The Pythagorean Syllogism by Maclain Hunter

Authored by Maclain Hunter · first-party-authored · authored-foundational-record

Stored 4P source diagnostic: clear · finding no_violation_detected · admission admitted_provisionally. This intake result is separate from the theorem checks of the exact propositional sequent.

The Pythagorean Syllogism by Maclain Hunter: Since: A: The sides of a given right triangle must be connected (to close the shape and fit the definition of the shape) and therefore have a coherent relationship that is determined by the length of each side and the criteria that the triangle in question is a right triangle with an angle of 90 degrees. The logical structure the Pythagorean Theorem represents algebraically is logically equivalent to the inherent logical relationship between the sides of a right triangle to its hypotenuse (geometrically). B: Assume that the algebraic equation a ^ 2 + b ^ 2 = c ^ 2 did not describe the logical and geometric relationships between sides of a right triangle, with c representing the hypotenuse. That assumption leads to logical incoherence and geometric paradox where a theorem that has been observed to work consistently and which has been proven over 100 times and ways is not valid, coherent, or proven. Therefore we know that the Pythagorean Theorem cannot be considered false or invalid in a coherent fashion. Like the syllogism itself it is valid and logically contingent. It is going to output a proper value for c when given the proper (valid and true) inputs for an and b, the same as an Aristotelian syllogism consistently does and certainly will. Logic itself would collapse if this did not hold true, so it is logically certain that it is Logical Truth. Necessarily: C: Given A and B, and the logically equivalent relationship between the geometric properties of a right triangle and the Pythagorean Theorem, the theorem is proven - through its alignment with the Reality and Logic (Universal Logos) of what it aims to describe. Assuming otherwise leads to contradiction and absurdity where an expression that aligns perfectly with observed Reality and mathematical theory, and Logic, might for some reason need an arbitrary amount of further validation (than those objective measures already achieved). Truly, the functionality of such a tautological structure (ie - the Pythagorean Theorem and its self-evident alignment with the actual structure of a given right triangle in geometry and beyond the algebraic equation) shows (proves) not only the EXISTENCE but the FUNCTIONALITY and PRACTICALITY of the Universal Logos (the Origin of Existence also called God or Pure Reason). Formal Argument: {A, A <-> B, B <-> C} |= (C -> B) -> A Conclusion: {A, B, C}: Truly, the functionality of such a tautological structure (ie - the Pythagorean Theorem and its self-evident alignment with the actual structure of a given right triangle in geometry and beyond the algebraic equation) shows (proves) not only it’s own validity, but the EXISTENCE, the FUNCTIONALITY, and PRACTICALITY of the Universal Logos (the Origin of Existence also called God or Pure Reason) that is manifest in such a concept like the tautological trigonometric axiom known as the Pythagorean Theorem. In other words, the theorem is proven by First Cause (Logos - Logic - God) and also proves the existence of (acknowledges) that same logical and coherent “First Cause of Reality, Logic, and Trigonometry” (rightly/reasonably called “the One True God”, on account of being the singular Logical Origin of Logical Existence that defines what can exist and be valid). Truth Table Analysis: valid ↓ ↓ ↓ ↓ ABC⎪A A↔B B↔C╞(C→B)→A ―――――――――――――――――――――― TTT⎪T TTT TTT TTT TT TTF⎪T TTT TFF FTT TT TFT⎪T TFF FFT TFF TT TFF⎪T TFF FTF FTF TT FTT⎪F FFT TTT TTT FF FTF⎪F FFT TFF FTT FF FFT⎪F FTF FFT TFF TF FFF⎪F FTF FTF FTF FF Q.E.D. Elaboration: The Pythagorean Theorem can validly be understood as a particular form of the syllogism. This is why I have chosen this form, which amounts to one of my renditions of the syllogism, to prove that this valid logical structure is tautological and can prove itself universally in any valid instance or application of itself. Note that I use “since” and “necessarily” because I avoid adding contingency and uncertainty into a proof structure that is meant to provide certainty by introducing such terms and conditionals as “if” and “then”. The proof shows the nuances of Logic and that tautological circular reasoning, especially as it relates to logical necessities and foundational principles confirmed by the very functionality of the systems they are axioms of (ie - the Pythagorean Theorem and Trigonometry) isn’t fallacious. That runs contrary to what the unwise may have said in the past - about proving things with tautologies, and with themselves, being “impossible” (when it’s actually “logically necessary”). Here I showed how we can use the universal syllogistic form to prove that the Pythagorean Theorem, as a particular use of it, is as logical and proven (supported by tautological Logic) as can be.

Lean 4 verification and timestamps

The machine formalization date is separate from the original authored work. See the verification record for exact checked scope and any unresolved parts.

Logic-centered continuity

This authored proof is connected to Logic, Logos, God, and the self-evident Ground of Reality, and to the distinct systems 4P Engine, MCB Core, 4P-MCB, and the source-first Canon. Formal proof, empirical evidence, and system-generated records remain separately typed.