Geometric aspects of Lukasiewicz logic A short excursion - - PowerPoint PPT Presentation

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Geometric aspects of Lukasiewicz logic

A short excursion

❱✐♥❝❡♥③♦ ▼❛rr❛

vincenzo.marra@unimi.it

❉✐♣❛rt✐♠❡♥t♦ ❞✐ ▼❛t❡♠❛t✐❝❛ ❋❡❞❡r✐❣♦ ❊♥r✐q✉❡s ❯♥✐✈❡rs✐t✒ ❛ ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ▼✐❧❛♥♦ ■t❛❧②

❙●❙▲P❙ ❲♦r❦s❤♦♣ ♦♥ ▼❛♥②✲✈❛❧✉❡❞ ▲♦❣✐❝s ❇❡r♥✱ ❙✇✐t③❡r❧❛♥❞ ▼❛② ✷✷♥❞✱ ✷✵✶✺

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

The basics of Lukasiewicz logic

❏❛♥ ✥ ▲✉❦❛s✐❡✇✐❝③✱ ✶✽✼✽④✶✾✺✻✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ st❛rt ✇✐t❤ ❛ ✭☞♥✐t❡ ♦r ✐♥☞♥✐t❡✮ s❡t ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✱ ♦r ❛t♦♠✐❝ ❢♦r♠✉❧✚✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ♣r♦♣♦s✐t✐♦♥s✳ ❙❛②✱ ✐❢ ✇❡ ❝♦♥t❡♥t ♦✉rs❡❧✈❡s ✇✐t❤ ❝♦✉♥t❛❜❧② ♠❛♥②✿ ❳✶, ❳✷, . . . , ❳♥, . . . . ✭❲❡ ❝❛♥ ✉s❡ ♣✱ q✱ ❡t❝✳ ❛s ❛ ❧✐❣❤t❡r s❤♦rt✲❤❛♥❞ ♥♦t❛t✐♦♥✳✮ ❚♦ t❤❡s❡ ✇❡ ❛❞❥♦✐♥ t✇♦ s②♠❜♦❧s ❛♥❞ ✱ s❛②✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ❛ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✐s ❛❧✇❛②s tr✉❡ ✭t❤❡ ✈❡r✉♠✮✱ ❛♥❞ ♦♥❡ t❤❛t ✐s ❛❧✇❛②s ❢❛❧s❡ ✭t❤❡ ❢❛❧s✉♠✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ❝♦♥str✉❝t ❝♦♠♣♦✉♥❞ ❢♦r♠✉❧✚ ✇❡ ✉s❡ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥♥❡❝t✐✈❡s✿ ✱ ❢♦r ❞✐s❥✉♥❝t✐♦♥ ✭❭✐♥❝❧✉s✐✈❡ ♦r✧✱ ▲❛t✐♥ ✈❡❧✮❀ ✱ ❢♦r ❝♦♥❥✉♥❝t✐♦♥ ✭❭❛♥❞✧✱ ▲❛t✐♥ ❡t✮❀ ✱ ❢♦r ✐♠♣❧✐❝❛t✐♦♥ ✭❭✐❢✳ ✳ ✳ t❤❡♥✳ ✳ ✳ ✧✱ ❝♦♥❞✐t✐♦♥❛❧ ❛ss❡rt✐♦♥s✮❀ ✱ ❢♦r ♥❡❣❛t✐♦♥ ✭❭♥♦t✧✱ ♥❡❣❛t✐✈❡ ❛ss❡rt✐♦♥s✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ st❛rt ✇✐t❤ ❛ ✭☞♥✐t❡ ♦r ✐♥☞♥✐t❡✮ s❡t ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✱ ♦r ❛t♦♠✐❝ ❢♦r♠✉❧✚✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ♣r♦♣♦s✐t✐♦♥s✳ ❙❛②✱ ✐❢ ✇❡ ❝♦♥t❡♥t ♦✉rs❡❧✈❡s ✇✐t❤ ❝♦✉♥t❛❜❧② ♠❛♥②✿ ❳✶, ❳✷, . . . , ❳♥, . . . . ✭❲❡ ❝❛♥ ✉s❡ ♣✱ q✱ ❡t❝✳ ❛s ❛ ❧✐❣❤t❡r s❤♦rt✲❤❛♥❞ ♥♦t❛t✐♦♥✳✮ ❚♦ t❤❡s❡ ✇❡ ❛❞❥♦✐♥ t✇♦ s②♠❜♦❧s ⊤ ❛♥❞ ⊥✱ s❛②✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ❛ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✐s ❛❧✇❛②s tr✉❡ ✭t❤❡ ✈❡r✉♠✮✱ ❛♥❞ ♦♥❡ t❤❛t ✐s ❛❧✇❛②s ❢❛❧s❡ ✭t❤❡ ❢❛❧s✉♠✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ❝♦♥str✉❝t ❝♦♠♣♦✉♥❞ ❢♦r♠✉❧✚ ✇❡ ✉s❡ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥♥❡❝t✐✈❡s✿ ✱ ❢♦r ❞✐s❥✉♥❝t✐♦♥ ✭❭✐♥❝❧✉s✐✈❡ ♦r✧✱ ▲❛t✐♥ ✈❡❧✮❀ ✱ ❢♦r ❝♦♥❥✉♥❝t✐♦♥ ✭❭❛♥❞✧✱ ▲❛t✐♥ ❡t✮❀ ✱ ❢♦r ✐♠♣❧✐❝❛t✐♦♥ ✭❭✐❢✳ ✳ ✳ t❤❡♥✳ ✳ ✳ ✧✱ ❝♦♥❞✐t✐♦♥❛❧ ❛ss❡rt✐♦♥s✮❀ ✱ ❢♦r ♥❡❣❛t✐♦♥ ✭❭♥♦t✧✱ ♥❡❣❛t✐✈❡ ❛ss❡rt✐♦♥s✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ st❛rt ✇✐t❤ ❛ ✭☞♥✐t❡ ♦r ✐♥☞♥✐t❡✮ s❡t ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s✱ ♦r ❛t♦♠✐❝ ❢♦r♠✉❧✚✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ♣r♦♣♦s✐t✐♦♥s✳ ❙❛②✱ ✐❢ ✇❡ ❝♦♥t❡♥t ♦✉rs❡❧✈❡s ✇✐t❤ ❝♦✉♥t❛❜❧② ♠❛♥②✿ ❳✶, ❳✷, . . . , ❳♥, . . . . ✭❲❡ ❝❛♥ ✉s❡ ♣✱ q✱ ❡t❝✳ ❛s ❛ ❧✐❣❤t❡r s❤♦rt✲❤❛♥❞ ♥♦t❛t✐♦♥✳✮ ❚♦ t❤❡s❡ ✇❡ ❛❞❥♦✐♥ t✇♦ s②♠❜♦❧s ⊤ ❛♥❞ ⊥✱ s❛②✱ t❤❛t ❛r❡ t♦ st❛♥❞ ❢♦r ❛ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✐s ❛❧✇❛②s tr✉❡ ✭t❤❡ ✈❡r✉♠✮✱ ❛♥❞ ♦♥❡ t❤❛t ✐s ❛❧✇❛②s ❢❛❧s❡ ✭t❤❡ ❢❛❧s✉♠✮✱ r❡s♣❡❝t✐✈❡❧②✳ ❚♦ ❝♦♥str✉❝t ❝♦♠♣♦✉♥❞ ❢♦r♠✉❧✚ ✇❡ ✉s❡ t❤❡ ❧♦❣✐❝❛❧ ❝♦♥♥❡❝t✐✈❡s✿ ∨✱ ❢♦r ❞✐s❥✉♥❝t✐♦♥ ✭❭✐♥❝❧✉s✐✈❡ ♦r✧✱ ▲❛t✐♥ ✈❡❧✮❀ ∧✱ ❢♦r ❝♦♥❥✉♥❝t✐♦♥ ✭❭❛♥❞✧✱ ▲❛t✐♥ ❡t✮❀ →✱ ❢♦r ✐♠♣❧✐❝❛t✐♦♥ ✭❭✐❢✳ ✳ ✳ t❤❡♥✳ ✳ ✳ ✧✱ ❝♦♥❞✐t✐♦♥❛❧ ❛ss❡rt✐♦♥s✮❀ ¬✱ ❢♦r ♥❡❣❛t✐♦♥ ✭❭♥♦t✧✱ ♥❡❣❛t✐✈❡ ❛ss❡rt✐♦♥s✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ✉s✉❛❧ r❡❝✉rs✐✈❡ ❞❡☞♥✐t✐♦♥ ♦❢ ❣❡♥❡r❛❧ ❢♦r♠✉❧✚ ♥♦✇ r❡❛❞s ❛s ❢♦❧❧♦✇s✳ ⊤ ❛♥❞ ⊥ ❛r❡ ❢♦r♠✉❧✚✳ ❆❧❧ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❢♦r♠✉❧✚✳ ■❢ α ❛♥❞ β ❛r❡ ❢♦r♠✉❧✚✱ s♦ ❛r❡ (α ∨ β)✱ (α ∧ β)✱ (α → β)✱ ❛♥❞ ¬α✳ ◆♦t❤✐♥❣ ❡❧s❡ ✐s ❛ ❢♦r♠✉❧❛✳ ▲❡t ✉s ✇r✐t❡ ❋♦r♠ ❢♦r t❤❡ s❡t ♦❢ ❛❧❧ ❢♦r♠✉❧✚ ❝♦♥str✉❝t❡❞ ♦✈❡r t❤❡ ❝♦✉♥t❛❜❧❡ ❧❛♥❣✉❛❣❡ ❳✶ ❳♥ ✳ ❖❜s❡r✈❡ t❤❛t ❢♦r♠✉❧✚ ❛r❡ ❞❡☞♥❡❞ ❡①❛❝t❧② ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ✉s✉❛❧ r❡❝✉rs✐✈❡ ❞❡☞♥✐t✐♦♥ ♦❢ ❣❡♥❡r❛❧ ❢♦r♠✉❧✚ ♥♦✇ r❡❛❞s ❛s ❢♦❧❧♦✇s✳ ⊤ ❛♥❞ ⊥ ❛r❡ ❢♦r♠✉❧✚✳ ❆❧❧ ♣r♦♣♦s✐t✐♦♥❛❧ ✈❛r✐❛❜❧❡s ❛r❡ ❢♦r♠✉❧✚✳ ■❢ α ❛♥❞ β ❛r❡ ❢♦r♠✉❧✚✱ s♦ ❛r❡ (α ∨ β)✱ (α ∧ β)✱ (α → β)✱ ❛♥❞ ¬α✳ ◆♦t❤✐♥❣ ❡❧s❡ ✐s ❛ ❢♦r♠✉❧❛✳ ▲❡t ✉s ✇r✐t❡ ❋♦r♠ ❢♦r t❤❡ s❡t ♦❢ ❛❧❧ ❢♦r♠✉❧✚ ❝♦♥str✉❝t❡❞ ♦✈❡r t❤❡ ❝♦✉♥t❛❜❧❡ ❧❛♥❣✉❛❣❡ ❳✶, . . . , ❳♥, . . .✳ ❖❜s❡r✈❡ t❤❛t ❢♦r♠✉❧✚ ❛r❡ ❞❡☞♥❡❞ ❡①❛❝t❧② ✐♥ t❤❡ s❛♠❡ ♠❛♥♥❡r ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ❢♦r ♦✉r ❧♦❣✐❝✳ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❤❛s ❛ ♠❛♥②✲✈❛❧✉❡❞ s❡♠❛♥t✐❝s✿ s♣❡❝✐☞❝❛❧❧②✱ ✇❡ t❛❦❡ [✵, ✶] ⊆ R ❛s ❛ s❡t ♦❢ ❭tr✉t❤ ✈❛❧✉❡s✧✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s✱ ♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✱ ♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇ ❋♦r♠ ✵ ✶ s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✉t❤✲❢✉♥❝t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥② ❢♦r♠✉❧✚ ❛♥❞ ✳ ✇ ✵✳ ✇ ✶ ✇ ✳ ✇ ✶ ✐❢ ✇ ✇ ✶ ✇ ✇ ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ❢♦r ♦✉r ❧♦❣✐❝✳ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❤❛s ❛ ♠❛♥②✲✈❛❧✉❡❞ s❡♠❛♥t✐❝s✿ s♣❡❝✐☞❝❛❧❧②✱ ✇❡ t❛❦❡ [✵, ✶] ⊆ R ❛s ❛ s❡t ♦❢ ❭tr✉t❤ ✈❛❧✉❡s✧✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s✱ ♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✱ ♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇ : ❋♦r♠ → [✵, ✶] s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✉t❤✲❢✉♥❝t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥② ❢♦r♠✉❧✚ α ❛♥❞ β✳ ✇ ✵✳ ✇ ✶ ✇ ✳ ✇ ✶ ✐❢ ✇ ✇ ✶ ✇ ✇ ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ❢♦r ♦✉r ❧♦❣✐❝✳ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❤❛s ❛ ♠❛♥②✲✈❛❧✉❡❞ s❡♠❛♥t✐❝s✿ s♣❡❝✐☞❝❛❧❧②✱ ✇❡ t❛❦❡ [✵, ✶] ⊆ R ❛s ❛ s❡t ♦❢ ❭tr✉t❤ ✈❛❧✉❡s✧✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s✱ ♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✱ ♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇ : ❋♦r♠ → [✵, ✶] s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✉t❤✲❢✉♥❝t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥② ❢♦r♠✉❧✚ α ❛♥❞ β✳ ✇(⊥) = ✵✳ ✇ ✶ ✇ ✳ ✇ ✶ ✐❢ ✇ ✇ ✶ ✇ ✇ ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ❢♦r ♦✉r ❧♦❣✐❝✳ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❤❛s ❛ ♠❛♥②✲✈❛❧✉❡❞ s❡♠❛♥t✐❝s✿ s♣❡❝✐☞❝❛❧❧②✱ ✇❡ t❛❦❡ [✵, ✶] ⊆ R ❛s ❛ s❡t ♦❢ ❭tr✉t❤ ✈❛❧✉❡s✧✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s✱ ♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✱ ♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇ : ❋♦r♠ → [✵, ✶] s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✉t❤✲❢✉♥❝t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥② ❢♦r♠✉❧✚ α ❛♥❞ β✳ ✇(⊥) = ✵✳ ✇(¬α) = ✶ − ✇(α)✳ ✇ ✶ ✐❢ ✇ ✇ ✶ ✇ ✇ ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ♥♦✇ ❞❡☞♥❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ❢♦r ♦✉r ❧♦❣✐❝✳ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❤❛s ❛ ♠❛♥②✲✈❛❧✉❡❞ s❡♠❛♥t✐❝s✿ s♣❡❝✐☞❝❛❧❧②✱ ✇❡ t❛❦❡ [✵, ✶] ⊆ R ❛s ❛ s❡t ♦❢ ❭tr✉t❤ ✈❛❧✉❡s✧✳ ❆♥ ❛ss✐❣♥♠❡♥t ♦❢ tr✉t❤ ✈❛❧✉❡s✱ ♦r ❛♥ ❡✈❛❧✉❛t✐♦♥✱ ♦r ❛ ♣♦ss✐❜❧❡ ✇♦r❧❞ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ✇ : ❋♦r♠ → [✵, ✶] s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ tr✉t❤✲❢✉♥❝t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥s ❢♦r ❛♥② ❢♦r♠✉❧✚ α ❛♥❞ β✳ ✇(⊥) = ✵✳ ✇(¬α) = ✶ − ✇(α)✳ ✇(α → β) = ✶ ✐❢ ✇(α) ✇(β) ✶ − (✇(α) − ✇(β)) ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚r✉t❤✲❢✉♥❝t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥✳

✇(α → β) = ✶ ✐❢ ✇(α) ✇(β) ✶ − (✇(α) − ✇(β)) ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚r✉t❤✲❢✉♥❝t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥✳

✇(α → β) = ♠✐♥ {✶, ✶ − (✇(α) − ✇(β))}

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❛r❡ ✉s✐♥❣ {⊥, ¬, →} ♦♥❧② ❛s ♣r✐♠✐t✐✈❡ ❝♦♥♥❡❝t✐✈❡s✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡s ✭⊤✱ ∨✱ ❛♥❞ ∧✮ ❛r❡ ❞❡☞♥❛❜❧❡ ❛s ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ❆♥❞ ✐t ✐s ❝✉st♦♠❛r② t♦ ❞❡☞♥❡ ♠♦r❡✳

④ ❋❛❧s✉♠ ❱❡r✉♠ ④ ◆❡❣❛t✐♦♥ ④ ■♠♣❧✐❝❛t✐♦♥ ✭▲❛tt✐❝❡✮ ❉✐s❥✉♥❝t✐♦♥ ✭▲❛tt✐❝❡✮ ❈♦♥❥✉♥❝t✐♦♥ ❇✐❝♦♥❞✐t✐♦♥❛❧ ❙tr♦♥❣ ❞✐s❥✉♥❝t✐♦♥ ❙tr♦♥❣ ❝♦♥❥✉♥❝t✐♦♥ ❈♦✲✐♠♣❧✐❝❛t✐♦♥ ❈♦♥♥❡❝t✐✈❡s ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❛r❡ ✉s✐♥❣ {⊥, ¬, →} ♦♥❧② ❛s ♣r✐♠✐t✐✈❡ ❝♦♥♥❡❝t✐✈❡s✳ ❚❤❡ r❡♠❛✐♥✐♥❣ ♦♥❡s ✭⊤✱ ∨✱ ❛♥❞ ∧✮ ❛r❡ ❞❡☞♥❛❜❧❡ ❛s ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ❆♥❞ ✐t ✐s ❝✉st♦♠❛r② t♦ ❞❡☞♥❡ ♠♦r❡✳

Notation Definition Name ⊥ ④ ❋❛❧s✉♠ ⊤ ¬⊥ ❱❡r✉♠ ¬α ④ ◆❡❣❛t✐♦♥ α → β ④ ■♠♣❧✐❝❛t✐♦♥ α ∨ β (α → β) → β ✭▲❛tt✐❝❡✮ ❉✐s❥✉♥❝t✐♦♥ α ∧ β ¬(¬α ∨ ¬β) ✭▲❛tt✐❝❡✮ ❈♦♥❥✉♥❝t✐♦♥ α ↔ β (α → β) ∧ (β → α) ❇✐❝♦♥❞✐t✐♦♥❛❧ α ⊕ β ¬α → β ❙tr♦♥❣ ❞✐s❥✉♥❝t✐♦♥ α ⊙ β ¬(α → ¬β) ❙tr♦♥❣ ❝♦♥❥✉♥❝t✐♦♥ α ⊖ β ¬(α → β) ❈♦✲✐♠♣❧✐❝❛t✐♦♥ Table: ❈♦♥♥❡❝t✐✈❡s ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢♦r♠❛❧ s❡♠❛♥t✐❝s ✐s ❛s ❢♦❧❧♦✇s✿

Notation Formal semantics ⊥ ✇(⊥) = ✵ ⊤ ✇(⊤) = ✶ ¬α ✇(¬α) = ✶ − ✇(α) α → β ✇(α → β) = ♠✐♥ {✶, ✶ − (✇(α) − ✇(β))} α ∨ β ✇(α ∨ β) = ♠❛① {✇(α), ✇(β)} α ∧ β ✇(α ∧ β) = ♠✐♥ {✇(α), ✇(β)} α ↔ β ✇(α ↔ β) = ✶ − |✇(α) − ✇(β)| α ⊕ β ✇(α ⊕ β) = ♠✐♥ {✶, ✇(α) + ✇(β)} α ⊙ β ✇(α ⊙ β) = ♠❛① {✵, ✇(α) + ✇(β) − ✶} α ⊖ β ✇(α ⊖ β) = ♠❛① {✵, ✇(α) − ✇(β)} Table: ❋♦r♠❛❧ s❡♠❛♥t✐❝s ♦❢ ❝♦♥♥❡❝t✐✈❡s ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚r✉t❤✲❢✉♥❝t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ❭str♦♥❣ ❝♦♥❥✉♥❝t✐♦♥✧ ⊙✳ ✭◆♦t❡✿ ◆♦♥✲✐❞❡♠♣♦t❡♥t ♦♣❡r❛t✐♦♥✳✮

✇(α ⊙ β) = ♠❛① {✵, ✇(α) + ✇(β) − ✶}

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥❛❧②t✐❝ tr✉t❤s✱ ♦r t❛✉t♦❧♦❣✐❡s ❛❢t❡r ▲✳ ❲✐tt❣❡♥st❡✐♥✱ ❛r❡ ♥♦✇ ❞❡☞♥❡❞ ❛s t❤♦s❡ ❢♦r♠✉❧✚ α ∈ ❋♦r♠ t❤❛t ❛r❡ tr✉❡ ✐♥ ❡✈❡r② ♣♦ss✐❜❧❡ ✇♦r❧❞✱ ✐✳❡✳ s✉❝❤ t❤❛t ✇(α) = ✶ ❢♦r ❛♥② ❛ss✐❣♥♠❡♥t ✇✳ ✭❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✮ ✭❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✮ ✭Pr✐♥❝✐♣❧❡ ♦❢ ♥♦♥✲❝♦♥tr❛❞✐❝t✐♦♥✮ ✭▲❛✇ ♦❢ ❞♦✉❜❧❡ ♥❡❣❛t✐♦♥✮ ✭❈♦♥s❡q✉❡♥t✐❛ ♠✐r❛❜✐❧✐s✮ ✭❈♦♥tr❛♣♦s✐t✐♦♥✮ ✭Pr❡✲❧✐♥❡❛r✐t②✮ ❉❡☞♥❡✿ ❚❛✉t ❋♦r♠ ✐s t❤❡ s❡t ♦❢ ❛❧❧ t❛✉t♦❧♦❣✐❡s✳ ❲r✐t❡✿ t♦ ♠❡❛♥ ❚❛✉t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥❛❧②t✐❝ tr✉t❤s✱ ♦r t❛✉t♦❧♦❣✐❡s ❛❢t❡r ▲✳ ❲✐tt❣❡♥st❡✐♥✱ ❛r❡ ♥♦✇ ❞❡☞♥❡❞ ❛s t❤♦s❡ ❢♦r♠✉❧✚ α ∈ ❋♦r♠ t❤❛t ❛r❡ tr✉❡ ✐♥ ❡✈❡r② ♣♦ss✐❜❧❡ ✇♦r❧❞✱ ✐✳❡✳ s✉❝❤ t❤❛t ✇(α) = ✶ ❢♦r ❛♥② ❛ss✐❣♥♠❡♥t ✇✳ ⊥ → α ✭❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✮ α ∨ ¬α ✭❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✮ ¬(α ∧ ¬α) ✭Pr✐♥❝✐♣❧❡ ♦❢ ♥♦♥✲❝♦♥tr❛❞✐❝t✐♦♥✮ ¬¬α → α ✭▲❛✇ ♦❢ ❞♦✉❜❧❡ ♥❡❣❛t✐♦♥✮ (¬α → α) → α ✭❈♦♥s❡q✉❡♥t✐❛ ♠✐r❛❜✐❧✐s✮ (α → β) → (¬β → ¬α) ✭❈♦♥tr❛♣♦s✐t✐♦♥✮ (α → β) ∨ (β → α) ✭Pr❡✲❧✐♥❡❛r✐t②✮ ❉❡☞♥❡✿ ❚❛✉t ⊆ ❋♦r♠ ✐s t❤❡ s❡t ♦❢ ❛❧❧ t❛✉t♦❧♦❣✐❡s✳ ❲r✐t❡✿ α t♦ ♠❡❛♥ α ∈ ❚❛✉t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❛✉t♦❧♦❣✐❡s ❛r❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝ ♥♦t✐♦♥✳ ▲♦❣✐❝ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ s②♥t❛① ✭t❤❡ ❧❛♥❣✉❛❣❡✮ ❛♥❞ s❡♠❛♥t✐❝s ✭t❤❡ ✇♦r❧❞✮✳ ❚❤❡ s②♥t❛❝t✐❝ ❝♦✉♥t❡r♣❛rt ♦❢ ❛ t❛✉t♦❧♦❣② ✐s ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡♦r❡♠ ♦❢ t❤❡ ❧♦❣✐❝✳ ❚♦ ❞❡☞♥❡ ♣r♦✈❛❜✐❧✐t②✱ ✇❡ s❡❧❡❝t ✭✇✐t❤ ❛ ❧♦t ♦❢ ❤✐♥❞s✐❣❤t✮ ❛ s❡t ♦❢ t❛✉t♦❧♦❣✐❡s✱ ❛♥❞ ❞❡❝❧❛r❡ t❤❛t t❤❡② ❛r❡ ❛①✐♦♠s✿ t❤❡② ❝♦✉♥t ❛s ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚ ❜② ❞❡☞♥✐t✐♦♥✳ ◆❡①t ✇❡ s❡❧❡❝t ❛ s❡t ♦❢ ❞❡❞✉❝t✐♦♥ r✉❧❡s t❤❛t t❡❧❧ ✉s t❤❛t ✐❢ ✇❡ ❛❧r❡❛❞② ❡st❛❜❧✐s❤❡❞ t❤❛t ❢♦r♠✉❧✚

✶ ♥ ❛r❡ ♣r♦✈❛❜❧❡✱ ❛♥❞

t❤❡s❡ ❤❛✈❡ ❛ ❝❡rt❛✐♥ s❤❛♣❡✱ t❤❡♥ ❛ s♣❡❝✐☞❝ ❢♦r♠✉❧❛ ✐s ❛❧s♦ ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❛✉t♦❧♦❣✐❡s ❛r❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝ ♥♦t✐♦♥✳ ▲♦❣✐❝ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ s②♥t❛① ✭t❤❡ ❧❛♥❣✉❛❣❡✮ ❛♥❞ s❡♠❛♥t✐❝s ✭t❤❡ ✇♦r❧❞✮✳ ❚❤❡ s②♥t❛❝t✐❝ ❝♦✉♥t❡r♣❛rt ♦❢ ❛ t❛✉t♦❧♦❣② ✐s ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡♦r❡♠ ♦❢ t❤❡ ❧♦❣✐❝✳ ❚♦ ❞❡☞♥❡ ♣r♦✈❛❜✐❧✐t②✱ ✇❡ s❡❧❡❝t ✭✇✐t❤ ❛ ❧♦t ♦❢ ❤✐♥❞s✐❣❤t✮ ❛ s❡t ♦❢ t❛✉t♦❧♦❣✐❡s✱ ❛♥❞ ❞❡❝❧❛r❡ t❤❛t t❤❡② ❛r❡ ❛①✐♦♠s✿ t❤❡② ❝♦✉♥t ❛s ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚ ❜② ❞❡☞♥✐t✐♦♥✳ ◆❡①t ✇❡ s❡❧❡❝t ❛ s❡t ♦❢ ❞❡❞✉❝t✐♦♥ r✉❧❡s t❤❛t t❡❧❧ ✉s t❤❛t ✐❢ ✇❡ ❛❧r❡❛❞② ❡st❛❜❧✐s❤❡❞ t❤❛t ❢♦r♠✉❧✚

✶ ♥ ❛r❡ ♣r♦✈❛❜❧❡✱ ❛♥❞

t❤❡s❡ ❤❛✈❡ ❛ ❝❡rt❛✐♥ s❤❛♣❡✱ t❤❡♥ ❛ s♣❡❝✐☞❝ ❢♦r♠✉❧❛ ✐s ❛❧s♦ ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❛✉t♦❧♦❣✐❡s ❛r❡ ❛ ❢♦r♠❛❧ s❡♠❛♥t✐❝ ♥♦t✐♦♥✳ ▲♦❣✐❝ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ s②♥t❛① ✭t❤❡ ❧❛♥❣✉❛❣❡✮ ❛♥❞ s❡♠❛♥t✐❝s ✭t❤❡ ✇♦r❧❞✮✳ ❚❤❡ s②♥t❛❝t✐❝ ❝♦✉♥t❡r♣❛rt ♦❢ ❛ t❛✉t♦❧♦❣② ✐s ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✱ ❛❧s♦ ❝❛❧❧❡❞ t❤❡♦r❡♠ ♦❢ t❤❡ ❧♦❣✐❝✳ ❚♦ ❞❡☞♥❡ ♣r♦✈❛❜✐❧✐t②✱ ✇❡ s❡❧❡❝t ✭✇✐t❤ ❛ ❧♦t ♦❢ ❤✐♥❞s✐❣❤t✮ ❛ s❡t ♦❢ t❛✉t♦❧♦❣✐❡s✱ ❛♥❞ ❞❡❝❧❛r❡ t❤❛t t❤❡② ❛r❡ ❛①✐♦♠s✿ t❤❡② ❝♦✉♥t ❛s ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚ ❜② ❞❡☞♥✐t✐♦♥✳ ◆❡①t ✇❡ s❡❧❡❝t ❛ s❡t ♦❢ ❞❡❞✉❝t✐♦♥ r✉❧❡s t❤❛t t❡❧❧ ✉s t❤❛t ✐❢ ✇❡ ❛❧r❡❛❞② ❡st❛❜❧✐s❤❡❞ t❤❛t ❢♦r♠✉❧✚ α✶, . . . , α♥ ❛r❡ ♣r♦✈❛❜❧❡✱ ❛♥❞ t❤❡s❡ ❤❛✈❡ ❛ ❝❡rt❛✐♥ s❤❛♣❡✱ t❤❡♥ ❛ s♣❡❝✐☞❝ ❢♦r♠✉❧❛ β ✐s ❛❧s♦ ❛ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧❛✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▼♦st ✐♠♣♦rt❛♥t ❞❡❞✉❝t✐♦♥ r✉❧❡ ✭♦♥❧② ♦♥❡ ✇❡ ✉s❡✮✿ ♠♦❞✉s ♣♦♥❡♥s✳ α α → β β (♠♣) ◆♦✇ ✇❡ ❞❡❝❧❛r❡ t❤❛t ❛ ❢♦r♠✉❧❛ ❋♦r♠ ✐s ♣r♦✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣r♦♦❢ ♦❢ ✱ t❤❛t ✐s✱ ❛ ☞♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❢♦r♠✉❧✚

✶ ❧ ❛ s✉❝❤ t❤❛t✿ ❧

✳ ❊❛❝❤

✐✱ ✐

❧ ✐s ❡✐t❤❡r ❛♥ ❛①✐♦♠✱ ♦r ✐s ♦❜t❛✐♥❛❜❧❡ ❢r♦♠

❥

❛♥❞

❦✱ ❥ ❦

✐✱ ✈✐❛ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♠♦❞✉s ♣♦♥❡♥s✳ ❉❡☞♥❡✿ ❚❤♠ ❋♦r♠ ✐s t❤❡ s❡t ♦❢ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚✳ ❲r✐t❡✿ t♦ ♠❡❛♥ ❚❤♠✳ ❲❡ st✐❧❧ ♥❡❡❞ t♦ ❞❡☞♥❡ t❤❡ ❛①✐♦♠s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▼♦st ✐♠♣♦rt❛♥t ❞❡❞✉❝t✐♦♥ r✉❧❡ ✭♦♥❧② ♦♥❡ ✇❡ ✉s❡✮✿ ♠♦❞✉s ♣♦♥❡♥s✳ α α → β β (♠♣) ◆♦✇ ✇❡ ❞❡❝❧❛r❡ t❤❛t ❛ ❢♦r♠✉❧❛ α ∈ ❋♦r♠ ✐s ♣r♦✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣r♦♦❢ ♦❢ α✱ t❤❛t ✐s✱ ❛ ☞♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❢♦r♠✉❧✚ α✶, . . . , α❧ ❛ s✉❝❤ t❤❛t✿ α❧ = α✳ ❊❛❝❤ α✐✱ ✐ < ❧ ✐s ❡✐t❤❡r ❛♥ ❛①✐♦♠✱ ♦r ✐s ♦❜t❛✐♥❛❜❧❡ ❢r♦♠ α❥ ❛♥❞ α❦✱ ❥ , ❦ < ✐✱ ✈✐❛ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♠♦❞✉s ♣♦♥❡♥s✳ ❉❡☞♥❡✿ ❚❤♠ ⊆ ❋♦r♠ ✐s t❤❡ s❡t ♦❢ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚✳ ❲r✐t❡✿ ⊢ α t♦ ♠❡❛♥ α ∈ ❚❤♠✳ ❲❡ st✐❧❧ ♥❡❡❞ t♦ ❞❡☞♥❡ t❤❡ ❛①✐♦♠s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▼♦st ✐♠♣♦rt❛♥t ❞❡❞✉❝t✐♦♥ r✉❧❡ ✭♦♥❧② ♦♥❡ ✇❡ ✉s❡✮✿ ♠♦❞✉s ♣♦♥❡♥s✳ α α → β β (♠♣) ◆♦✇ ✇❡ ❞❡❝❧❛r❡ t❤❛t ❛ ❢♦r♠✉❧❛ α ∈ ❋♦r♠ ✐s ♣r♦✈❛❜❧❡ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣r♦♦❢ ♦❢ α✱ t❤❛t ✐s✱ ❛ ☞♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❢♦r♠✉❧✚ α✶, . . . , α❧ ❛ s✉❝❤ t❤❛t✿ α❧ = α✳ ❊❛❝❤ α✐✱ ✐ < ❧ ✐s ❡✐t❤❡r ❛♥ ❛①✐♦♠✱ ♦r ✐s ♦❜t❛✐♥❛❜❧❡ ❢r♦♠ α❥ ❛♥❞ α❦✱ ❥ , ❦ < ✐✱ ✈✐❛ ❛♥ ❛♣♣❧✐❝❛t✐♦♥ ♦❢ ♠♦❞✉s ♣♦♥❡♥s✳ ❉❡☞♥❡✿ ❚❤♠ ⊆ ❋♦r♠ ✐s t❤❡ s❡t ♦❢ ♣r♦✈❛❜❧❡ ❢♦r♠✉❧✚✳ ❲r✐t❡✿ ⊢ α t♦ ♠❡❛♥ α ∈ ❚❤♠✳ ❲❡ st✐❧❧ ♥❡❡❞ t♦ ❞❡☞♥❡ t❤❡ ❛①✐♦♠s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ ❆ ❢♦rt✐♦r✐✳ ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ ❄ ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ ❆ ❢♦rt✐♦r✐✳ ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ ❄ ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ ❄ ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ ❄ ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) ((α → β) → β) → ((β → α) → α) ❄ ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) ((α → β) → β) → ((β → α) → α) ❄ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) ((α → β) → β) → ((β → α) → α) ❄ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ (A5) (¬α → α) → α ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) ((α → β) → β) → ((β → α) → α) ❄ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ (A5) (¬α → α) → α ❈♦♥s❡q✉❡♥t✐❛ ▼✐r❛❜✐❧✐s✳ ❯♣♦♥ ❞❡☞♥✐♥❣ α ∨ β ≡ (α → β) → β ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) (α ∨ β) → (β ∨ α) ❉✐s❥✉♥❝t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✳ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ (A5) α ∨ ¬α ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✳ ❯♣♦♥ ❞❡☞♥✐♥❣ α ∨ β ≡ (α → β) → β ✭❆✵④❆✺✮ r❡❛❞ ❛s s❤♦✇♥ ♥❡①t✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) (α ∨ β) → (β ∨ α) ❉✐s❥✉♥❝t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✳ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ (A5) α ∨ ¬α ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✳ α ∨ β ≡ (α → β) → β ❉❡❞✉❝t✐♦♥ r✉❧❡ ❢♦r ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ (R1)

α α→β β

▼♦❞✉s ♣♦♥❡♥s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆①✐♦♠ s②st❡♠ ❢♦r ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳ (A0) ⊥ → α ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t✳ (A1) α → (β → α) ❆ ❢♦rt✐♦r✐✳ (A2) (α → β) → ((β → γ) → (α → γ)) ■♠♣❧✐❝❛t✐♦♥ ✐s tr❛♥s✐t✐✈❡✳ (A3) (α ∨ β) → (β ∨ α) ❉✐s❥✉♥❝t✐♦♥ ✐s ❝♦♠♠✉t❛t✐✈❡✳ (A4) (α → β) → (¬β → ¬α) ❈♦♥tr❛♣♦s✐t✐♦♥✳ (A5) α ∨ ¬α ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✳ α ∨ β ≡ (α → β) → β ❉❡❞✉❝t✐♦♥ r✉❧❡ ❢♦r ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳ (R1)

α α→β β

▼♦❞✉s ♣♦♥❡♥s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❝❛♥ ❜❡ s✉❝❝✐♥❝t❧② ❞❡s❝r✐❜❡❞ ❛s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✇✐t❤♦✉t t❤❡ ❆r✐st♦t❡❧✐❛♥ ❧❛✇ ♦❢ ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✱ ❜✉t ✇✐t❤ t❤❡ ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t ❧❛✇✳ ❙✉❝❤ ❭s✉❝❝✐♥t ❞❡s❝r✐♣t✐♦♥s✧ ❝❛♥ ❜❡ ♣♦❧②s❡♠♦✉s t♦ ❛ s✉r♣r✐s✐♥❣ ❡①t❡♥t ✐♥❞❡❡❞✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❝❛♥ ❜❡ s✉❝❝✐♥❝t❧② ❞❡s❝r✐❜❡❞ ❛s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✇✐t❤♦✉t t❤❡ ❆r✐st♦t❡❧✐❛♥ ❧❛✇ ♦❢ ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✱ ❜✉t ✇✐t❤ t❤❡ ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t ❧❛✇✳ ❙✉❝❤ ❭s✉❝❝✐♥t ❞❡s❝r✐♣t✐♦♥s✧ ❝❛♥ ❜❡ ♣♦❧②s❡♠♦✉s t♦ ❛ s✉r♣r✐s✐♥❣ ❡①t❡♥t ✐♥❞❡❡❞✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

✥ ▲✉❦❛s✐❡✇✐❝③ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝ ❝❛♥ ❜❡ s✉❝❝✐♥❝t❧② ❞❡s❝r✐❜❡❞ ❛s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✇✐t❤♦✉t t❤❡ ❆r✐st♦t❡❧✐❛♥ ❧❛✇ ♦❢ ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✱ ❜✉t ✇✐t❤ t❤❡ ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t ❧❛✇✳✶ ❙✉❝❤ ❭s✉❝❝✐♥t ❞❡s❝r✐♣t✐♦♥s✧ ❝❛♥ ❜❡ ♣♦❧②s❡♠♦✉s t♦ ❛ s✉r♣r✐s✐♥❣ ❡①t❡♥t ✐♥❞❡❡❞✳ ▼♦r❛❧✿ ❚❤❡ ✐♠♣♦rt ♦❢ r❡♠♦✈✐♥❣ ♦♥❡ ❛①✐♦♠ ❢r♦♠ ❛♥ ❛①✐♦♠ s②st❡♠ ❞❡♣❡♥❞s ♦♥ t❤❡ ❛①✐♦♠ s②st❡♠ ✐ts❡❧❢✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❍✐❧❜❡rt✲st②❧❡ s②st❡♠s ❛r❡ ♦❢ ❧✐tt❧❡ ✉s❡ t♦ ❛♥❛❧②s❡ t❤❡ str✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❧♦❣✐❝s ✐♥ t❡r♠s ♦❢ ❛ s♣❡❝✐☞❝ ❛①✐♦♠❛t✐s❛t✐♦♥✳ ✭❋♦r t❤✐s✱ t❤❡ ●❡♥t③❡♥✲st②❧❡ s②st❡♠s ✉s❡❞ ✐♥ ♣r♦♦❢ t❤❡♦r② ❛r❡ ♠♦r❡ ✉s❡❢✉❧✳✮

✶❆❧♠♦st ✈❡r❜❛t✐♠ ❢r♦♠ ❏✳ ▼♦s❝❤♦✈❛❦✐s✱ ■♥t✉✐t✐♦♥✐st✐❝ ▲♦❣✐❝✱ ❚❤❡

❙t❛♥❢♦r❞ ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ P❤✐❧♦s♦♣❤②✱ ✷✵✶✵✱ ❊❞✇❛r❞ ◆✳ ❩❛❧t❛ ✭❡❞✳✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

✥ ▲✉❦❛s✐❡✇✐❝③ ■♥t✉✐t✐♦♥✐st✐❝ ❧♦❣✐❝ ❝❛♥ ❜❡ s✉❝❝✐♥❝t❧② ❞❡s❝r✐❜❡❞ ❛s ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ✇✐t❤♦✉t t❤❡ ❆r✐st♦t❡❧✐❛♥ ❧❛✇ ♦❢ ❚❡rt✐✉♠ ♥♦♥ ❞❛t✉r✱ ❜✉t ✇✐t❤ t❤❡ ❊① ❢❛❧s♦ q✉♦❞❧✐❜❡t ❧❛✇✳✶ ❙✉❝❤ ❭s✉❝❝✐♥t ❞❡s❝r✐♣t✐♦♥s✧ ❝❛♥ ❜❡ ♣♦❧②s❡♠♦✉s t♦ ❛ s✉r♣r✐s✐♥❣ ❡①t❡♥t ✐♥❞❡❡❞✳

✿✿✿✿✿✿

▼♦r❛❧✿ ❚❤❡ ✐♠♣♦rt ♦❢ r❡♠♦✈✐♥❣ ♦♥❡ ❛①✐♦♠ ❢r♦♠ ❛♥ ❛①✐♦♠ s②st❡♠ ❞❡♣❡♥❞s ♦♥ t❤❡ ❛①✐♦♠ s②st❡♠ ✐ts❡❧❢✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❍✐❧❜❡rt✲st②❧❡ s②st❡♠s ❛r❡ ♦❢ ❧✐tt❧❡ ✉s❡ t♦ ❛♥❛❧②s❡ t❤❡ str✉❝t✉r❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❧♦❣✐❝s ✐♥ t❡r♠s ♦❢ ❛ s♣❡❝✐☞❝ ❛①✐♦♠❛t✐s❛t✐♦♥✳ ✭❋♦r t❤✐s✱ t❤❡ ●❡♥t③❡♥✲st②❧❡ s②st❡♠s ✉s❡❞ ✐♥ ♣r♦♦❢ t❤❡♦r② ❛r❡ ♠♦r❡ ✉s❡❢✉❧✳✮

✶❆❧♠♦st ✈❡r❜❛t✐♠ ❢r♦♠ ❏✳ ▼♦s❝❤♦✈❛❦✐s✱ ■♥t✉✐t✐♦♥✐st✐❝ ▲♦❣✐❝✱ ❚❤❡

❙t❛♥❢♦r❞ ❊♥❝②❝❧♦♣❡❞✐❛ ♦❢ P❤✐❧♦s♦♣❤②✱ ✷✵✶✵✱ ❊❞✇❛r❞ ◆✳ ❩❛❧t❛ ✭❡❞✳✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤✐s ❝♦♥❝❧✉❞❡s ♦✉r ❞❡☞♥✐t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ✭♣r♦♣♦s✐t✐♦♥❛❧✮ ❧♦❣✐❝✳ ❆ ☞rst ✐♠♣♦rt❛♥t r❡s✉❧t✳ ■♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❛✉t♦❧♦❣✐❡s ❛♥❞ t❤❡♦r❡♠s ✐s ❡♥t✐r❡❧② ❛♥❛❧♦❣♦✉s t♦ t❤❡ ♦♥❡ t❤❛t ❤♦❧❞s ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ■t ✐s st❛t❡❞ ✐♥ t❤❡ ♥❡①t r❡s✉❧t✱ ❛ s✉❜st❛♥t✐❛❧ ♣✐❡❝❡ ♦❢ ♠❛t❤❡♠❛t✐❝s✿ ❚❛✉t ❚❤♠

❆✳ ❘♦s❡ ❛♥❞ ❏✳ ❇❛r❦❧❡② ❘♦ss❡r✱ ❚r❛♥s✳ ♦❢ t❤❡ ❆▼❙✱ ✶✾✺✽✳

Pr♦♦❢ ✐s s②♥t❛❝t✐❝✳ ❆❧❣❡❜r❛✐❝ ♣r♦♦❢ ❣✐✈❡♥ s❤♦rt❧② t❤❡r❡❛❢t❡r ❜② ❈✳❈✳ ❈❤❛♥❣✱ ✇❤✐❝❤ ✐♥tr♦❞✉❝❡❞ ▼❱✲❛❧❣❡❜r❛s ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❲❡ ✇✐❧❧ r❡t✉r♥ t♦ t❤❡♠ ✐❢ t✐♠❡ ❛❧❧♦✇s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤✐s ❝♦♥❝❧✉❞❡s ♦✉r ❞❡☞♥✐t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ✭♣r♦♣♦s✐t✐♦♥❛❧✮ ❧♦❣✐❝✳ ❆ ☞rst ✐♠♣♦rt❛♥t r❡s✉❧t✳ ■♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❛✉t♦❧♦❣✐❡s ❛♥❞ t❤❡♦r❡♠s ✐s ❡♥t✐r❡❧② ❛♥❛❧♦❣♦✉s t♦ t❤❡ ♦♥❡ t❤❛t ❤♦❧❞s ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✳ ■t ✐s st❛t❡❞ ✐♥ t❤❡ ♥❡①t r❡s✉❧t✱ ❛ s✉❜st❛♥t✐❛❧ ♣✐❡❝❡ ♦❢ ♠❛t❤❡♠❛t✐❝s✿ Soundness and Completeness Theorem for L ❚❛✉t = ❚❤♠ .

❆✳ ❘♦s❡ ❛♥❞ ❏✳ ❇❛r❦❧❡② ❘♦ss❡r✱ ❚r❛♥s✳ ♦❢ t❤❡ ❆▼❙✱ ✶✾✺✽✳

Pr♦♦❢ ✐s s②♥t❛❝t✐❝✳ ❆❧❣❡❜r❛✐❝ ♣r♦♦❢ ❣✐✈❡♥ s❤♦rt❧② t❤❡r❡❛❢t❡r ❜② ❈✳❈✳ ❈❤❛♥❣✱ ✇❤✐❝❤ ✐♥tr♦❞✉❝❡❞ ▼❱✲❛❧❣❡❜r❛s ❢♦r t❤✐s ♣✉r♣♦s❡✳ ❲❡ ✇✐❧❧ r❡t✉r♥ t♦ t❤❡♠ ✐❢ t✐♠❡ ❛❧❧♦✇s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❈❧❛ss✐❝❛❧ ❧♦❣✐❝ s❛t✐s☞❡s ❛ str♦♥❣❡r ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✳ ❋♦r ❙, {α} ⊆ ❋♦r♠✱ ✇r✐t❡ ❙ ⊢ α ✐❢ α ✐s ♣r♦✈❛❜❧❡ ❢♦r♠ t❤❡ ❧♦❣✐❝❛❧ ❛①✐♦♠s ❛✉❣♠❡♥t❡❞ ❜② ❙✱ ❛♥❞ ❙ α ✐❢ α ❤♦❧❞s ✐♥ ❡❛❝❤ ♠♦❞❡❧ ✭❂♣♦ss✐❜❧❡ ✇♦r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❤❡r❡✐♥ ❡❛❝❤ ❢♦r♠✉❧❛ ♦❢ ❙ ❤♦❧❞s✳ ❋♦r ❛♥② ❋♦r♠✱ ❛♥❞ ❛♥② s❡t ❙ ❋♦r♠✱ ❙ ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❙ ■♥ t❤❡ ❛❝t✉❛❧ ✉s❡ ♦❢ ❛♥② ❧♦❣✐❝✱ ✐t ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ t♦ ❤❛✈❡ ❝♦♠♣❧❡t❡♥❡ss ✉♥❞❡r ❛❞❞✐t✐♦♥❛❧ s❡ts ❙ ♦❢ ❛ss✉♠♣t✐♦♥s✳ ■t ✐s ❙ t❤❛t ❡♥❝♦❞❡s ♦✉r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t ❛ s♣❡❝✐☞❝ ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥✳ P✉r❡ ❧♦❣✐❝ ✭❙ ✮ ❝❛♥ t❡❛❝❤ ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ✇♦r❧❞✱ ❜② ❞❡☞♥✐t✐♦♥✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❈❧❛ss✐❝❛❧ ❧♦❣✐❝ s❛t✐s☞❡s ❛ str♦♥❣❡r ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✳ ❋♦r ❙, {α} ⊆ ❋♦r♠✱ ✇r✐t❡ ❙ ⊢ α ✐❢ α ✐s ♣r♦✈❛❜❧❡ ❢♦r♠ t❤❡ ❧♦❣✐❝❛❧ ❛①✐♦♠s ❛✉❣♠❡♥t❡❞ ❜② ❙✱ ❛♥❞ ❙ α ✐❢ α ❤♦❧❞s ✐♥ ❡❛❝❤ ♠♦❞❡❧ ✭❂♣♦ss✐❜❧❡ ✇♦r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❤❡r❡✐♥ ❡❛❝❤ ❢♦r♠✉❧❛ ♦❢ ❙ ❤♦❧❞s✳ Strong Completeness Theorem for CL ❋♦r ❛♥② α ∈ ❋♦r♠✱ ❛♥❞ ❛♥② s❡t ❙ ⊆ ❋♦r♠✱ ❙ α ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❙ ⊢ α . ■♥ t❤❡ ❛❝t✉❛❧ ✉s❡ ♦❢ ❛♥② ❧♦❣✐❝✱ ✐t ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ t♦ ❤❛✈❡ ❝♦♠♣❧❡t❡♥❡ss ✉♥❞❡r ❛❞❞✐t✐♦♥❛❧ s❡ts ❙ ♦❢ ❛ss✉♠♣t✐♦♥s✳ ■t ✐s ❙ t❤❛t ❡♥❝♦❞❡s ♦✉r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t ❛ s♣❡❝✐☞❝ ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥✳ P✉r❡ ❧♦❣✐❝ ✭❙ ✮ ❝❛♥ t❡❛❝❤ ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ✇♦r❧❞✱ ❜② ❞❡☞♥✐t✐♦♥✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❈❧❛ss✐❝❛❧ ❧♦❣✐❝ s❛t✐s☞❡s ❛ str♦♥❣❡r ❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✳ ❋♦r ❙, {α} ⊆ ❋♦r♠✱ ✇r✐t❡ ❙ ⊢ α ✐❢ α ✐s ♣r♦✈❛❜❧❡ ❢♦r♠ t❤❡ ❧♦❣✐❝❛❧ ❛①✐♦♠s ❛✉❣♠❡♥t❡❞ ❜② ❙✱ ❛♥❞ ❙ α ✐❢ α ❤♦❧❞s ✐♥ ❡❛❝❤ ♠♦❞❡❧ ✭❂♣♦ss✐❜❧❡ ✇♦r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❤❡r❡✐♥ ❡❛❝❤ ❢♦r♠✉❧❛ ♦❢ ❙ ❤♦❧❞s✳ Strong Completeness Theorem for CL ❋♦r ❛♥② α ∈ ❋♦r♠✱ ❛♥❞ ❛♥② s❡t ❙ ⊆ ❋♦r♠✱ ❙ α ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❙ ⊢ α . ■♥ t❤❡ ❛❝t✉❛❧ ✉s❡ ♦❢ ❛♥② ❧♦❣✐❝✱ ✐t ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ t♦ ❤❛✈❡ ❝♦♠♣❧❡t❡♥❡ss ✉♥❞❡r ❛❞❞✐t✐♦♥❛❧ s❡ts ❙ ♦❢ ❛ss✉♠♣t✐♦♥s✳ ■t ✐s ❙ t❤❛t ❡♥❝♦❞❡s ♦✉r ❦♥♦✇❧❡❞❣❡ ❛❜♦✉t ❛ s♣❡❝✐☞❝ ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥✳ P✉r❡ ❧♦❣✐❝ ✭❙ = ∅✮ ❝❛♥ t❡❛❝❤ ✉s ♥♦t❤✐♥❣ ❛❜♦✉t t❤❡ ✇♦r❧❞✱ ❜② ❞❡☞♥✐t✐♦♥✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❢❛✐❧s str♦♥❣ ❝♦♠♣❧❡t❡♥❡ss✳ ▲❡t ❙ ❜❡ t❤❡ s❡t ♦❢ ❢♦r♠✉❧✚ ✐♥ ♦♥❡ ✈❛r✐❛❜❧❡ ♣✿ ϕ♥(♣) := ((♥ + ✶)(♣♥ ∧ ¬♣)) ⊕ ♣♥+✶ , ❢♦r ❡❛❝❤ ✐♥t❡❣❡r ♥ ✶✱ ✇❤❡r❡ ♣❦ := ♣ ⊙ · · · ⊙ ♣

• ❦ t✐♠❡s

, ❦♣ := ♣ ⊕ · · · ⊕ ♣

• ❦ t✐♠❡s

. ❚❤❡♥ ❙ ⊢✥

▲ ♣✱ ❜✉t ❙ ✥ ▲ ♣✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❙ ⊢✥

▲ ♣✱ ❜✉t ❙ ✥ ▲ ♣✳

■♥t✉✐t✐✈❡❧②✱ ②♦✉ ❝❛♥ t❤✐♥❦ ♦❢ ❙ ❛s ❡♠❜♦❞②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♦❢ ❛ss✉♠♣t✐♦♥s✿

1 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✶/✷✳ 2 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✷/✸✳ 3 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✸/✹✳ 4 ✳ ✳ ✳

❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♦❡s ♥♦t ❢♦❧❧♦✇ t❤❛t✿ ❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✶✳✧✱ ✐✳❡✳ ❙

✥ ▲ ♣✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❛♥② ♣r♦♦❢ ♦❢ ♣

❢r♦♠ ❙ ❝❛♥ ♦♥❧② ✉s❡s ❛ ☞♥✐t❡ s✉❜s❡t ♦❢ ❙✳ ❙❡♠❛♥t✐❝❛❧❧②✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✇♦r❧❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛❧❧ ♦❢ ❙ ✐s t❤❡ ♦♥❡ s✉❝❤ t❤❛t ✇ ♣ ✶✱ ✐✳❡✳ ❙

✥ ▲ ♣✳

❚❛❦✐♥❣ st♦❝❦✳

✥ ▲ ✐s

✱ ❜✉t

✥ ▲ ✐s ♥♦t✳

◆♦t❡✳ ❙

✥ ▲

❙

✥ ▲

❛❧✇❛②s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❙ ⊢✥

▲ ♣✱ ❜✉t ❙ ✥ ▲ ♣✳

■♥t✉✐t✐✈❡❧②✱ ②♦✉ ❝❛♥ t❤✐♥❦ ♦❢ ❙ ❛s ❡♠❜♦❞②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♦❢ ❛ss✉♠♣t✐♦♥s✿

1 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✶/✷✳ 2 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✷/✸✳ 3 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✸/✹✳ 4 ✳ ✳ ✳

❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♦❡s ♥♦t ❢♦❧❧♦✇ t❤❛t✿ ❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ⊢✥

▲ ♣✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❛♥② ♣r♦♦❢ ♦❢ ♣

❢r♦♠ ❙ ❝❛♥ ♦♥❧② ✉s❡s ❛ ☞♥✐t❡ s✉❜s❡t ♦❢ ❙✳ ❙❡♠❛♥t✐❝❛❧❧②✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✇♦r❧❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛❧❧ ♦❢ ❙ ✐s t❤❡ ♦♥❡ s✉❝❤ t❤❛t ✇ ♣ ✶✱ ✐✳❡✳ ❙

✥ ▲ ♣✳

❚❛❦✐♥❣ st♦❝❦✳

✥ ▲ ✐s

✱ ❜✉t

✥ ▲ ✐s ♥♦t✳

◆♦t❡✳ ❙

✥ ▲

❙

✥ ▲

❛❧✇❛②s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❙ ⊢✥

▲ ♣✱ ❜✉t ❙ ✥ ▲ ♣✳

■♥t✉✐t✐✈❡❧②✱ ②♦✉ ❝❛♥ t❤✐♥❦ ♦❢ ❙ ❛s ❡♠❜♦❞②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♦❢ ❛ss✉♠♣t✐♦♥s✿

1 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✶/✷✳ 2 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✷/✸✳ 3 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✸/✹✳ 4 ✳ ✳ ✳

❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♦❡s ♥♦t ❢♦❧❧♦✇ t❤❛t✿ ❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ⊢✥

▲ ♣✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❛♥② ♣r♦♦❢ ♦❢ ♣

❢r♦♠ ❙ ❝❛♥ ♦♥❧② ✉s❡s ❛ ☞♥✐t❡ s✉❜s❡t ♦❢ ❙✳ ❙❡♠❛♥t✐❝❛❧❧②✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✇♦r❧❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛❧❧ ♦❢ ❙ ✐s t❤❡ ♦♥❡ s✉❝❤ t❤❛t ✇(♣) = ✶✱ ✐✳❡✳ ❙ ✥

▲ ♣✳

❚❛❦✐♥❣ st♦❝❦✳

✥ ▲ ✐s

✱ ❜✉t

✥ ▲ ✐s ♥♦t✳

◆♦t❡✳ ❙

✥ ▲

❙

✥ ▲

❛❧✇❛②s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❙ ⊢✥

▲ ♣✱ ❜✉t ❙ ✥ ▲ ♣✳

■♥t✉✐t✐✈❡❧②✱ ②♦✉ ❝❛♥ t❤✐♥❦ ♦❢ ❙ ❛s ❡♠❜♦❞②✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♦❢ ❛ss✉♠♣t✐♦♥s✿

1 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✶/✷✳ 2 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✷/✸✳ 3 ♣ :=❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ ✸/✹✳ 4 ✳ ✳ ✳

❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♦❡s ♥♦t ❢♦❧❧♦✇ t❤❛t✿ ❭❊♥③♦ ✐s t❛❧❧✧ ✐s tr✉❡ t♦ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ⊢✥

▲ ♣✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❛♥② ♣r♦♦❢ ♦❢ ♣

❢r♦♠ ❙ ❝❛♥ ♦♥❧② ✉s❡s ❛ ☞♥✐t❡ s✉❜s❡t ♦❢ ❙✳ ❙❡♠❛♥t✐❝❛❧❧②✱ t❤❡ ♦♥❧② ♣♦ss✐❜❧❡ ✇♦r❧❞ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ ❛❧❧ ♦❢ ❙ ✐s t❤❡ ♦♥❡ s✉❝❤ t❤❛t ✇(♣) = ✶✱ ✐✳❡✳ ❙ ✥

▲ ♣✳

❚❛❦✐♥❣ st♦❝❦✳ ⊢✥

▲ ✐s compact✱ ❜✉t ✥ ▲ ✐s ♥♦t✳

◆♦t❡✳ ❙ ⊢✥

▲ α ⇒ ❙ ✥ ▲ α ❛❧✇❛②s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❍❛②✲❲✓ ♦❥❝✐❝❦✐ ❚❤❡♦r❡♠✿ Completeness Theorem for f.a. theories in L ❋♦r ❛♥② α ∈ ❋♦r♠✱ ❛♥❞ ❛♥② ☞♥✐t❡ s❡t ❋ ⊆ ❋♦r♠✱ ❋ ✥

▲ α

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❋ ⊢✥

▲ α .

❆ ❢♦❧❦❧♦r❡ t❤❡♦r❡♠✿ ❋♦r ❛♥② ❋♦r♠✱ ❛♥❞ ❛♥② ♠❛①✐♠❛❧ ❝♦♥s✐st❡♥t s❡t ▼ ❋♦r♠✱ ▼

✥ ▲

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ▼

✥ ▲

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❍❛②✲❲✓ ♦❥❝✐❝❦✐ ❚❤❡♦r❡♠✿ Completeness Theorem for f.a. theories in L ❋♦r ❛♥② α ∈ ❋♦r♠✱ ❛♥❞ ❛♥② ☞♥✐t❡ s❡t ❋ ⊆ ❋♦r♠✱ ❋ ✥

▲ α

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ❋ ⊢✥

▲ α .

❆ ❢♦❧❦❧♦r❡ t❤❡♦r❡♠✿ Completeness Theorem for maximal theories in L ❋♦r ❛♥② α ∈ ❋♦r♠✱ ❛♥❞ ❛♥② ♠❛①✐♠❛❧ ❝♦♥s✐st❡♥t s❡t ▼ ⊆ ❋♦r♠✱ ▼ ✥

▲ α

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ▼ ⊢✥

▲ α .

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Satisfiability and consistency in L

Notion Definition Description α ✐s s❛t✐s☞❛❜❧❡ ∃✇ s✉❝❤ t❤❛t ✇(α) = ✶ α ✐s ✶✲s❛t✐s☞❛❜✐❧❡ α ✐s ❝♦♥s✐st❡♥t ∃β s✉❝❤ t❤❛t α ⊢✥

▲ β

α ❞♦❡s ♥♦t ♣r♦✈❡ s♠t❤❣✳ α ✐s ✉♥s❛t✐s☞❛❜❧❡ ∀✇ ✇❡ ❤❛✈❡ ✇(α) < ✶ α ✐s ♥♦t ✶✲s❛t✐s☞❛❜❧❡ α ✐s ✐♥❝♦♥s✐st❡♥t ∀β ✇❡ ❤❛✈❡ α ⊢✥

▲ β

α ♣r♦✈❡s ❡✈❡r②t❤✐♥❣ α ✐s str♦♥❣❧② ✉♥s❛t✳ ∀✇ ✇❡ ❤❛✈❡ ✇(α) = ✵ α ✐s ❛❧✇❛②s ❢❛❧s❡ α ✐s str♦♥❣❧② ✐♥❝♦♥✳ ∀β ✇❡ ❤❛✈❡ ⊢✥

▲ α → β

α ✐♠♣❧✐❡s ❡✈❡r②t❤✐♥❣ ◆♦t❛ ❇❡♥❡✳ ❚❤❡ t❡r♠✐♥♦❧♦❣② ❭❙tr♦♥❣❧② ✉♥s❛t✐s☞❛❜❧❡✴✐♥❝♦♥s✐st❡♥t✧ ✐s ♥♦t st❛♥❞❛r❞✳ ■ ♦♥❧② ✉s❡ ✐t ❢♦r ❡❛s❡ ♦❢ ❡①♣♦s✐t✐♦♥✳ ■ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ st❛♥❞❛r❞ t❡r♠✐♥♦❧♦❣② ❢♦r t❤❡s❡ ❝♦♥❝❡♣ts✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Satisfiability and consistency in L

Notion Definition Description α ✐s s❛t✐s☞❛❜❧❡ ∃✇ s✉❝❤ t❤❛t ✇(α) = ✶ α ✐s ✶✲s❛t✐s☞❛❜✐❧❡ α ✐s ❝♦♥s✐st❡♥t ∃β s✉❝❤ t❤❛t α ⊢✥

▲ β

α ❞♦❡s ♥♦t ♣r♦✈❡ s♠t❤❣✳ α ✐s ✉♥s❛t✐s☞❛❜❧❡ ∀✇ ✇❡ ❤❛✈❡ ✇(α) < ✶ α ✐s ♥♦t ✶✲s❛t✐s☞❛❜❧❡ α ✐s ✐♥❝♦♥s✐st❡♥t ∀β ✇❡ ❤❛✈❡ α ⊢✥

▲ β

α ♣r♦✈❡s ❡✈❡r②t❤✐♥❣ α ✐s str♦♥❣❧② ✉♥s❛t✳ ∀✇ ✇❡ ❤❛✈❡ ✇(α) = ✵ α ✐s ❛❧✇❛②s ❢❛❧s❡ α ✐s str♦♥❣❧② ✐♥❝♦♥✳ ∀β ✇❡ ❤❛✈❡ ⊢✥

▲ α → β

α ✐♠♣❧✐❡s ❡✈❡r②t❤✐♥❣ ◆♦t❛ ❇❡♥❡✳ ❚❤❡ t❡r♠✐♥♦❧♦❣② ❭❙tr♦♥❣❧② ✉♥s❛t✐s☞❛❜❧❡✴✐♥❝♦♥s✐st❡♥t✧ ✐s ♥♦t st❛♥❞❛r❞✳ ■ ♦♥❧② ✉s❡ ✐t ❢♦r ❡❛s❡ ♦❢ ❡①♣♦s✐t✐♦♥✳ ■ ❞♦ ♥♦t ❦♥♦✇ ♦❢ ❛ st❛♥❞❛r❞ t❡r♠✐♥♦❧♦❣② ❢♦r t❤❡s❡ ❝♦♥❝❡♣ts✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Satisfiability and consistency in L

Notion Definition Description α ✐s s❛t✐s☞❛❜❧❡ ∃✇ s✉❝❤ t❤❛t ✇(α) = ✶ α ✐s ✶✲s❛t✐s☞❛❜✐❧❡ α ✐s ❝♦♥s✐st❡♥t ∃β s✉❝❤ t❤❛t α ⊢✥

▲ β

α ❞♦❡s ♥♦t ♣r♦✈❡ s♠t❤❣✳ α ✐s ✉♥s❛t✐s☞❛❜❧❡ ∀✇ ✇❡ ❤❛✈❡ ✇(α) < ✶ α ✐s ♥♦t ✶✲s❛t✐s☞❛❜❧❡ α ✐s ✐♥❝♦♥s✐st❡♥t ∀β ✇❡ ❤❛✈❡ α ⊢✥

▲ β

α ♣r♦✈❡s ❡✈❡r②t❤✐♥❣ α ✐s str♦♥❣❧② ✉♥s❛t✳ ∀✇ ✇❡ ❤❛✈❡ ✇(α) = ✵ α ✐s ❛❧✇❛②s ❢❛❧s❡ α ✐s str♦♥❣❧② ✐♥❝♦♥✳ ∀β ✇❡ ❤❛✈❡ ⊢✥

▲ α → β

α ✐♠♣❧✐❡s ❡✈❡r②t❤✐♥❣ ❊q✉✐✈❛❧❡♥t ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❜② t❤❡ Pr✐♥❝✐♣❧❡ ♦❢ ❇✐✈❛❧❡♥❝❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Satisfiability and consistency in L

Notion Definition Description α ✐s s❛t✐s☞❛❜❧❡ ∃✇ s✉❝❤ t❤❛t ✇(α) = ✶ α ✐s ✶✲s❛t✐s☞❛❜✐❧❡ α ✐s ❝♦♥s✐st❡♥t ∃β s✉❝❤ t❤❛t α ⊢✥

▲ β

α ❞♦❡s ♥♦t ♣r♦✈❡ s♠t❤❣✳ α ✐s ✉♥s❛t✐s☞❛❜❧❡ ∀✇ ✇❡ ❤❛✈❡ ✇(α) < ✶ α ✐s ♥♦t ✶✲s❛t✐s☞❛❜❧❡ α ✐s ✐♥❝♦♥s✐st❡♥t ∀β ✇❡ ❤❛✈❡ α ⊢✥

▲ β

α ♣r♦✈❡s ❡✈❡r②t❤✐♥❣ α ✐s str♦♥❣❧② ✉♥s❛t✳ ∀✇ ✇❡ ❤❛✈❡ ✇(α) = ✵ α ✐s ❛❧✇❛②s ❢❛❧s❡ α ✐s str♦♥❣❧② ✐♥❝♦♥✳ ∀β ✇❡ ❤❛✈❡ ⊢✥

▲ α → β

α ✐♠♣❧✐❡s ❡✈❡r②t❤✐♥❣ ❊q✉✐✈❛❧❡♥t ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝ ❜② t❤❡ ❉❡❞✉❝t✐♦♥ ❚❤❡♦r❡♠✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Deduction Theorem for CL ❋♦r ❛♥② α, β ∈ ❋♦r♠✱ α ⊢ β ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ⊢ α → β . ❚❤❡ ❞✐r❡❝t✐♦♥ ❢❛✐❧s ✐♥ ✥ ▲✿

✥ ▲

✱ ❜✉t

✥ ▲

✳ ❋♦r ❛♥② ❋♦r♠✱

✥ ▲

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ♥ ✶ s✉❝❤ t❤❛t

✥ ▲ ♥

✭◆♦t❛t✐♦♥✿

♥ ♥ t✐♠❡s

✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Deduction Theorem for CL ❋♦r ❛♥② α, β ∈ ❋♦r♠✱ α ⊢ β ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ⊢ α → β . ❚❤❡ ❞✐r❡❝t✐♦♥ ⇒ ❢❛✐❧s ✐♥ ✥ ▲✿ α ⊢✥

▲ α ⊙ α✱ ❜✉t ⊢✥ ▲ α → α ⊙ α✳

❋♦r ❛♥② ❋♦r♠✱

✥ ▲

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ♥ ✶ s✉❝❤ t❤❛t

✥ ▲ ♥

✭◆♦t❛t✐♦♥✿

♥ ♥ t✐♠❡s

✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Deduction Theorem for CL ❋♦r ❛♥② α, β ∈ ❋♦r♠✱ α ⊢ β ✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ⊢ α → β . ❚❤❡ ❞✐r❡❝t✐♦♥ ⇒ ❢❛✐❧s ✐♥ ✥ ▲✿ α ⊢✥

▲ α ⊙ α✱ ❜✉t ⊢✥ ▲ α → α ⊙ α✳

Local Deduction Theorem for L ❋♦r ❛♥② α, β ∈ ❋♦r♠✱ α ⊢✥

▲ β

✐❢✱ ❛♥❞ ♦♥❧② ✐❢✱ ∃♥ ✶ s✉❝❤ t❤❛t ⊢✥

▲ α♥ → β .

✭◆♦t❛t✐♦♥✿ α♥ := α ⊙ · · · ⊙ α

• ♥ t✐♠❡s

.✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ✐s ♦❢ ♣❛r❛♠♦✉♥t ❝♦♥❝❡♣t✉❛❧ ✐♠♣♦rt❛♥❝❡✿ ❲❡ ❝❛♥♥♦t t❤✐♥❦ ♦❢ ❛s ❭❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ✱ t❤❡r❡ ❢♦❧❧♦✇s ✧✱ ✐✳❡✳ ❛s ✳ ❚❤❡ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥ ✐s ♥♦t ❛ ❝♦♥❞✐t✐♦♥❛❧✳ ❚❤❡ ✭❋r❡❣❡❛♥✮ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ss❡rt✐♥❣ ❛ ♣r♦♣♦s✐t✐♦♥ ❛♥❞ ❝♦♥t❡♠♣❧❛t✐♥❣ t❤❛t ♣r♦♣♦s✐t✐♦♥ ❜❡❝♦♠❡s ❡ss❡♥t✐❛❧✿ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✱ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠✰❜✐✈❛❧❡♥❝❡ ♠❛❦❡ t❤❛t ❞✐st✐♥❝t✐♦♥ ❢❛r ❧❡ss ✐♠♣♦rt❛♥t✳ ✭❈❢✳ t❤❡ ❚❛rs❦✐❛♥ ✐❞❡♥t✐☞❝❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❛ ♣r♦♣♦s✐t✐♦♥ ✇✐t❤ ✐ts tr✉t❤ ❝♦♥❞✐t✐♦♥s✿ t❤✐s ❢❛✐❧s ❜❛❞❧② ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳✮ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s ✐t❡♠s✱ ✇❤✐❧❡ ✐t ✐s ❡❛s② t♦ s❛② ✇❤❛t t❤❡ ❛ss❡rt✐♦♥ ♠❡❛♥s✱ ✐t ✐s ❢❛r ❤❛r❞❡r t♦ s❛② ✇❤❛t t❤❡ ♣❧❛✐♥ ♣r♦♣♦s✐t✐♦♥ ♠❡❛♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡♥❞❡❞ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈❡ ✐s ✉♥❝❧❡❛r✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ✐s ♦❢ ♣❛r❛♠♦✉♥t ❝♦♥❝❡♣t✉❛❧ ✐♠♣♦rt❛♥❝❡✿

1 ❲❡ ❝❛♥♥♦t t❤✐♥❦ ♦❢ α → β ❛s ❭❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ α✱

t❤❡r❡ ❢♦❧❧♦✇s β✧✱ ✐✳❡✳ ❛s α ⊢ β✳ ❚❤❡ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥ ✐s ♥♦t ❛ ❝♦♥❞✐t✐♦♥❛❧✳ ❚❤❡ ✭❋r❡❣❡❛♥✮ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ss❡rt✐♥❣ ❛ ♣r♦♣♦s✐t✐♦♥ ❛♥❞ ❝♦♥t❡♠♣❧❛t✐♥❣ t❤❛t ♣r♦♣♦s✐t✐♦♥ ❜❡❝♦♠❡s ❡ss❡♥t✐❛❧✿ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✱ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠✰❜✐✈❛❧❡♥❝❡ ♠❛❦❡ t❤❛t ❞✐st✐♥❝t✐♦♥ ❢❛r ❧❡ss ✐♠♣♦rt❛♥t✳ ✭❈❢✳ t❤❡ ❚❛rs❦✐❛♥ ✐❞❡♥t✐☞❝❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❛ ♣r♦♣♦s✐t✐♦♥ ✇✐t❤ ✐ts tr✉t❤ ❝♦♥❞✐t✐♦♥s✿ t❤✐s ❢❛✐❧s ❜❛❞❧② ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳✮ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s ✐t❡♠s✱ ✇❤✐❧❡ ✐t ✐s ❡❛s② t♦ s❛② ✇❤❛t t❤❡ ❛ss❡rt✐♦♥ ♠❡❛♥s✱ ✐t ✐s ❢❛r ❤❛r❞❡r t♦ s❛② ✇❤❛t t❤❡ ♣❧❛✐♥ ♣r♦♣♦s✐t✐♦♥ ♠❡❛♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡♥❞❡❞ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈❡ ✐s ✉♥❝❧❡❛r✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ✐s ♦❢ ♣❛r❛♠♦✉♥t ❝♦♥❝❡♣t✉❛❧ ✐♠♣♦rt❛♥❝❡✿

1 ❲❡ ❝❛♥♥♦t t❤✐♥❦ ♦❢ α → β ❛s ❭❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ α✱

t❤❡r❡ ❢♦❧❧♦✇s β✧✱ ✐✳❡✳ ❛s α ⊢ β✳ ❚❤❡ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥ ✐s ♥♦t ❛ ❝♦♥❞✐t✐♦♥❛❧✳

2 ❚❤❡ ✭❋r❡❣❡❛♥✮ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ss❡rt✐♥❣ ❛ ♣r♦♣♦s✐t✐♦♥

❛♥❞ ❝♦♥t❡♠♣❧❛t✐♥❣ t❤❛t ♣r♦♣♦s✐t✐♦♥ ❜❡❝♦♠❡s ❡ss❡♥t✐❛❧✿ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✱ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠✰❜✐✈❛❧❡♥❝❡ ♠❛❦❡ t❤❛t ❞✐st✐♥❝t✐♦♥ ❢❛r ❧❡ss ✐♠♣♦rt❛♥t✳ ✭❈❢✳ t❤❡ ❚❛rs❦✐❛♥ ✐❞❡♥t✐☞❝❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❛ ♣r♦♣♦s✐t✐♦♥ α ✇✐t❤ ✐ts tr✉t❤ ❝♦♥❞✐t✐♦♥s✿ t❤✐s ❢❛✐❧s ❜❛❞❧② ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳✮ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s ✐t❡♠s✱ ✇❤✐❧❡ ✐t ✐s ❡❛s② t♦ s❛② ✇❤❛t t❤❡ ❛ss❡rt✐♦♥ ♠❡❛♥s✱ ✐t ✐s ❢❛r ❤❛r❞❡r t♦ s❛② ✇❤❛t t❤❡ ♣❧❛✐♥ ♣r♦♣♦s✐t✐♦♥ ♠❡❛♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡♥❞❡❞ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈❡ ✐s ✉♥❝❧❡❛r✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ❢❛✐❧✉r❡ ♦❢ t❤❡ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ✐s ♦❢ ♣❛r❛♠♦✉♥t ❝♦♥❝❡♣t✉❛❧ ✐♠♣♦rt❛♥❝❡✿

1 ❲❡ ❝❛♥♥♦t t❤✐♥❦ ♦❢ α → β ❛s ❭❢r♦♠ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ α✱

t❤❡r❡ ❢♦❧❧♦✇s β✧✱ ✐✳❡✳ ❛s α ⊢ β✳ ❚❤❡ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥ ✐s ♥♦t ❛ ❝♦♥❞✐t✐♦♥❛❧✳

2 ❚❤❡ ✭❋r❡❣❡❛♥✮ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ss❡rt✐♥❣ ❛ ♣r♦♣♦s✐t✐♦♥

❛♥❞ ❝♦♥t❡♠♣❧❛t✐♥❣ t❤❛t ♣r♦♣♦s✐t✐♦♥ ❜❡❝♦♠❡s ❡ss❡♥t✐❛❧✿ ✐♥ ❝❧❛ss✐❝❛❧ ❧♦❣✐❝✱ ❞❡❞✉❝t✐♦♥ t❤❡♦r❡♠✰❜✐✈❛❧❡♥❝❡ ♠❛❦❡ t❤❛t ❞✐st✐♥❝t✐♦♥ ❢❛r ❧❡ss ✐♠♣♦rt❛♥t✳ ✭❈❢✳ t❤❡ ❚❛rs❦✐❛♥ ✐❞❡♥t✐☞❝❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ❛ ♣r♦♣♦s✐t✐♦♥ α ✇✐t❤ ✐ts tr✉t❤ ❝♦♥❞✐t✐♦♥s✿ t❤✐s ❢❛✐❧s ❜❛❞❧② ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳✮

3 ❆s ❛ ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ t✇♦ ♣r❡✈✐♦✉s ✐t❡♠s✱ ✇❤✐❧❡ ✐t ✐s ❡❛s②

t♦ s❛② ✇❤❛t t❤❡ ❛ss❡rt✐♦♥ ⊢ α → β ♠❡❛♥s✱ ✐t ✐s ❢❛r ❤❛r❞❡r t♦ s❛② ✇❤❛t t❤❡ ♣❧❛✐♥ ♣r♦♣♦s✐t✐♦♥ α → β ♠❡❛♥s✳ ■♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✐♥t❡♥❞❡❞ ♠❡❛♥✐♥❣ ♦❢ t❤❡ ❝♦♥♥❡❝t✐✈❡ → ✐s ✉♥❝❧❡❛r✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Symbol Name Classically read ⊤ ✈❡r✉♠ ❆❧✇❛②s tr✉❡ ⊥ ❢❛❧s✉♠ ❆❧✇❛②s ❢❛❧s❡ ∨ ❞✐s❥✉♥❝t✐♦♥ ■♥❝❧✉s✐✈❡ ♦r ✭✈❡❧✮ ∧ ❝♦♥❥✉♥❝t✐♦♥ ❆♥❞ → ✐♠♣❧✐❝❛t✐♦♥ ■❢✳ ✳ ✳ t❤❡♥✳ ✳ ✳ ¬ ♥❡❣❛t✐♦♥ ◆♦t Notation Definition Formal Semantics ⊤ ¬⊥ ✇(⊤) = ✶ α ∨ β (α → β) → β ✇(α ∨ β) = ♠❛① {✇(α), ✇(β)} α ∧ β ¬(¬α ∨ ¬β) ✇(α ∧ β) = ♠✐♥ {✇(α), ✇(β)} α ↔ β (α → β) ∧ (β → α) ✇(α ↔ β) = ✶ − |✇(α) − ✇(β)| α ⊕ β ¬α → β ✇(α ⊕ β) = ♠✐♥ {✇(α) + ✇(β), ✶} α ⊖ β ¬(α → β) ✇(α ⊖ β) = ♠❛① {✇(α) − ✇(β), ✵} Table: ❈♦♥♥❡❝t✐✈❡s ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚r✉t❤✲❢✉♥❝t✐♦♥ ♦❢ ✥ ▲✉❦❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♦♥✳

✇(α → β) = ♠✐♥ {✶, ✶ − (✇(α) − ✇(β))} ✇(α → β) = ✶ ✐❢ ✇(α) ✇(β) ✶ − (✇(α) − ✇(β)) ♦t❤❡r✇✐s❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

MV-algebras

❈✳ ❈✳ ❈❤❛♥❣ ✐♥ ❘♦♠❡✱ ✶✾✻✾✳

❙❛② ❋♦r♠ ❛r❡ ❧♦❣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t ✐❢ ✳ ❲r✐t❡ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

MV-algebras

❈✳ ❈✳ ❈❤❛♥❣ ✐♥ ❘♦♠❡✱ ✶✾✻✾✳

Lindenbaum’s Equivalence Relation ❙❛② α, β ∈ ❋♦r♠ ❛r❡ ❧♦❣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t ✐❢ ⊢ α ↔ β✳ ❲r✐t❡ α ≡ β✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❖♥ t❤❡ q✉♦t✐❡♥t s❡t ❋♦r♠

≡ ✱ t❤❡ ❝♦♥♥❡❝t✐✈❡s ✐♥❞✉❝❡ ♦♣❡r❛t✐♦♥s✿

✵ := [⊥]≡ ¬[α]≡ := [¬α]≡ [α]≡ ⊕ [β]≡ := [α ⊕ β]≡ ❚❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡

❋♦r♠

✵ ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳

❵▼❱✲❛❧❣❡❜r❛✬ ✐s s❤♦rt ❢♦r ❵▼❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬✱ ❭❢♦r ❧❛❝❦ ♦❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❤❛♥❣✱ ✶✾✽✻✮✳ ▼❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❂ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝

❆❜str❛❝t❧②✿ ▼ ✵ ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛ ✐❢ ▼ ✵ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✱ ① ①✱ ✶ ✵ ✐s ❛❜s♦r❜✐♥❣ ❢♦r ✭① ✶ ✶✮✱ ❛♥❞✱ ❝❤❛r❛❝t❡r✐st✐❝❛❧❧②✱ ① ② ② ② ① ① ✭✯✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❖♥ t❤❡ q✉♦t✐❡♥t s❡t ❋♦r♠

≡ ✱ t❤❡ ❝♦♥♥❡❝t✐✈❡s ✐♥❞✉❝❡ ♦♣❡r❛t✐♦♥s✿

✵ := [⊥]≡ ¬[α]≡ := [¬α]≡ [α]≡ ⊕ [β]≡ := [α ⊕ β]≡ ❚❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ (❋♦r♠

≡ , ⊕, ¬, ✵) ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳

❵▼❱✲❛❧❣❡❜r❛✬ ✐s s❤♦rt ❢♦r ❵▼❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬✱ ❭❢♦r ❧❛❝❦ ♦❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❤❛♥❣✱ ✶✾✽✻✮✳ ▼❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❂ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝

❆❜str❛❝t❧②✿ ▼ ✵ ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛ ✐❢ ▼ ✵ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✱ ① ①✱ ✶ ✵ ✐s ❛❜s♦r❜✐♥❣ ❢♦r ✭① ✶ ✶✮✱ ❛♥❞✱ ❝❤❛r❛❝t❡r✐st✐❝❛❧❧②✱ ① ② ② ② ① ① ✭✯✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❖♥ t❤❡ q✉♦t✐❡♥t s❡t ❋♦r♠

≡ ✱ t❤❡ ❝♦♥♥❡❝t✐✈❡s ✐♥❞✉❝❡ ♦♣❡r❛t✐♦♥s✿

✵ := [⊥]≡ ¬[α]≡ := [¬α]≡ [α]≡ ⊕ [β]≡ := [α ⊕ β]≡ ❚❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ (❋♦r♠

≡ , ⊕, ¬, ✵) ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳

❵▼❱✲❛❧❣❡❜r❛✬ ✐s s❤♦rt ❢♦r ❵▼❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬✱ ❭❢♦r ❧❛❝❦ ♦❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❤❛♥❣✱ ✶✾✽✻✮✳ ▼❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❂ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝

❆❜str❛❝t❧②✿ ▼ ✵ ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛ ✐❢ ▼ ✵ ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✱ ① ①✱ ✶ ✵ ✐s ❛❜s♦r❜✐♥❣ ❢♦r ✭① ✶ ✶✮✱ ❛♥❞✱ ❝❤❛r❛❝t❡r✐st✐❝❛❧❧②✱ ① ② ② ② ① ① ✭✯✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❖♥ t❤❡ q✉♦t✐❡♥t s❡t ❋♦r♠

≡ ✱ t❤❡ ❝♦♥♥❡❝t✐✈❡s ✐♥❞✉❝❡ ♦♣❡r❛t✐♦♥s✿

✵ := [⊥]≡ ¬[α]≡ := [¬α]≡ [α]≡ ⊕ [β]≡ := [α ⊕ β]≡ ❚❤❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ (❋♦r♠

≡ , ⊕, ¬, ✵) ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳

❵▼❱✲❛❧❣❡❜r❛✬ ✐s s❤♦rt ❢♦r ❵▼❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬✱ ❭❢♦r ❧❛❝❦ ♦❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❤❛♥❣✱ ✶✾✽✻✮✳ ▼❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❂ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♦❣✐❝

❆❜str❛❝t❧②✿ (▼, ⊕, ¬, ✵) ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛ ✐❢ (▼, ⊕, ✵) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞✱ ¬¬① = ①✱ ✶ := ¬✵ ✐s ❛❜s♦r❜✐♥❣ ❢♦r ⊕ ✭① ⊕ ✶ = ✶✮✱ ❛♥❞✱ ❝❤❛r❛❝t❡r✐st✐❝❛❧❧②✱ ¬(¬① ⊕ ②) ⊕ ② = ¬(¬② ⊕ ①) ⊕ ① ✭✯✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥② ▼❱✲❛❧❣❡❜r❛ ❤❛s ❛♥ ✉♥❞❡r❧②✐♥❣ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② ✵ ❛♥❞ ❛❜♦✈❡ ❜② ✶✳ ❏♦✐♥s ❛r❡ ❣✐✈❡♥ ❜② ① ∨ ② := ¬(¬① ⊕ ②) ⊕ ② ❚❤✉s✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❧❛✇ ✭✯✮ st❛t❡s t❤❛t ❥♦✐♥s ❝♦♠♠✉t❡✿ ① ② ② ① ▼❡❡ts ❛r❡ ❞❡☞♥❡❞ ❜② t❤❡ ❞❡ ▼♦r❣❛♥ ❝♦♥❞✐t✐♦♥ ① ② ① ② ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s❂■❞❡♠♣♦t❡♥t ▼❱✲❛❧❣❡❜r❛s✿ ① ① ①✳ ❊q✉✐✈❛❧❡♥t❧②✿ ▼❱✲❛❧❣❡❜r❛s t❤❛t s❛t✐s❢② t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❧❛✇ ① ① ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥② ▼❱✲❛❧❣❡❜r❛ ❤❛s ❛♥ ✉♥❞❡r❧②✐♥❣ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② ✵ ❛♥❞ ❛❜♦✈❡ ❜② ✶✳ ❏♦✐♥s ❛r❡ ❣✐✈❡♥ ❜② ① ∨ ② := ¬(¬① ⊕ ②) ⊕ ② ❚❤✉s✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❧❛✇ ✭✯✮ st❛t❡s t❤❛t ❥♦✐♥s ❝♦♠♠✉t❡✿ ① ∨ ② = ② ∨ ① ▼❡❡ts ❛r❡ ❞❡☞♥❡❞ ❜② t❤❡ ❞❡ ▼♦r❣❛♥ ❝♦♥❞✐t✐♦♥ ① ② ① ② ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s❂■❞❡♠♣♦t❡♥t ▼❱✲❛❧❣❡❜r❛s✿ ① ① ①✳ ❊q✉✐✈❛❧❡♥t❧②✿ ▼❱✲❛❧❣❡❜r❛s t❤❛t s❛t✐s❢② t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❧❛✇ ① ① ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥② ▼❱✲❛❧❣❡❜r❛ ❤❛s ❛♥ ✉♥❞❡r❧②✐♥❣ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② ✵ ❛♥❞ ❛❜♦✈❡ ❜② ✶✳ ❏♦✐♥s ❛r❡ ❣✐✈❡♥ ❜② ① ∨ ② := ¬(¬① ⊕ ②) ⊕ ② ❚❤✉s✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❧❛✇ ✭✯✮ st❛t❡s t❤❛t ❥♦✐♥s ❝♦♠♠✉t❡✿ ① ∨ ② = ② ∨ ① ▼❡❡ts ❛r❡ ❞❡☞♥❡❞ ❜② t❤❡ ❞❡ ▼♦r❣❛♥ ❝♦♥❞✐t✐♦♥ ① ∧ ② := ¬(¬① ∨ ¬②) ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s❂■❞❡♠♣♦t❡♥t ▼❱✲❛❧❣❡❜r❛s✿ ① ① ①✳ ❊q✉✐✈❛❧❡♥t❧②✿ ▼❱✲❛❧❣❡❜r❛s t❤❛t s❛t✐s❢② t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❧❛✇ ① ① ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆♥② ▼❱✲❛❧❣❡❜r❛ ❤❛s ❛♥ ✉♥❞❡r❧②✐♥❣ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② ✵ ❛♥❞ ❛❜♦✈❡ ❜② ✶✳ ❏♦✐♥s ❛r❡ ❣✐✈❡♥ ❜② ① ∨ ② := ¬(¬① ⊕ ②) ⊕ ② ❚❤✉s✱ t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❧❛✇ ✭✯✮ st❛t❡s t❤❛t ❥♦✐♥s ❝♦♠♠✉t❡✿ ① ∨ ② = ② ∨ ① ▼❡❡ts ❛r❡ ❞❡☞♥❡❞ ❜② t❤❡ ❞❡ ▼♦r❣❛♥ ❝♦♥❞✐t✐♦♥ ① ∧ ② := ¬(¬① ∨ ¬②) ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛s❂■❞❡♠♣♦t❡♥t ▼❱✲❛❧❣❡❜r❛s✿ ① ⊕ ① = ①✳ ❊q✉✐✈❛❧❡♥t❧②✿ ▼❱✲❛❧❣❡❜r❛s t❤❛t s❛t✐s❢② t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❧❛✇ ① ∨ ¬① = ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ✐♥t❡r✈❛❧ [✵, ✶] ⊆ R ❝❛♥ ❜❡ ♠❛❞❡ ✐♥t♦ ❛♥ ▼❱✲❛❧❣❡❜r❛ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ✵ ❜② ❞❡☞♥✐♥❣ ① ⊕ ② := ♠✐♥ {① + ②, ✶} , ¬① := ✶ − ① . ❚❤❡ ✉♥❞❡r❧②✐♥❣ ❧❛tt✐❝❡ ♦r❞❡r ♦❢ t❤✐s ▼❱✲❛❧❣❡❜r❛ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♥❛t✉r❛❧ ♦r❞❡r ♦❢ [✵, ✶]✳ ❚❤❡ ✈❛r✐❡t② ♦❢ ▼❱✲❛❧❣❡❜r❛s ✐s ❣❡♥❡r❛t❡❞ ❜② ✵ ✶ ✳

❈✳❈✳ ❈❤❛♥❣✱ ❚r❛♥s✳ ♦❢ t❤❡ ❆▼❙✱ ✶✾✺✾✳ ❚❤✐s ♠❡❛♥s✿ ❚❤❡ ❝❧❛ss ♦❢ ▼❱✲❛❧❣❡❜r❛s ❝♦✐♥❝✐❞❡s ✇✐t❤ ❍❙P ✵ ✶ ⑤ ❛♥② ▼❱✲❛❧❣❡❜r❛ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉❜❛❧❣❡❜r❛ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ ❝♦♣✐❡s ♦❢ ✵ ✶ ✳ ❖r✿ ❚❤❡ ❡q✉❛t✐♦♥s ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❱✲❛❧❣❡❜r❛s✮ t❤❛t ❤♦❧❞ ✐♥ ❛❧❧ ▼❱✲❛❧❣❡❜r❛s ❛r❡ ❡①❛❝t❧② t❤♦s❡ t❤❛t ❤♦❧❞ ✐♥ ✵ ✶ ✳ ❖r✿ ❆♥② ❋♦r♠ t❤❛t ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ s♦♠❡ ▼❱✲❛❧❣❡❜r❛✱ ❛❧r❡❛❞② ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ ✵ ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ✐♥t❡r✈❛❧ [✵, ✶] ⊆ R ❝❛♥ ❜❡ ♠❛❞❡ ✐♥t♦ ❛♥ ▼❱✲❛❧❣❡❜r❛ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ✵ ❜② ❞❡☞♥✐♥❣ ① ⊕ ② := ♠✐♥ {① + ②, ✶} , ¬① := ✶ − ① . ❚❤❡ ✉♥❞❡r❧②✐♥❣ ❧❛tt✐❝❡ ♦r❞❡r ♦❢ t❤✐s ▼❱✲❛❧❣❡❜r❛ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♥❛t✉r❛❧ ♦r❞❡r ♦❢ [✵, ✶]✳ Theorem (Chang’s completeness theorem, 1959) ❚❤❡ ✈❛r✐❡t② ♦❢ ▼❱✲❛❧❣❡❜r❛s ✐s ❣❡♥❡r❛t❡❞ ❜② [✵, ✶]✳

❈✳❈✳ ❈❤❛♥❣✱ ❚r❛♥s✳ ♦❢ t❤❡ ❆▼❙✱ ✶✾✺✾✳ ❚❤✐s ♠❡❛♥s✿ ❚❤❡ ❝❧❛ss ♦❢ ▼❱✲❛❧❣❡❜r❛s ❝♦✐♥❝✐❞❡s ✇✐t❤ ❍❙P ✵ ✶ ⑤ ❛♥② ▼❱✲❛❧❣❡❜r❛ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉❜❛❧❣❡❜r❛ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ ❝♦♣✐❡s ♦❢ ✵ ✶ ✳ ❖r✿ ❚❤❡ ❡q✉❛t✐♦♥s ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❱✲❛❧❣❡❜r❛s✮ t❤❛t ❤♦❧❞ ✐♥ ❛❧❧ ▼❱✲❛❧❣❡❜r❛s ❛r❡ ❡①❛❝t❧② t❤♦s❡ t❤❛t ❤♦❧❞ ✐♥ ✵ ✶ ✳ ❖r✿ ❆♥② ❋♦r♠ t❤❛t ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ s♦♠❡ ▼❱✲❛❧❣❡❜r❛✱ ❛❧r❡❛❞② ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ ✵ ✶ ✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ ✐♥t❡r✈❛❧ [✵, ✶] ⊆ R ❝❛♥ ❜❡ ♠❛❞❡ ✐♥t♦ ❛♥ ▼❱✲❛❧❣❡❜r❛ ✇✐t❤ ♥❡✉tr❛❧ ❡❧❡♠❡♥t ✵ ❜② ❞❡☞♥✐♥❣ ① ⊕ ② := ♠✐♥ {① + ②, ✶} , ¬① := ✶ − ① . ❚❤❡ ✉♥❞❡r❧②✐♥❣ ❧❛tt✐❝❡ ♦r❞❡r ♦❢ t❤✐s ▼❱✲❛❧❣❡❜r❛ ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ ♥❛t✉r❛❧ ♦r❞❡r ♦❢ [✵, ✶]✳ Theorem (Chang’s completeness theorem, 1959) ❚❤❡ ✈❛r✐❡t② ♦❢ ▼❱✲❛❧❣❡❜r❛s ✐s ❣❡♥❡r❛t❡❞ ❜② [✵, ✶]✳

❈✳❈✳ ❈❤❛♥❣✱ ❚r❛♥s✳ ♦❢ t❤❡ ❆▼❙✱ ✶✾✺✾✳ ❚❤✐s ♠❡❛♥s✿ ❚❤❡ ❝❧❛ss ♦❢ ▼❱✲❛❧❣❡❜r❛s ❝♦✐♥❝✐❞❡s ✇✐t❤ ❍❙P ([✵, ✶]) ⑤ ❛♥② ▼❱✲❛❧❣❡❜r❛ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❛s ❛ ❤♦♠♦♠♦r♣❤✐❝ ✐♠❛❣❡ ♦❢ ❛ s✉❜❛❧❣❡❜r❛ ♦❢ ❛ ♣r♦❞✉❝t ♦❢ ❝♦♣✐❡s ♦❢ [✵, ✶]✳ ❖r✿ ❚❤❡ ❡q✉❛t✐♦♥s ✭✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❱✲❛❧❣❡❜r❛s✮ t❤❛t ❤♦❧❞ ✐♥ ❛❧❧ ▼❱✲❛❧❣❡❜r❛s ❛r❡ ❡①❛❝t❧② t❤♦s❡ t❤❛t ❤♦❧❞ ✐♥ [✵, ✶]✳ ❖r✿ ❆♥② α ∈ ❋♦r♠ t❤❛t ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ s♦♠❡ ▼❱✲❛❧❣❡❜r❛✱ ❛❧r❡❛❞② ❤❛s ❛ ❝♦✉♥t❡r✲♠♦❞❡❧ ✐♥ [✵, ✶]✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❡q✉❛t✐♦♥✿ ① ∨ ¬① = ✶ . ✭⋆✮ ❚❤❡♥ ✭⋆✮ ✐s ♥♦t ❛♥ ✐❞❡♥t✐t② ♦✈❡r [✵, ✶]✿ t❤❡ ♦♥❧② ❡✈❛❧✉❛t✐♦♥s ✐♥t♦ [✵, ✶] t❤❛t s❛t✐s❢② ✭⋆✮ ❛r❡ ① → ✵ ❛♥❞ ① → ✶ ⑤ t❤❡ ❇♦♦❧❡❛♥✱ ♦r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♦♥s✳ ❍❡r❡ ✐s ❛ ✷✲✈❛r✐❛❜❧❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r t❡r♠✿ ① ① ② ② ✶ ✭ ✮ ❚❤❡ ❡✈❛❧✉❛t✐♦♥s ♦❢ ① ❛♥❞ ② ✐♥t♦ ✵ ✶ ✱ ✐✳❡✳ t❤❡ ♣❛✐rs r s ✵ ✶ ✷✱ t❤❛t s❛t✐s❢② ✱ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♣♦✐♥ts ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✿ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❡q✉❛t✐♦♥✿ ① ∨ ¬① = ✶ . ✭⋆✮ ❚❤❡♥ ✭⋆✮ ✐s ♥♦t ❛♥ ✐❞❡♥t✐t② ♦✈❡r [✵, ✶]✿ t❤❡ ♦♥❧② ❡✈❛❧✉❛t✐♦♥s ✐♥t♦ [✵, ✶] t❤❛t s❛t✐s❢② ✭⋆✮ ❛r❡ ① → ✵ ❛♥❞ ① → ✶ ⑤ t❤❡ ❇♦♦❧❡❛♥✱ ♦r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♦♥s✳ ❍❡r❡ ✐s ❛ ✷✲✈❛r✐❛❜❧❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r t❡r♠✿ ① ∨ ¬① ∨ ② ∨ ¬② = ✶ ✭⋆⋆✮ ❚❤❡ ❡✈❛❧✉❛t✐♦♥s ♦❢ ① ❛♥❞ ② ✐♥t♦ ✵ ✶ ✱ ✐✳❡✳ t❤❡ ♣❛✐rs r s ✵ ✶ ✷✱ t❤❛t s❛t✐s❢② ✱ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♣♦✐♥ts ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✿ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❡q✉❛t✐♦♥✿ ① ∨ ¬① = ✶ . ✭⋆✮ ❚❤❡♥ ✭⋆✮ ✐s ♥♦t ❛♥ ✐❞❡♥t✐t② ♦✈❡r [✵, ✶]✿ t❤❡ ♦♥❧② ❡✈❛❧✉❛t✐♦♥s ✐♥t♦ [✵, ✶] t❤❛t s❛t✐s❢② ✭⋆✮ ❛r❡ ① → ✵ ❛♥❞ ① → ✶ ⑤ t❤❡ ❇♦♦❧❡❛♥✱ ♦r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♦♥s✳ ❍❡r❡ ✐s ❛ ✷✲✈❛r✐❛❜❧❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r t❡r♠✿ ① ∨ ¬① ∨ ② ∨ ¬② = ✶ ✭⋆⋆✮ ❚❤❡ ❡✈❛❧✉❛t✐♦♥s ♦❢ ① ❛♥❞ ② ✐♥t♦ [✵, ✶]✱ ✐✳❡✳ t❤❡ ♣❛✐rs (r, s) ∈ [✵, ✶]✷✱ t❤❛t s❛t✐s❢② (⋆⋆)✱ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♣♦✐♥ts ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✿ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r ❡q✉❛t✐♦♥✿ ① ∨ ¬① = ✶ . ✭⋆✮ ❚❤❡♥ ✭⋆✮ ✐s ♥♦t ❛♥ ✐❞❡♥t✐t② ♦✈❡r [✵, ✶]✿ t❤❡ ♦♥❧② ❡✈❛❧✉❛t✐♦♥s ✐♥t♦ [✵, ✶] t❤❛t s❛t✐s❢② ✭⋆✮ ❛r❡ ① → ✵ ❛♥❞ ① → ✶ ⑤ t❤❡ ❇♦♦❧❡❛♥✱ ♦r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♦♥s✳ ❍❡r❡ ✐s ❛ ✷✲✈❛r✐❛❜❧❡ ❣❡♥❡r❛❧✐s❛t✐♦♥ ♦❢ t❤❡ t❡rt✐✉♠ ♥♦♥ ❞❛t✉r t❡r♠✿ ① ∨ ¬① ∨ ② ∨ ¬② = ✶ ✭⋆⋆✮ ❚❤❡ ❡✈❛❧✉❛t✐♦♥s ♦❢ ① ❛♥❞ ② ✐♥t♦ [✵, ✶]✱ ✐✳❡✳ t❤❡ ♣❛✐rs (r, s) ∈ [✵, ✶]✷✱ t❤❛t s❛t✐s❢② (⋆⋆)✱ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ♣♦✐♥ts ❧②✐♥❣ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✿ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❳ ∨ ¬❳ = ✶ ✭⋆✮ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t ✐♥t❡r✈❛❧✳ ❳ ∨ ¬❳ ∨ ❨ ∨ ¬❨ = ✶ ✭⋆⋆✮ ❚❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ✉♥✐t sq✉❛r❡✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❡ t✇✐st❡❞ ❝✉❜✐❝✿ V ({② − ① ✷, ③ − ① ✸}) ✭P❛r❛♠❡tr✐s❛t✐♦♥✿ t − → (t, t✷, t✸)✳✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Rational polyhedra ▲❡♦♥❛r❞♦✬s ❚r✉♥❝❛t❡❞ ■❝♦s❛❤❡❞r♦♥

✭■❧❧✉str❛t✐♦♥ ❢♦r ▲✉❝❛ P❛❝✐♦❧✐✬s ❚❤❡ ❉✐✈✐♥❡ Pr♦♣♦rt✐♦♥✱ ✶✺✵✾✳✮

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❝♦♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ F ♥ /θ✱ ✇✐t❤ θ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♦♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ θ ✐s ❢❛r ❢r♦♠ ✐♠♠❛t❡r✐❛❧✿ t❤❡r❡ ✐s ♥♦ ❍✐❧❜❡rt✬s ❇❛s✐s ❚❤❡♦r❡♠ ❢♦r ▼❱✲❛❧❣❡❜r❛s✳ ❚❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❛ s❡t P

♥✱ ✇r✐tt❡♥ ❝♦♥✈ P✱ ✐s t❤❡

❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ P✿ ❝♦♥✈ P

♠ ✐ ✶

r✐✈✐ ✈✐ P ❛♥❞ ✵ r✐ ✇✐t❤

♠ ✐ ✶

r✐ ✶ ❙✉❝❤ ❛ s❡t ✐s ❝♦♥✈❡① ✐❢ P ❝♦♥✈ P✳ ❚❤❡ s❡t P ✐s ❝❛❧❧❡❞✿ ❛ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋

♥ ✇✐t❤ P

❝♦♥✈ ❋❀ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋

♥ ✇✐t❤

P ❝♦♥✈ ❋✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❝♦♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ F ♥ /θ✱ ✇✐t❤ θ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♦♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ θ ✐s ❢❛r ❢r♦♠ ✐♠♠❛t❡r✐❛❧✿ t❤❡r❡ ✐s ♥♦ ❍✐❧❜❡rt✬s ❇❛s✐s ❚❤❡♦r❡♠ ❢♦r ▼❱✲❛❧❣❡❜r❛s✳ ❚❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❛ s❡t P ⊆ R♥✱ ✇r✐tt❡♥ ❝♦♥✈ P✱ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ P✿ ❝♦♥✈ P = ♠

• ✐=✶

r✐✈✐ | ✈✐ ∈ P ❛♥❞ ✵ r✐ ∈ R ✇✐t❤

♠

• ✐=✶

r✐ = ✶

• .

❙✉❝❤ ❛ s❡t ✐s ❝♦♥✈❡① ✐❢ P = ❝♦♥✈ P✳ ❚❤❡ s❡t P ✐s ❝❛❧❧❡❞✿ ❛ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋

♥ ✇✐t❤ P

❝♦♥✈ ❋❀ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋

♥ ✇✐t❤

P ❝♦♥✈ ❋✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❝♦♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ F ♥ /θ✱ ✇✐t❤ θ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♦♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ θ ✐s ❢❛r ❢r♦♠ ✐♠♠❛t❡r✐❛❧✿ t❤❡r❡ ✐s ♥♦ ❍✐❧❜❡rt✬s ❇❛s✐s ❚❤❡♦r❡♠ ❢♦r ▼❱✲❛❧❣❡❜r❛s✳ ❚❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❛ s❡t P ⊆ R♥✱ ✇r✐tt❡♥ ❝♦♥✈ P✱ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ P✿ ❝♦♥✈ P = ♠

• ✐=✶

r✐✈✐ | ✈✐ ∈ P ❛♥❞ ✵ r✐ ∈ R ✇✐t❤

♠

• ✐=✶

r✐ = ✶

• .

❙✉❝❤ ❛ s❡t ✐s ❝♦♥✈❡① ✐❢ P = ❝♦♥✈ P✳ ❚❤❡ s❡t P ✐s ❝❛❧❧❡❞✿ ❛ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ⊆ R♥ ✇✐t❤ P = ❝♦♥✈ ❋❀ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋

♥ ✇✐t❤

P ❝♦♥✈ ❋✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❲❡ ❝♦♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❤♦s❡ ♦❢ t❤❡ ❢♦r♠ F ♥ /θ✱ ✇✐t❤ θ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♦♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❤❡ ❛ss✉♠♣t✐♦♥ ♦♥ θ ✐s ❢❛r ❢r♦♠ ✐♠♠❛t❡r✐❛❧✿ t❤❡r❡ ✐s ♥♦ ❍✐❧❜❡rt✬s ❇❛s✐s ❚❤❡♦r❡♠ ❢♦r ▼❱✲❛❧❣❡❜r❛s✳ ❚❤❡ ❝♦♥✈❡① ❤✉❧❧ ♦❢ ❛ s❡t P ⊆ R♥✱ ✇r✐tt❡♥ ❝♦♥✈ P✱ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥ts ♦❢ P✿ ❝♦♥✈ P = ♠

• ✐=✶

r✐✈✐ | ✈✐ ∈ P ❛♥❞ ✵ r✐ ∈ R ✇✐t❤

♠

• ✐=✶

r✐ = ✶

• .

❙✉❝❤ ❛ s❡t ✐s ❝♦♥✈❡① ✐❢ P = ❝♦♥✈ P✳ ❚❤❡ s❡t P ✐s ❝❛❧❧❡❞✿ ❛ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ⊆ R♥ ✇✐t❤ P = ❝♦♥✈ ❋❀ ❛ r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡✱ ✐❢ t❤❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ⊆ Q♥ ✇✐t❤ P = ❝♦♥✈ ❋✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆ ♣♦❧②t♦♣❡ ✐♥ R✷✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆ ♣♦❧②t♦♣❡ ✐♥ R✷ ✭❛ s✐♠♣❧❡①✮✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❆ ✭❝♦♠♣❛❝t✮ ♣♦❧②❤❡❞r♦♥ ✐♥ R♥ ✐s ❛ ✉♥✐♦♥ ♦❢ ☞♥✐t❡❧② ♠❛♥② ♣♦❧②t♦♣❡s ✐♥ R♥✳ ❆ ♣♦❧②❤❡❞r♦♥ ✐♥ R✷✳ ❙✐♠✐❧❛r❧②✱ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ ✐s ❛ ✉♥✐♦♥ ♦❢ ☞♥✐t❡❧② ♠❛♥② r❛t✐♦♥❛❧ ♣♦❧②t♦♣❡s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t P ⊆ R♥ ❜❡ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥✳ ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❢ : P → R ✐s ❛ Z✲♠❛♣ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

1 ❚❤❡r❡ ✐s ❛ ☞♥✐t❡ s❡t {▲✶, . . . , ▲♠} ♦❢ ❛✍♥❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s

▲✐ : R♥ → R s✉❝❤ t❤❛t ❢ (①) = ▲✐① (①) ❢♦r s♦♠❡ ✶ ✐① ♠✳ ❊❛❝❤ ▲✐ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ✐♥t❡❣❡r ❝♦❡✍❝✐❡♥ts✳ ❆ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ [✵, ✶] → R✳ ❆ ♠❛♣ ❋ P

♥

◗

♠ ❜❡t✇❡❡♥ ♣♦❧②❤❡❞r❛ ❛❧✇❛②s ✐s ♦❢ t❤❡

❢♦r♠ ❋ ❢✶ ❢♠ ✱ ❢✐ P ✳ ❚❤❡♥ ❋ ✐s ❛ ✲♠❛♣ ✐❢ ❡❛❝❤ ♦♥❡ ♦❢ ✐ts s❝❛❧❛r ❝♦♠♣♦♥❡♥ts ❢✐ ✐s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t P ⊆ R♥ ❜❡ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥✳ ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❢ : P → R ✐s ❛ Z✲♠❛♣ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

1 ❚❤❡r❡ ✐s ❛ ☞♥✐t❡ s❡t {▲✶, . . . , ▲♠} ♦❢ ❛✍♥❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s

▲✐ : R♥ → R s✉❝❤ t❤❛t ❢ (①) = ▲✐① (①) ❢♦r s♦♠❡ ✶ ✐① ♠✳

2 ❊❛❝❤ ▲✐ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ✐♥t❡❣❡r

❝♦❡✍❝✐❡♥ts✳ ❆ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ [✵, ✶] → R✳ ❆ ♠❛♣ ❋ P

♥

◗

♠ ❜❡t✇❡❡♥ ♣♦❧②❤❡❞r❛ ❛❧✇❛②s ✐s ♦❢ t❤❡

❢♦r♠ ❋ ❢✶ ❢♠ ✱ ❢✐ P ✳ ❚❤❡♥ ❋ ✐s ❛ ✲♠❛♣ ✐❢ ❡❛❝❤ ♦♥❡ ♦❢ ✐ts s❝❛❧❛r ❝♦♠♣♦♥❡♥ts ❢✐ ✐s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

▲❡t P ⊆ R♥ ❜❡ ❛ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥✳ ❆ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ❢ : P → R ✐s ❛ Z✲♠❛♣ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✳

1 ❚❤❡r❡ ✐s ❛ ☞♥✐t❡ s❡t {▲✶, . . . , ▲♠} ♦❢ ❛✍♥❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s

▲✐ : R♥ → R s✉❝❤ t❤❛t ❢ (①) = ▲✐① (①) ❢♦r s♦♠❡ ✶ ✐① ♠✳

2 ❊❛❝❤ ▲✐ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ♣♦❧②♥♦♠✐❛❧ ✇✐t❤ ✐♥t❡❣❡r

❝♦❡✍❝✐❡♥ts✳ ❆ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ [✵, ✶] → R✳ ❆ ♠❛♣ ❋ : P ⊆ R♥ → ◗ ⊆ R♠ ❜❡t✇❡❡♥ ♣♦❧②❤❡❞r❛ ❛❧✇❛②s ✐s ♦❢ t❤❡ ❢♦r♠ ❋ = (❢✶, . . . , ❢♠)✱ ❢✐ : P → R✳ ❚❤❡♥ ❋ ✐s ❛ Z✲♠❛♣ ✐❢ ❡❛❝❤ ♦♥❡ ♦❢ ✐ts s❝❛❧❛r ❝♦♠♣♦♥❡♥ts ❢✐ ✐s✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❘❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ s✉❜s❡ts ♦❢ R♥ t❤❛t ❛r❡ ❞❡☞♥❛❜❧❡ ❜② ❛ t❡r♠ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❱✲❛❧❣❡❜r❛s❀ ❛♥❞ Z✲♠❛♣s ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ❛r❡ ❞❡☞♥❛❜❧❡ ❜② t✉♣❧❡s ♦❢ t❡r♠s ✐♥ t❤❛t ❧❛♥❣✉❛❣❡✳ ❚❤❡ ❝❛t❡❣♦r② ♦❢ ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ❛♥❞ t❤❡✐r ❤♦♠♦♠♦r♣❤✐s♠s✱ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❝❛t❡❣♦r② ♦❢ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✱ ❛♥❞ t❤❡ ✲♠❛♣s ❛♠♦♥❣st t❤❡♠✳

❱✳▼✳ ✫ ▲✳ ❙♣❛❞❛✱ ❉✉❛❧✐t②✱ ♣r♦❥❡❝t✐✈✐t②✱ ❛♥❞ ✉♥✐☞❝❛t✐♦♥ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❛♥❞ ▼❱✲❛❧❣❡❜r❛s✱ ❆♥♥❛❧s ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▲♦❣✐❝✱ ✷✵✶✷✳

♦♣ ❢♣

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❘❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ❛r❡ ♣r❡❝✐s❡❧② t❤❡ s✉❜s❡ts ♦❢ R♥ t❤❛t ❛r❡ ❞❡☞♥❛❜❧❡ ❜② ❛ t❡r♠ ✐♥ t❤❡ ❧❛♥❣✉❛❣❡ ♦❢ ▼❱✲❛❧❣❡❜r❛s❀ ❛♥❞ Z✲♠❛♣s ❛r❡ ♣r❡❝✐s❡❧② t❤❡ ❝♦♥t✐♥✉♦✉s tr❛♥s❢♦r♠❛t✐♦♥s t❤❛t ❛r❡ ❞❡☞♥❛❜❧❡ ❜② t✉♣❧❡s ♦❢ t❡r♠s ✐♥ t❤❛t ❧❛♥❣✉❛❣❡✳ Stone-type duality for finitely presented MV-algebras ❚❤❡ ❝❛t❡❣♦r② ♦❢ ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▼❱✲❛❧❣❡❜r❛s✱ ❛♥❞ t❤❡✐r ❤♦♠♦♠♦r♣❤✐s♠s✱ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♦♣♣♦s✐t❡ ♦❢ t❤❡ ❝❛t❡❣♦r② ♦❢ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✱ ❛♥❞ t❤❡ Z✲♠❛♣s ❛♠♦♥❣st t❤❡♠✳

❱✳▼✳ ✫ ▲✳ ❙♣❛❞❛✱ ❉✉❛❧✐t②✱ ♣r♦❥❡❝t✐✈✐t②✱ ❛♥❞ ✉♥✐☞❝❛t✐♦♥ ✐♥ ✥ ▲✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝ ❛♥❞ ▼❱✲❛❧❣❡❜r❛s✱ ❆♥♥❛❧s ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▲♦❣✐❝✱ ✷✵✶✷✳

Poly♦♣

Q

MV❢♣

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❋r♦♠ ▼❱✲❛❧❣❡❜r❛s t♦ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✿ ●✐✈❡♥ F ♥ /τ(①✶, . . . , ①♥)✱ t❤❡ ❛ss♦❝✐❛t❡❞ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ V (τ) ✐s t❤❡ s❡t ♦❢ ♥✲t✉♣❧❡s (r✶, . . . , r♥) ∈ [✵, ✶]♥ s✉❝❤ t❤❛t τ(r✶, . . . , r♥) = ✵ ✐♥ [✵, ✶]✳ ❋r♦♠ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ t♦ ▼❱✲❛❧❣❡❜r❛s✿ ●✐✈❡♥ P

♥✱ t❤❡

❝♦❧❧❡❝t✐♦♥ P ♦❢ ❛❧❧ ✲♠❛♣s P ✵ ✶ ✐s ❛ ✭☞♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡✮ ▼❱✲❛❧❣❡❜r❛ ✉♥❞❡r t❤❡ ♣♦✐♥t✇✐s❡ ♦♣❡r❛t✐♦♥ ✐♥❤❡r✐t❡❞ ❢r♦♠ ✵ ✶ ✳ ❊①❛♠♣❧❡✳ ■❢ ①✶ ①♥ ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♦ ✵ ✐♥ ❛♥② ▼❱✲❛❧❣❡❜r❛✱ t❤❡♥ ✐t ❣❡♥❡r❛t❡s t❤❡ tr✐✈✐❛❧ ✐❞❡❛❧ ✵ ✳ ■♥ t❤✐s ❝❛s❡✱

♥ ♥✱ ❛♥❞

✵ ✶ ♥✳ ❍❡♥❝❡ t❤❡ ❞✉❛❧s ♦❢ ❢r❡❡ ❛❧❣❡❜r❛s ❛r❡ t❤❡ ✉♥✐t ❝✉❜❡s✳

❘❡♠❛r❦✳ ❚❤❡ s✉❜s♣❛❝❡ ✵ ✶ ♥ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❛①✐♠❛❧ s♣❡❝tr❛❧ s♣❛❝❡ ♦❢

♥

✱ t♦♣♦❧♦❣✐s❡❞ ❜② t❤❡ ✭❛♥❛❧♦❣✉❡ ♦❢✮ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ▼❱✲❛❧❣❡❜r❛ P ✐s t❤❡ ❡①❛❝t ❛♥❛❧♦❣✉❡ ❢♦r r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ r✐♥❣ ♦❢ ❛♥ ❛✍♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❋r♦♠ ▼❱✲❛❧❣❡❜r❛s t♦ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✿ ●✐✈❡♥ F ♥ /τ(①✶, . . . , ①♥)✱ t❤❡ ❛ss♦❝✐❛t❡❞ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ V (τ) ✐s t❤❡ s❡t ♦❢ ♥✲t✉♣❧❡s (r✶, . . . , r♥) ∈ [✵, ✶]♥ s✉❝❤ t❤❛t τ(r✶, . . . , r♥) = ✵ ✐♥ [✵, ✶]✳ ❋r♦♠ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ t♦ ▼❱✲❛❧❣❡❜r❛s✿ ●✐✈❡♥ P ⊆ R♥✱ t❤❡ ❝♦❧❧❡❝t✐♦♥ ∇ (P) ♦❢ ❛❧❧ Z✲♠❛♣s P → [✵, ✶] ✐s ❛ ✭☞♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡✮ ▼❱✲❛❧❣❡❜r❛ ✉♥❞❡r t❤❡ ♣♦✐♥t✇✐s❡ ♦♣❡r❛t✐♦♥ ✐♥❤❡r✐t❡❞ ❢r♦♠ [✵, ✶]✳ ❊①❛♠♣❧❡✳ ■❢ ①✶ ①♥ ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♦ ✵ ✐♥ ❛♥② ▼❱✲❛❧❣❡❜r❛✱ t❤❡♥ ✐t ❣❡♥❡r❛t❡s t❤❡ tr✐✈✐❛❧ ✐❞❡❛❧ ✵ ✳ ■♥ t❤✐s ❝❛s❡✱

♥ ♥✱ ❛♥❞

✵ ✶ ♥✳ ❍❡♥❝❡ t❤❡ ❞✉❛❧s ♦❢ ❢r❡❡ ❛❧❣❡❜r❛s ❛r❡ t❤❡ ✉♥✐t ❝✉❜❡s✳

❘❡♠❛r❦✳ ❚❤❡ s✉❜s♣❛❝❡ ✵ ✶ ♥ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❛①✐♠❛❧ s♣❡❝tr❛❧ s♣❛❝❡ ♦❢

♥

✱ t♦♣♦❧♦❣✐s❡❞ ❜② t❤❡ ✭❛♥❛❧♦❣✉❡ ♦❢✮ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ▼❱✲❛❧❣❡❜r❛ P ✐s t❤❡ ❡①❛❝t ❛♥❛❧♦❣✉❡ ❢♦r r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ r✐♥❣ ♦❢ ❛♥ ❛✍♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❋r♦♠ ▼❱✲❛❧❣❡❜r❛s t♦ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✿ ●✐✈❡♥ F ♥ /τ(①✶, . . . , ①♥)✱ t❤❡ ❛ss♦❝✐❛t❡❞ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ V (τ) ✐s t❤❡ s❡t ♦❢ ♥✲t✉♣❧❡s (r✶, . . . , r♥) ∈ [✵, ✶]♥ s✉❝❤ t❤❛t τ(r✶, . . . , r♥) = ✵ ✐♥ [✵, ✶]✳ ❋r♦♠ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ t♦ ▼❱✲❛❧❣❡❜r❛s✿ ●✐✈❡♥ P ⊆ R♥✱ t❤❡ ❝♦❧❧❡❝t✐♦♥ ∇ (P) ♦❢ ❛❧❧ Z✲♠❛♣s P → [✵, ✶] ✐s ❛ ✭☞♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡✮ ▼❱✲❛❧❣❡❜r❛ ✉♥❞❡r t❤❡ ♣♦✐♥t✇✐s❡ ♦♣❡r❛t✐♦♥ ✐♥❤❡r✐t❡❞ ❢r♦♠ [✵, ✶]✳ ❊①❛♠♣❧❡✳ ■❢ τ(①✶, . . . , ①♥) ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♦ ✵ ✐♥ ❛♥② ▼❱✲❛❧❣❡❜r❛✱ t❤❡♥ ✐t ❣❡♥❡r❛t❡s t❤❡ tr✐✈✐❛❧ ✐❞❡❛❧ {✵}✳ ■♥ t❤✐s ❝❛s❡✱ F ♥ /τ = F ♥✱ ❛♥❞ V (τ) = [✵, ✶]♥✳ ❍❡♥❝❡ t❤❡ ❞✉❛❧s ♦❢ ❢r❡❡ ❛❧❣❡❜r❛s ❛r❡ t❤❡ ✉♥✐t ❝✉❜❡s✳

❘❡♠❛r❦✳ ❚❤❡ s✉❜s♣❛❝❡ ✵ ✶ ♥ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❛①✐♠❛❧ s♣❡❝tr❛❧ s♣❛❝❡ ♦❢

♥

✱ t♦♣♦❧♦❣✐s❡❞ ❜② t❤❡ ✭❛♥❛❧♦❣✉❡ ♦❢✮ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ▼❱✲❛❧❣❡❜r❛ P ✐s t❤❡ ❡①❛❝t ❛♥❛❧♦❣✉❡ ❢♦r r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ r✐♥❣ ♦❢ ❛♥ ❛✍♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❋r♦♠ ▼❱✲❛❧❣❡❜r❛s t♦ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛✿ ●✐✈❡♥ F ♥ /τ(①✶, . . . , ①♥)✱ t❤❡ ❛ss♦❝✐❛t❡❞ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ V (τ) ✐s t❤❡ s❡t ♦❢ ♥✲t✉♣❧❡s (r✶, . . . , r♥) ∈ [✵, ✶]♥ s✉❝❤ t❤❛t τ(r✶, . . . , r♥) = ✵ ✐♥ [✵, ✶]✳ ❋r♦♠ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ t♦ ▼❱✲❛❧❣❡❜r❛s✿ ●✐✈❡♥ P ⊆ R♥✱ t❤❡ ❝♦❧❧❡❝t✐♦♥ ∇ (P) ♦❢ ❛❧❧ Z✲♠❛♣s P → [✵, ✶] ✐s ❛ ✭☞♥✐t❡❧② ♣r❡s❡♥t❛❜❧❡✮ ▼❱✲❛❧❣❡❜r❛ ✉♥❞❡r t❤❡ ♣♦✐♥t✇✐s❡ ♦♣❡r❛t✐♦♥ ✐♥❤❡r✐t❡❞ ❢r♦♠ [✵, ✶]✳ ❊①❛♠♣❧❡✳ ■❢ τ(①✶, . . . , ①♥) ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♦ ✵ ✐♥ ❛♥② ▼❱✲❛❧❣❡❜r❛✱ t❤❡♥ ✐t ❣❡♥❡r❛t❡s t❤❡ tr✐✈✐❛❧ ✐❞❡❛❧ {✵}✳ ■♥ t❤✐s ❝❛s❡✱ F ♥ /τ = F ♥✱ ❛♥❞ V (τ) = [✵, ✶]♥✳ ❍❡♥❝❡ t❤❡ ❞✉❛❧s ♦❢ ❢r❡❡ ❛❧❣❡❜r❛s ❛r❡ t❤❡ ✉♥✐t ❝✉❜❡s✳

❘❡♠❛r❦✳ ❚❤❡ s✉❜s♣❛❝❡ V (τ) ⊆ [✵, ✶]♥ ❤♦♠❡♦♠♦r♣❤✐❝ t♦ t❤❡ ♠❛①✐♠❛❧ s♣❡❝tr❛❧ s♣❛❝❡ ♦❢ F ♥ /τ✱ t♦♣♦❧♦❣✐s❡❞ ❜② t❤❡ ✭❛♥❛❧♦❣✉❡ ♦❢✮ t❤❡ ❩❛r✐s❦✐ t♦♣♦❧♦❣②✳ ❚❤❡ ▼❱✲❛❧❣❡❜r❛ ∇ (P) ✐s t❤❡ ❡①❛❝t ❛♥❛❧♦❣✉❡ ❢♦r r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r❛ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ r✐♥❣ ♦❢ ❛♥ ❛✍♥❡ ❛❧❣❡❜r❛✐❝ ✈❛r✐❡t②✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❙②♥t❛①✿ ❊q✉❛t✐♦♥s✱ ♦r ❋♦r♠✉❧✚ ⑤ ❆❧❣❡❜r❛✐❝

❳ ∨ ¬❳ = ✶ ❳ ∨ ¬❳ ∨ ❨ ∨ ¬❨ = ✶

❙❡♠❛♥t✐❝s✿ ❙♦❧✉t✐♦♥s✱ ♦r ▼♦❞❡❧s ⑤ ●❡♦♠❡tr✐❝❛❧

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❳ ∨ ¬❳ = ✶ ❳ ∨ ¬❳ ∨ ❨ ∨ ¬❨ = ✶ Stone-type duality for finitely presented MV-algebras Poly♦♣

Q

MV❢♣

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Epilogue StoneSp♦♣ BoolAlg

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

■♥ t❤❡ t❤✐rt✐❡s✱ ❙t♦♥❡ ❞✐s❝♦✈❡r❡❞ t❤❛t t❤❡ s❡t ♦❢ ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❝❛rr✐❡s ❛ ♥❛t✉r❛❧ t♦♣♦❧♦❣②✿ ♦♣❡♥ s❡ts ❝♦rr❡s♣♦♥❞ t♦ ❛r❜✐tr❛r② ✐❞❡❛❧s✳ ■♥ t❤❡ ■♥tr♦❞✉❝t✐♦♥ t♦ ❤✐s ❜♦♦❦ ♦♥ ❙t♦♥❡ s♣❛❝❡s✱ P✳ ❏♦❤♥st♦♥❡ ✇r✐t❡s✿ ◆♦✇ t❤✐s ✇❛s ❛ r❡❛❧❧② ❜♦❧❞ ✐❞❡❛✳ ❆❧t❤♦✉❣❤ t❤❡ ♣r❛❝t✐t✐♦♥❡rs ♦❢ ❛❜str❛❝t ❣❡♥❡r❛❧ t♦♣♦❧♦❣② ❬✳✳✳❪ ❤❛❞ ❜② t❤❡ ❡❛r❧② t❤✐rt✐❡s ❞❡✈❡❧♦♣❡❞ ❝♦♥s✐❞❡r❛❜❧❡ ❡①♣❡rt✐s❡ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ s♣❛❝❡s ✇✐t❤ ♣❛rt✐❝✉❧❛r ♣r♦♣❡rt✐❡s✱ t❤❡ ♠♦t✐✈❛t✐♦♥ ♦❢ t❤❡ s✉❜❥❡❝t ✇❛s st✐❧❧ ❣❡♦♠❡tr✐❝❛❧ ❬✳✳✳❪ ❛♥❞ ✭❛s ❢❛r ❛s ■ ❦♥♦✇✮ ♥♦❜♦❞② ❤❛❞ ♣r❡✈✐♦✉s❧② ❤❛❞ t❤❡ ✐❞❡❛ ♦❢ ❛♣♣❧②✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s t♦ t❤❡ st✉❞② ♦❢ s♣❛❝❡s ❝♦♥str✉❝t❡❞ ❢r♦♠ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝ ❞❛t❛✳ ❚❤❡ ❡♥s✉✐♥❣ s♣❛❝❡s ❛r❡ ♥♦✇❛❞❛②s ❝❛❧❧❡❞ ❙t♦♥❡ s♣❛❝❡s✳ ❚❤❡ ❝❧♦♣❡♥ s❡ts ⑤ t❤♦s❡ s❡ts ✇❤✐❝❤ ❛r❡ ❜♦t❤ ❝❧♦s❡❞ ❛♥❞ ♦♣❡♥ ✐♥ t❤❡ t♦♣♦❧♦❣② ⑤ ❝♦rr❡s♣♦♥❞ t♦ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧s✴☞❧t❡rs✱ ❛♥❞ ❤❡♥❝❡ t♦ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❛❧❣❡❜r❛✳ ❚❤✉s✱ t❤❡ ♦r✐❣✐♥❛❧ ❛❧❣❡❜r❛ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ ✐ts s♣❛❝❡ ♦❢ ♠❛①✐♠❛❧ ✐❞❡❛❧s❀ ❙t♦♥❡✬s ❝♦♥str✉❝t✐♦♥ ✐s ✐♥ ❢❛❝t ❛ t✇♦✲✇❛② r♦❛❞✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

■♥ t❤❡ t❤✐rt✐❡s✱ ❙t♦♥❡ ❞✐s❝♦✈❡r❡❞ t❤❛t t❤❡ s❡t ♦❢ ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❝❛rr✐❡s ❛ ♥❛t✉r❛❧ t♦♣♦❧♦❣②✿ ♦♣❡♥ s❡ts ❝♦rr❡s♣♦♥❞ t♦ ❛r❜✐tr❛r② ✐❞❡❛❧s✳ ■♥ t❤❡ ■♥tr♦❞✉❝t✐♦♥ t♦ ❤✐s ❜♦♦❦ ♦♥ ❙t♦♥❡ s♣❛❝❡s✱ P✳ ❏♦❤♥st♦♥❡ ✇r✐t❡s✿ ◆♦✇ t❤✐s ✇❛s ❛ r❡❛❧❧② ❜♦❧❞ ✐❞❡❛✳ ❆❧t❤♦✉❣❤ t❤❡ ♣r❛❝t✐t✐♦♥❡rs ♦❢ ❛❜str❛❝t ❣❡♥❡r❛❧ t♦♣♦❧♦❣② ❬✳✳✳❪ ❤❛❞ ❜② t❤❡ ❡❛r❧② t❤✐rt✐❡s ❞❡✈❡❧♦♣❡❞ ❝♦♥s✐❞❡r❛❜❧❡ ❡①♣❡rt✐s❡ ✐♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ s♣❛❝❡s ✇✐t❤ ♣❛rt✐❝✉❧❛r ♣r♦♣❡rt✐❡s✱ t❤❡ ♠♦t✐✈❛t✐♦♥ ♦❢ t❤❡ s✉❜❥❡❝t ✇❛s st✐❧❧ ❣❡♦♠❡tr✐❝❛❧ ❬✳✳✳❪ ❛♥❞ ✭❛s ❢❛r ❛s ■ ❦♥♦✇✮ ♥♦❜♦❞② ❤❛❞ ♣r❡✈✐♦✉s❧② ❤❛❞ t❤❡ ✐❞❡❛ ♦❢ ❛♣♣❧②✐♥❣ t❤❡s❡ t❡❝❤♥✐q✉❡s t♦ t❤❡ st✉❞② ♦❢ s♣❛❝❡s ❝♦♥str✉❝t❡❞ ❢r♦♠ ♣✉r❡❧② ❛❧❣❡❜r❛✐❝ ❞❛t❛✳ ❚❤❡ ❡♥s✉✐♥❣ s♣❛❝❡s ❛r❡ ♥♦✇❛❞❛②s ❝❛❧❧❡❞ ❙t♦♥❡ s♣❛❝❡s✳ ❚❤❡ ❝❧♦♣❡♥ s❡ts ⑤ t❤♦s❡ s❡ts ✇❤✐❝❤ ❛r❡ ❜♦t❤ ❝❧♦s❡❞ ❛♥❞ ♦♣❡♥ ✐♥ t❤❡ t♦♣♦❧♦❣② ⑤ ❝♦rr❡s♣♦♥❞ t♦ ♣r✐♥❝✐♣❛❧ ✐❞❡❛❧s✴☞❧t❡rs✱ ❛♥❞ ❤❡♥❝❡ t♦ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❛❧❣❡❜r❛✳ ❚❤✉s✱ t❤❡ ♦r✐❣✐♥❛❧ ❛❧❣❡❜r❛ ❝❛♥ ❜❡ r❡❝♦✈❡r❡❞ ❢r♦♠ ✐ts s♣❛❝❡ ♦❢ ♠❛①✐♠❛❧ ✐❞❡❛❧s❀ ❙t♦♥❡✬s ❝♦♥str✉❝t✐♦♥ ✐s ✐♥ ❢❛❝t ❛ t✇♦✲✇❛② r♦❛❞✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

The syntax-semantics dictionary✳

Algebra, or Syntax. Topology, or Semantics. ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❙t♦♥❡ ✭♦r ❇♦♦❧❡❛♥✮ s♣❛❝❡ ❍♦♠♦♠♦r♣❤✐s♠ ❈♦♥t✐♥✉♦✉s ♠❛♣ ❋✐♥✐t❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❋✐♥✐t❡ s❡t ❋✐♥✐t❡ ❛❧❣❡❜r❛ ❤♦♠♦♠♦r♣❤✐s♠ ❋✉♥❝t✐♦♥ ❋r❡❡ ♥✲❣❡♥✳ ❛❧❣❡❜r❛ {✵, ✶}♥ ▼❛①✐♠❛❧ ✐❞❡❛❧ P♦✐♥t ♦❢ ❙t♦♥❡ s♣❛❝❡ ■❞❡❛❧ ❈❧♦s❡❞ s✉❜s❡t ♦❢ ❙t♦♥❡ s♣❛❝❡ Pr✐♥❝✐♣❛❧ ✐❞❡❛❧ ❈❧♦♣❡♥ s✉❜s❡t ♦❢ ❙t♦♥❡ s♣❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳

StoneSp♦♣ BoolAlg

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

The syntax-semantics dictionary✳

Algebra, or Syntax. Geometry, or Semantics. ❋✳♣✳ ❛❧❣❡❜r❛ ❘❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ ❍♦♠♦♠♦r♣❤✐s♠ Z✲♠❛♣ ❋✳♣✳ s✉❜❛❧❣❡❜r❛ ❈♦♥t✐♥✉♦✉s ✐♠❛❣❡ ❜② Z✲♠❛♣ ❋✳♣✳ q✉♦t✐❡♥t ❛❧❣❡❜r❛ ❘❛t✐♦♥❛❧ s✉❜♣♦❧②❤❡❞r♦♥ ❋✳♣✳ ♣r♦❥❡❝t✐✈❡ ❛❧❣❡❜r❛ ❘❡tr❛❝t ♦❢ ❝✉❜❡ ❜② Z✲♠❛♣s ❋r❡❡ ♥✲❣❡♥✳ ❛❧❣❡❜r❛ [✵, ✶]♥ ▼❛①✐♠❛❧ ❝♦♥❣r✉❡♥❝❡ P♦✐♥t ♦❢ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ ■♥t❡rs❡❝t✐♦♥ ♦❢ ♠❛①✐♠❛❧ ❝♦♥❣✳ ❈❧♦s❡❞ s✉❜s❡t ♦❢ r❛t✐♦♥❛❧ ♣♦❧②❤❡❞r♦♥ ❋✐♥✐t❡ ♣r♦❞✉❝t ❆ × ❇ ❋✐♥✐t❡ ❞✐s❥♦✐♥t ✉♥✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ Poly♦♣

Q

MV❢♣

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

Algebraic geometry General algebra

• r♦✉♥❞ ☞❡❧❞ ❦

❆ ❦[①✶, . . . , ①♥] F♥ ❆✍♥❡ s♣❛❝❡ ❦ ♥ ❆♥ ■❞❡❛❧ ♦❢ ❦[①✶, . . . , ①♥] ❈♦♥❣r✉❡♥❝❡ ♦♥ F♥ ❆✍♥❡ ✈❛r✐❡t② ✐♥ ❦ ♥

• ❛❧♦✐s✲☞①❡❞ s✉❜s❡t ♦❢ ❆♥

❈♦♦r❞✳ r✐♥❣ ❦[①✐]/ I (V (❙)) ◗✉♦t✐❡♥t F♥/I (V (❙)) ❍♦♠♦♠♦r♣❤✐s♠ ♦❢ ❦✲❛❧❣✳ V✲❤♦♠♦♠♦r♣❤✐s♠ ▼❛♣ ♦❢ ❛✍♥❡ ✈❛r✐❡t✐❡s ❚❡r♠✲❞❡☞♥❛❜❧❡ ♠❛♣ ◆✉❧❧st❡❧❧❡♥s❛t③ ❱✳▼✳ ✫ ▲✳ ❙♣❛❞❛✱ ✷✵✶✷ ❝♦✲◆✉❧❧st❡❧❧❡♥s❛t③ ? ▼❛①✐♠❛❧ ✐❞❡❛❧ ▼❛①✐♠❛❧ ❝♦♥❣r✉❡♥❝❡ ✳ ✳ ✳ ✳ ✳ ✳

❱✳▼✳ ❛♥❞ ▲✳ ❙♣❛❞❛✱ ❚❤❡ ❉✉❛❧ ❆❞❥✉♥❝t✐♦♥ ❜❡t✇❡❡♥ ▼❱✲❛❧❣❡❜r❛s ❛♥❞ ❚②❝❤♦♥♦☛ s♣❛❝❡s✱ ❙t✉❞✐❛ ▲♦❣✐❝❛ ✶✵✵✱ ✐♥ ♠❡♠♦r✐❛♠ ▲❡♦ ❊s❛❦✐❛✱ ✷✵✶✷✳ ❖✳ ❈❛r❛♠❡❧❧♦✱ ❱✳▼✳✱ ❛♥❞ ▲✳ ❙♣❛❞❛✱ ●❡♥❡r❛❧ ❛✍♥❡ ❛❞❥✉♥❝t✐♦♥s✱ ❛♥❞ ◆✉❧❧st❡❧❧❡♥s⑧ ❛t③❡✱ ♣r❡❧✐♠✐♥❛r② ❛r❳✐✈ ✈❡rs✐♦♥✱ ✷✵✶✹✳

✥ ▲✉❦❛s✐❡✇✐❝③ ❈❤❛♥❣ P♦❧②❤❡❞r❛ ❊♣✐❧♦❣✉❡

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✳