Categorical syllogisms: 2D and 3D atlas

Ivan B. Kurpishev · KLT–SYL–DK v1.1. This short illustrated atlas accompanies the Russian master article; it is not a full translation. All four moods use one normalized universe and a common injective transport.

Русская мастер-статья / Russian master

Barbara · Full inclusion

M is contained in P and S is contained in M; hence S is contained in P.

Barbara · AAA–1 · 2DM ⊆ P ∧ S ⊆ M ⇒ S ⊆ PBarbara · AAA–1 · 2DM ⊆ P ∧ S ⊆ M ⇒ S ⊆ PΠ₀: S, M, PPMSΠ₁: h(S), h(M), h(P)h(P)h(M)h(S)hh(x,y) = ( x / (1+x/8), y / (1+x/8) )S = subject M = middle term P = predicateIvan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
2D: source sets and their images under the same projectivity.
Barbara · AAA–1 · 3DM ⊆ P ∧ S ⊆ M ⇒ S ⊆ PBarbara · AAA–1 · 3DM ⊆ P ∧ S ⊆ M ⇒ S ⊆ PΠ₀: z=4Π₁: z=8P̂M̂ŜV ∉ Ŝ ∪ M̂ ∪ P̂Â = { (t x, t y, 4t) : (x,y) ∈ A, 1 ≤ t ≤ 2 }Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
3D: injective truncated-cone lift; the common apex is excluded.

Celarent · Separation

M and P are disjoint and S is contained in M; hence S and P are disjoint.

Celarent · EAE–1 · 2DM ∩ P = ∅ ∧ S ⊆ M ⇒ S ∩ P = ∅Celarent · EAE–1 · 2DM ∩ P = ∅ ∧ S ⊆ M ⇒ S ∩ P = ∅Π₀: S, M, PPMSΠ₁: h(S), h(M), h(P)h(P)h(M)h(S)hh(x,y) = ( x / (1+x/8), y / (1+x/8) )S = subject M = middle term P = predicateIvan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
2D: source sets and their images under the same projectivity.
Celarent · EAE–1 · 3DM ∩ P = ∅ ∧ S ⊆ M ⇒ S ∩ P = ∅Celarent · EAE–1 · 3DM ∩ P = ∅ ∧ S ⊆ M ⇒ S ∩ P = ∅Π₀: z=4Π₁: z=8P̂M̂ŜV ∉ Ŝ ∪ M̂ ∪ P̂Â = { (t x, t y, 4t) : (x,y) ∈ A, 1 ≤ t ≤ 2 }Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
3D: injective truncated-cone lift; the common apex is excluded.

Darii · Positive existential witness

Choose x in S intersect M. Since M is contained in P, the same x lies in S intersect P.

Darii · AII–1 · 2DM ⊆ P ∧ S ∩ M ≠ ∅ ⇒ S ∩ P ≠ ∅Darii · AII–1 · 2DM ⊆ P ∧ S ∩ M ≠ ∅ ⇒ S ∩ P ≠ ∅Π₀: S, M, PPMSxΠ₁: h(S), h(M), h(P)h(P)h(M)h(S)h(x)hh(x,y) = ( x / (1+x/8), y / (1+x/8) )S = subject M = middle term P = predicateIvan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
2D: source sets and their images under the same projectivity.
Darii · AII–1 · 3DM ⊆ P ∧ S ∩ M ≠ ∅ ⇒ S ∩ P ≠ ∅Darii · AII–1 · 3DM ⊆ P ∧ S ∩ M ≠ ∅ ⇒ S ∩ P ≠ ∅Π₀: z=4Π₁: z=8P̂M̂Ŝx₀x₁V ∉ Ŝ ∪ M̂ ∪ P̂Â = { (t x, t y, 4t) : (x,y) ∈ A, 1 ≤ t ≤ 2 }Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
3D: injective truncated-cone lift; the common apex is excluded.

Ferio · Negative existential witness

Choose x in S intersect M. Since M and P are disjoint, x lies in S outside P.

Ferio · EIO–1 · 2DM ∩ P = ∅ ∧ S ∩ M ≠ ∅ ⇒ S ∖ P ≠ ∅Ferio · EIO–1 · 2DM ∩ P = ∅ ∧ S ∩ M ≠ ∅ ⇒ S ∖ P ≠ ∅Π₀: S, M, PPMSxΠ₁: h(S), h(M), h(P)h(P)h(M)h(S)h(x)hh(x,y) = ( x / (1+x/8), y / (1+x/8) )S = subject M = middle term P = predicateIvan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
2D: source sets and their images under the same projectivity.
Ferio · EIO–1 · 3DM ∩ P = ∅ ∧ S ∩ M ≠ ∅ ⇒ S ∖ P ≠ ∅Ferio · EIO–1 · 3DM ∩ P = ∅ ∧ S ∩ M ≠ ∅ ⇒ S ∖ P ≠ ∅Π₀: z=4Π₁: z=8P̂M̂Ŝx₀x₁V ∉ Ŝ ∪ M̂ ∪ P̂Â = { (t x, t y, 4t) : (x,y) ∈ A, 1 ≤ t ≤ 2 }Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
3D: injective truncated-cone lift; the common apex is excluded.

Desargues–Kurpishev

The Desargues–Kurpishev example is a chosen compatible model. The homology is H=I+(1/12)vℓᵀ; the selected harmonic tuple is (−1,1;1/3,3). The canonical conic-to-triangle construction and automatic harmonicity remain OPEN. Geometric inversion does not reverse a syllogistic implication. SCIENCE=ABSTAIN; truth_layer_promotion=0; EXT_RUN=NOT_RUN.

Desargues–Kurpishev · 2D calibrationTwo conics + explicit homology + an independent harmonic checkDesargues–Kurpishev · 2D calibrationTwo conics + explicit homology + an independent harmonic checkΦ₁, Φ₂; Oᵢ = [Qᵢ⁻¹p]pℓ = O₁O₂O₁O₂ABCDgA=−1; B=1/3; C=1; Dg=3cr(A,C;B,Dg) = −1p: y=0; ℓ: x+3y−3=0Pⱼ′ = H PⱼP1P1′P2P2′P3P3′VX₁₂X₁₃X₂₃H = I₃ + (1/12) v ℓᵀv=(0,−2,1)ᵀ; det H=1/4dDes = ℓ is verified for this chosen H; no canonical conic-to-triangle map is claimed.Dg is a geometric point. The evidence foundation D_evidence is a separate typed record.Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
H=I₃+(1/12)vℓᵀ; det H=1/4; cr(−1,1;1/3,3)=−1.
Desargues–Kurpishev · 3D section modelCone surfaces → conic sections in Π; poles and trace are computed in ΠDesargues–Kurpishev · 3D section modelCone surfaces → conic sections in Π; poles and trace are computed in ΠV1Φ1V2Φ2O₁O₂DgΠ: z=0p: y=0 in ΠDg = O₁O₂ ∩ pThe cone vertices V₁,V₂ are geometric auxiliaries, not logical witnesses.The two-circle polar model uses a chosen support line p, not the Euclidean line at infinity.Ivan B. Kurpishev · KLT–SYL–DK v1.1 · coordinate-defined model
Π: z=0; Dg=O₁O₂∩p.