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Pattern Logic Primer
Logical Connectives
A logical statement with a union or overlap copula and existential or non-existential quantification of its logical terms will form a logical connective.
Junctives
Conjunction
Non-Conjunction (Alternative Denial)
Non-Conjunction (Alternative Denial)
ꓱα ∧ ꓱβ (α ∧ β)
The purple occasion at the top center of the diagram is rendered "alpha AND beta" and has necessary modality.
Non-Conjunction is the involution (opposite) of Conjunction.
Non-Conjunction (Alternative Denial)
Non-Conjunction (Alternative Denial)
Non-Conjunction (Alternative Denial)
∄α ∨ ∄β (α ↑ β)
The purple occasion at the top center of the diagram is rendered "NOT alpha AND beta" and has necessary modality.
Conjunction is the involution (opposite) of Non-Conjunction.
Disjunction
Non-Disjunction (Joint Denial)
Non-Disjunction (Joint Denial)
ꓱα ∨ ꓱβ (α ∨ β)
The purple occasion at the top center of the diagram is rendered "alpha OR beta" and has necessary modality.
Non-Disjunction is the involution (opposite) of Disjunction.
Non-Disjunction (Joint Denial)
Non-Disjunction (Joint Denial)
Non-Disjunction (Joint Denial)
∄α ∧ ∄β (α ↓ β)
The purple occasion at the top center of the diagram is rendered "NOT alpha OR beta" and has necessary modality.
Disjunction is the involution (opposite) of Non-Disjunction.
Implicatives
Material Implication
Material Implication
Material Implication
∄α ∨ ꓱβ (α → β)
The purple occasion at the top center of the diagram is rendered "IF alpha THEN beta" and has necessary modality.
Non-Implication is the involution (opposite) of Material Implication.
Non-Implication
Material Implication
Material Implication
ꓱα ∧ ∄β (α ↛ β)
The purple occasion at the top center of the diagram is rendered "alpha BUT NOT beta" and has necessary modality.
Material Implication is the involution (opposite) Non-Implication
Converse Implication
Converse Non-Implication
Converse Non-Implication
ꓱα ∨ ∄β (α ← β)
The purple occasion at the top center of the diagram is rendered "IF beta THEN alpha" and has necessary modality.
Converse Non-Implication is the involution (opposite) of Converse Implication.
Converse Non-Implication
Converse Non-Implication
Converse Non-Implication
∄α ∧ ꓱβ (α ↚ β)
The purple occasion at the top center of the diagram is rendered "beta BUT NOT alpha" and has necessary modality.
Converse Implication is the involution (opposite) of Converse Non-Implication.
Here's the Twist
Negation is not an Operation. It's a Pattern Transformation.
The astute student of logic will likely be scratching their head when they see "non-existential quantifiers" (∄) where a complement/negation sign (¬) is expected. What is going on here?
Pattern Logic understands the negation of a term not as a unary operator but rather, as a type of quantification specific to that term. This is based in the fact that a pattern transformation of involution applied to an existentially-quantified logical term results in a non-existentially-quantified term in Pattern Logic.
But this idea holds for more than just the involution of quantified terms. The entire pattern of a logical statement may also be involuted to determine its opposite or "logical complement". As mentioned in the descriptions of the simple connectives above, this produces all of the expected results so far (.ie the opposite of conjunction is non-conjunction). It is just as if one were to apply De Morgan's Laws or work it out with Venn Diagrams. They just have to treat each "∄" like a "¬".
But this is all about to change when we move to statements about statements and what it means to find their opposites.
The idea in pattern logic is a straight forward one: involution of patterns that follow the general form of a logical statement produces the "logical complement" of that statement. But the result is quite surprising!
