and non-existential there are two transformational groupings involved and they "entangle" the modalities of necessity and contingency as shown.

Pattern opposites (involutions) are arranged diagonally from each other within each square.

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.

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*.

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