[CG] Re: Transformations of Logical Graphs

Transformations of Logical Graphs • 12 • https://inquiryintoinquiry.com/2024/05/16/transformations-of-logical-graphs… Re: Transformations of Logical Graphs • 8 • 9 • 10 • 11 • https://inquiryintoinquiry.com/2024/05/12/transformations-of-logical-graphs… • https://inquiryintoinquiry.com/2024/05/13/transformations-of-logical-graphs… • https://inquiryintoinquiry.com/2024/05/14/transformations-of-logical-graphs… • https://inquiryintoinquiry.com/2024/05/15/transformations-of-logical-graphs… All, Taking from our wallets an old schedule of orbits, let's review the classes of logical graphs we've covered so far. Episode 9. Self-Dual Logical Graphs — ==================================== Episode 9 dealt with four orbits of “self-dual logical graphs”, whose text expressions are x, y, (x), (y), respectively. The logical graphs whose text expressions are x, y, (x), (y) denote the boolean functions f₁₂, f₁₀, f₃, f₅, in that order, and the value of each function f on each point (x, y) of B × B is shown in the following Table. Truth Table for Self-Dual Logical Graphs • https://inquiryintoinquiry.files.wordpress.com/2021/04/self-dual-logical-gr… Episode 10. Constants and Amphecks — =================================== Episode 10 dealt with two orbits of logical graphs called “constants” and “amphecks”, respectively. The “constant” logical graphs denote the constant functions, defined as follows. • f₀ : B × B → 0 • f₁₅ : B × B → 1 Under the Existential Interpretation: • The function f₀ is denoted by the logical graph whose text form is “( )”. • The function f₁₅ is denoted by the logical graph whose text form is “ ”. Under the Entitative Interpretation: • The function f₀ is denoted by the logical graph whose text form is “ ”. • The function f₁₅ is denoted by the logical graph whose text form is “( )”. The “ampheck” logical graphs denote the ampheck functions, defined as follows. • f₁(x, y) = NNOR(x, y). • f₇(x, y) = NAND(x, y). Under the Existential Interpretation: • The function f₁(x, y) = NNOR(x, y) is denoted by the logical graph with text form (x)(y). • The function f₇(x, y) = NAND(x, y) is denoted by the logical graph with text form (xy). Under the Entitative Interpretation: • The function f₁(x, y) = NNOR(x, y) is denoted by the logical graph with text form (xy). • The function f₇(x, y) = NAND(x, y) is denoted by the logical graph with text form (x)(y). The values of the constant and ampheck functions on the points of B × B are tabulated below. Truth Table for Constants and Amphecks • https://inquiryintoinquiry.files.wordpress.com/2021/04/constants-and-amphec… Episode 11. Subtractions and Implications — ========================================== Episode 11 dealt with two orbits of logical graphs called “subtractions” and “implications”, respectively. The “subtraction” logical graphs denote the subtraction functions, defined as follows. • f₂(x, y) = y ¬ x. • f₄(x, y) = x ¬ y. The “implication” logical graphs denote the implication functions, defined as follows. • f₁₁(x, y) = x ⇒ y. • f₁₃(x, y) = y ⇒ x. Under the action of the Entitative ↔ Existential duality the logical graphs for the subtraction f₂ and the implication f₁₁ fall into one orbit while the logical graphs for the subtraction f₄ and the implication f₁₃ fall into another orbit, making these 2 partitions of the 4 functions “orthogonal” or “transversal” to each other. The values of the subtraction and implication functions on the points of B × B are tabulated below. Truth Table for Subtractions and Implications • https://inquiryintoinquiry.files.wordpress.com/2021/05/subtractions-and-imp… Regards, Jon cc: https://www.academia.edu/community/V0yAG4