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geometric aspects of lukasiewicz logic

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

Jan 04, 2023 •467 likes •1.59k views

s Pr Geometric aspects of Lukasiewicz logic A short excursion rr vincenzo.marra@unimi.it

  1. ✇ ✵✳ ✇ ✶ ✇ ✳ ✶ ✐❢ ✇ ✇ ✇ ✶ ♩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 ✳

  2. ✇ ✶ ✇ ✳ ✶ ✐❢ ✇ ✇ ✇ ✶ ♩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 ✳ ✇ ( ⊥ ) = ✵✳

  3. ✶ ✐❢ ✇ ✇ ✇ ✶ ♩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 ) ✳

  4. ✥ ▲✉❊❛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 ) ✳ ï¿œ ✶ ✐❢ ✇ ( α ) ï¿œ ✇ ( β ) ✇ ( α → β ) = ✶ − ( ✇ ( α ) − ✇ ( β )) ♩t❀❡r✇✐s❡✳

  5. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚r✉t❀✲❢✉♥❝t✐♊♥ ♊❢ ✥ ▲✉❊❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♊♥✳ ï¿œ ✶ ✐❢ ✇ ( α ) ï¿œ ✇ ( β ) ✇ ( α → β ) = ✶ − ( ✇ ( α ) − ✇ ( β )) ♩t❀❡r✇✐s❡✳

  6. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚r✉t❀✲❢✉♥❝t✐♊♥ ♊❢ ✥ ▲✉❊❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♊♥✳ ✇ ( α → β ) = ♠✐♥ { ✶ , ✶ − ( ✇ ( α ) − ✇ ( β )) }

  7. 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❡✳

  8. ✥ ▲✉❊❛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 ❧♊❣✐❝✳

  9. ✥ ▲✉❊❛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 ❧♊❣✐❝✳

  10. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚r✉t❀✲❢✉♥❝t✐♊♥ ♊❢ ✥ ▲✉❊❛s✐❡✇✐❝③ ❭str♊♥❣ ❝♊♥❥✉♥❝t✐♊♥✧ ⊙ ✳ ✭ ◆♩t❡✿ ◆♊♥✲✐❞❡♠♣♊t❡♥t ♊♣❡r❛t✐♊♥✳ ✮ ✇ ( α ⊙ β ) = ♠❛① { ✵ , ✇ ( α ) + ✇ ( β ) − ✶ }

  11. ✭ ❊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 ✇ ✳

  12. ✥ ▲✉❊❛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✐♊♥✮ ✭ ❈♊♥s❡q✉❡♥t✐❛ ♠✐r❛❜✐❧✐s ✮ ( ¬ α → α ) → α ( 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❡✿ ï¿œ α t♩ ♠❡❛♥ α ∈ ❚❛✉t ✳

  13. ❚❀❡ 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❧❞✮✳

  14. ◆❡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✐♊♥✳

  15. ✥ ▲✉❊❛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♠✉❧❛✳

  16. ◆♩✇ ✇❡ ❞❡❝❧❛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

  17. ❲❡ 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 ∈ ❚❀♠ ✳

  18. ✥ ▲✉❊❛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✳

  19. ❊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✐❝❛❧ ❧♊❣✐❝✳

  20. ❆ ❢♊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✳

  21. ■♠♣❧✐❝❛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✐✳

  22. ❄ ❈♊♥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✐✈❡✳

  23. ❈♊♥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 ) ❄

  24. ❈♊♥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✐♊♥✳

  25. ❯♣♊♥ ❞❡☞♥✐♥❣ ✭❆✵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✳

  26. ✥ ▲✉❊❛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✳

  27. ❯♣♊♥ ❞❡☞♥✐♥❣ ✭❆✵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

  28. ✥ ▲✉❊❛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✐❝❛❧ ❧♊❣✐❝✳ α α → β (R1) ▌♊❞✉s ♣♊♥❡♥s✳ β

  29. ✥ ▲✉❊❛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 ❧♊❣✐❝✳ α α → β (R1) ▌♊❞✉s ♣♊♥❡♥s✳ β

  30. ❙✉❝❀ ❭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 ❧❛✇✳

  31. ✥ ▲✉❊❛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 ✐♥❞❡❡❞✳

  32. ▌♊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❡❢✉❧✳✮ ✥ ▲✉❊❛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 ✐♥❞❡❡❞✳ ✶ ❆❧♠♩st ✈❡r❜❛t✐♠ ❢r♩♠ ❏✳ ▌♊s❝❀♊✈❛❊✐s✱ ■♥t✉✐t✐♊♥✐st✐❝ ▲♊❣✐❝ ✱ ❚❀❡ ❙t❛♥❢♊r❞ ❊♥❝②❝❧♊♣❡❞✐❛ ♊❢ P❀✐❧♊s♊♣❀②✱ ✷✵✶✵✱ ❊❞✇❛r❞ ◆✳ ❩❛❧t❛ ✭❡❞✳✮✳

  33. ✥ ▲✉❊❛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❛ ✭❡❞✳✮✳

  34. ❚❛✉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✿

  35. ✥ ▲✉❊❛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 ï¿œ 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✳

  36. ❋♩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 ❙ ✱ ❛♥❞ ❙ ï¿œ α ✐❢ α ❀♊❧❞s ✐♥ ❡❛❝❀ ♠♊❞❡❧ ✭❂♣♊ss✐❜❧❡ ✇♩r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❀❡r❡✐♥ ❡❛❝❀ ❢♊r♠✉❧❛ ♊❢ ❙ ❀♊❧❞s✳

  37. ■♥ 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 ❙ ✱ ❛♥❞ ❙ ï¿œ α ✐❢ α ❀♊❧❞s ✐♥ ❡❛❝❀ ♠♊❞❡❧ ✭❂♣♊ss✐❜❧❡ ✇♩r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❀❡r❡✐♥ ❡❛❝❀ ❢♊r♠✉❧❛ ♊❢ ❙ ❀♊❧❞s✳ Strong Completeness Theorem for CL ❋♩r ❛♥② α ∈ ❋♩r♠ ✱ ❛♥❞ ❛♥② s❡t ❙ ⊆ ❋♩r♠ ✱ ❙ ï¿œ α ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ ❙ ⊢ α .

  38. ✥ ▲✉❊❛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 ❙ ✱ ❛♥❞ ❙ ï¿œ α ✐❢ α ❀♊❧❞s ✐♥ ❡❛❝❀ ♠♊❞❡❧ ✭❂♣♊ss✐❜❧❡ ✇♩r❧❞✱ ❛ss✐❣♥♠❡♥t✮ ✇❀❡r❡✐♥ ❡❛❝❀ ❢♊r♠✉❧❛ ♊❢ ❙ ❀♊❧❞s✳ Strong Completeness Theorem for CL ❋♩r ❛♥② α ∈ ❋♩r♠ ✱ ❛♥❞ ❛♥② s❡t ❙ ⊆ ❋♩r♠ ✱ ❙ ï¿œ 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✐♊♥✳

  39. ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ▲✉❊❛s✐❡✇✐❝③ ❧♊❣✐❝ ❢❛✐❧s str♊♥❣ ❝♊♠♣❧❡t❡♥❡ss✳ ✥ ▲❡t ❙ ❜❡ t❀❡ s❡t ♊❢ ❢♊r♠✉❧✚ ✐♥ ♊♥❡ ✈❛r✐❛❜❧❡ ♣ ✿ ϕ ♥ ( ♣ ) := (( ♥ + ✶ )( ♣ ♥ ∧ ¬ ♣ )) ⊕ ♣ ♥ + ✶ , ❢♊r ❡❛❝❀ ✐♥t❡❣❡r ♥ ï¿œ ✶✱ ✇❀❡r❡ ♣ ❊ := ♣ ⊙ · · · ⊙ ♣ , ï¿œ ￜￜ ï¿œ ❊ t✐♠❡s ❊♣ := ♣ ⊕ · · · ⊕ ♣ . ï¿œ ￜￜ ï¿œ ❊ t✐♠❡s ❚❀❡♥ ❙ ￜ⊢ ✥ ▲ ♣ ✱ ❜✉t ❙ ï¿œ ✥ ▲ ♣ ✳

  40. ❙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❛ ❊♣✐❧♊❣✉❡ ❙ ￜ⊢ ✥ ▲ ♣ ✱ ❜✉t ❙ ï¿œ ✥ ▲ ♣ ✳ ■♥t✉✐t✐✈❡❧②✱ ②♊✉ ❝❛♥ t❀✐♥❊ ♊❢ ❙ ❛s ❡♠❜♊❞②✐♥❣ t❀❡ ❢♊❧❧♊✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♊❢ ❛ss✉♠♣t✐♊♥s✿ 1 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✶ / ✷✳ 2 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✷ / ✞✳ 3 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✾ / ✹✳ 4 ✳ ✳ ✳

  41. ❙❡♠❛♥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❛ ❊♣✐❧♊❣✉❡ ❙ ￜ⊢ ✥ ▲ ♣ ✱ ❜✉t ❙ ï¿œ ✥ ▲ ♣ ✳ ■♥t✉✐t✐✈❡❧②✱ ②♊✉ ❝❛♥ t❀✐♥❊ ♊❢ ❙ ❛s ❡♠❜♊❞②✐♥❣ t❀❡ ❢♊❧❧♊✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♊❢ ❛ss✉♠♣t✐♊♥s✿ 1 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✶ / ✷✳ 2 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✷ / ✞✳ 3 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✾ / ✹✳ 4 ✳ ✳ ✳ ❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♊❡s ♥♊t ❢♊❧❧♊✇ t❀❛t✿ ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ￜ⊢ ✥ ▲ ♣ ✳ ❚❀✐s ✐s ❜❡❝❛✉s❡ ❛♥② ♣r♊♊❢ ♊❢ ♣ ❢r♩♠ ❙ ❝❛♥ ♊♥❧② ✉s❡s ❛ ☞♥✐t❡ s✉❜s❡t ♊❢ ❙ ✳

  42. ❚❛❊✐♥❣ st♊❝❊✳ ▲ ✐s ✱ ❜✉t ▲ ✐s ♥♊t✳ ✥ ✥ ◆♩t❡✳ ❙ ❛❧✇❛②s✳ ❙ ▲ ✥ ▲ ✥ ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❙ ￜ⊢ ✥ ▲ ♣ ✱ ❜✉t ❙ ï¿œ ✥ ▲ ♣ ✳ ■♥t✉✐t✐✈❡❧②✱ ②♊✉ ❝❛♥ t❀✐♥❊ ♊❢ ❙ ❛s ❡♠❜♊❞②✐♥❣ t❀❡ ❢♊❧❧♊✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♊❢ ❛ss✉♠♣t✐♊♥s✿ 1 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✶ / ✷✳ 2 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✷ / ✞✳ 3 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✾ / ✹✳ 4 ✳ ✳ ✳ ❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♊❡s ♥♊t ❢♊❧❧♊✇ t❀❛t✿ ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ￜ⊢ ✥ ▲ ♣ ✳ ❚❀✐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 ✇ ( ♣ ) = ✶✱ ✐✳❡✳ ❙ ï¿œ ✥ ▲ ♣ ✳

  43. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❙ ￜ⊢ ✥ ▲ ♣ ✱ ❜✉t ❙ ï¿œ ✥ ▲ ♣ ✳ ■♥t✉✐t✐✈❡❧②✱ ②♊✉ ❝❛♥ t❀✐♥❊ ♊❢ ❙ ❛s ❡♠❜♊❞②✐♥❣ t❀❡ ❢♊❧❧♊✇✐♥❣ ✐♥☞♥✐t❡ s❡t ♊❢ ❛ss✉♠♣t✐♊♥s✿ 1 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✶ / ✷✳ 2 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✷ / ✞✳ 3 ♣ := ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ ï¿œ ✾ / ✹✳ 4 ✳ ✳ ✳ ❙②♥t❛❝t✐❝❛❧❧②✱ ✐t ❞♊❡s ♥♊t ❢♊❧❧♊✇ t❀❛t✿ ❭❊♥③♊ ✐s t❛❧❧✧ ✐s tr✉❡ t♩ ❞❡❣r❡❡ = ✶✳✧✱ ✐✳❡✳ ❙ ￜ⊢ ✥ ▲ ♣ ✳ ❚❀✐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 ✇ ( ♣ ) = ✶✱ ✐✳❡✳ ❙ ï¿œ ✥ ▲ ♣ ✳ ❚❛❊✐♥❣ st♊❝❊✳ ⊢ ✥ ▲ ✐s compact ✱ ❜✉t ï¿œ ✥ ▲ ✐s ♥♊t✳ ◆♩t❡✳ ❙ ⊢ ✥ ▲ α ⇒ ❙ ï¿œ ✥ ▲ α ❛❧✇❛②s✳

  44. ❆ ❢♊❧❊❧♊r❡ t❀❡♊r❡♠✿ ❋♩r ❛♥② ❋♩r♠ ✱ ❛♥❞ ❛♥② ♠❛①✐♠❛❧ ❝♊♥s✐st❡♥t s❡t â–Œ ❋♩r♠ ✱ ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ â–Œ â–Œ ▲ ✥ ▲ ✥ ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚❀❡ ❍❛②✲❲✓ ♊❥❝✐❝❊✐ ❚❀❡♊r❡♠✿ Completeness Theorem for f.a. theories in ï¿œ L ❋♩r ❛♥② α ∈ ❋♩r♠ ✱ ❛♥❞ ❛♥② ☞♥✐t❡ s❡t ❋ ⊆ ❋♩r♠ ✱ ▲ α ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ ❋ ⊢ ✥ ▲ α . ❋ ï¿œ ✥

  45. ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚❀❡ ❍❛②✲❲✓ ♊❥❝✐❝❊✐ ❚❀❡♊r❡♠✿ Completeness Theorem for f.a. theories in ï¿œ L ❋♩r ❛♥② α ∈ ❋♩r♠ ✱ ❛♥❞ ❛♥② ☞♥✐t❡ s❡t ❋ ⊆ ❋♩r♠ ✱ ▲ α ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ ❋ ⊢ ✥ ▲ α . ❋ ï¿œ ✥ ❆ ❢♊❧❊❧♊r❡ t❀❡♊r❡♠✿ Completeness Theorem for maximal theories in ï¿œ L ❋♩r ❛♥② α ∈ ❋♩r♠ ✱ ❛♥❞ ❛♥② ♠❛①✐♠❛❧ ❝♊♥s✐st❡♥t s❡t â–Œ ⊆ ❋♩r♠ ✱ ▲ α ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ â–Œ ⊢ ✥ ▲ α . â–Œ ï¿œ ✥

  46. ◆♩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 ï¿œ 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 ￜ⊢ ✥ α ❞♊❡s ♥♊t ♣r♊✈❡ s♠t❀❣✳ ▲ β ∀ ✇ ✇❡ ❀❛✈❡ ✇ ( α ) < ✶ α ✐s ✉♥s❛t✐s☞❛❜❧❡ 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❡ ∀ β ✇❡ ❀❛✈❡ ⊢ ✥ α ✐s str♊♥❣❧② ✐♥❝♊♥✳ ▲ α → β α ✐♠♣❧✐❡s ❡✈❡r②t❀✐♥❣

  47. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ Satisfiability and consistency in ï¿œ 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 ￜ⊢ ✥ α ❞♊❡s ♥♊t ♣r♊✈❡ s♠t❀❣✳ ▲ β ∀ ✇ ✇❡ ❀❛✈❡ ✇ ( α ) < ✶ α ✐s ✉♥s❛t✐s☞❛❜❧❡ 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❡ ∀ β ✇❡ ❀❛✈❡ ⊢ ✥ α ✐s str♊♥❣❧② ✐♥❝♊♥✳ ▲ 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✳

  48. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ Satisfiability and consistency in ï¿œ 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 ￜ⊢ ✥ α ❞♊❡s ♥♊t ♣r♊✈❡ s♠t❀❣✳ ▲ β ∀ ✇ ✇❡ ❀❛✈❡ ✇ ( α ) < ✶ α ✐s ✉♥s❛t✐s☞❛❜❧❡ 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❡ ∀ β ✇❡ ❀❛✈❡ ⊢ ✥ α ✐s str♊♥❣❧② ✐♥❝♊♥✳ ▲ α → β α ✐♠♣❧✐❡s ❡✈❡r②t❀✐♥❣ ❊q✉✐✈❛❧❡♥t ✐♥ ❝❧❛ss✐❝❛❧ ❧♊❣✐❝ ❜② t❀❡ Pr✐♥❝✐♣❧❡ ♊❢ ❇✐✈❛❧❡♥❝❡✳

  49. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ Satisfiability and consistency in ï¿œ 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 ￜ⊢ ✥ α ❞♊❡s ♥♊t ♣r♊✈❡ s♠t❀❣✳ ▲ β ∀ ✇ ✇❡ ❀❛✈❡ ✇ ( α ) < ✶ α ✐s ✉♥s❛t✐s☞❛❜❧❡ 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❡ ∀ β ✇❡ ❀❛✈❡ ⊢ ✥ α ✐s str♊♥❣❧② ✐♥❝♊♥✳ ▲ α → β α ✐♠♣❧✐❡s ❡✈❡r②t❀✐♥❣ ❊q✉✐✈❛❧❡♥t ✐♥ ❝❧❛ss✐❝❛❧ ❧♊❣✐❝ ❜② t❀❡ ❉❡❞✉❝t✐♊♥ ❚❀❡♊r❡♠✳

  50. ❚❀❡ ❞✐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 .

  51. ❋♩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 ￜ⊢ ✥ ▲ α → α ⊙ α ✳

  52. ✥ ▲✉❊❛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 ￜ⊢ ✥ ▲ α → α ⊙ α ✳ Local Deduction Theorem for ï¿œ L ❋♩r ❛♥② α, β ∈ ❋♩r♠ ✱ ▲ α ♥ → β . α ⊢ ✥ ▲ β ✐❢✱ ❛♥❞ ♊♥❧② ✐❢✱ ∃ ♥ ï¿œ ✶ s✉❝❀ t❀❛t ⊢ ✥ ✭◆♩t❛t✐♊♥✿ α ♥ := α ⊙ · · · ⊙ α . ✮ ï¿œ ￜￜ ï¿œ ♥ t✐♠❡s

  53. ❲❡ ❝❛♥♥♊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❛♥❝❡✿

  54. ❚❀❡ ✭❋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✐♊♥❛❧✳

  55. ❆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 ❧♊❣✐❝✳✮

  56. ✥ ▲✉❊❛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✳

  57. ✥ ▲✉❊❛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 ) = ♠❛① { ✇ ( α ) − ✇ ( β ) , ✵ } ¬ ( α → β ) Table: ❈♊♥♥❡❝t✐✈❡s ✐♥ ✥ ▲✉❊❛s✐❡✇✐❝③ ❧♊❣✐❝✳

  58. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚r✉t❀✲❢✉♥❝t✐♊♥ ♊❢ ✥ ▲✉❊❛s✐❡✇✐❝③ ✐♠♣❧✐❝❛t✐♊♥✳ ✇ ( α → β ) = ♠✐♥ { ✶ , ✶ − ( ✇ ( α ) − ✇ ( β )) } ï¿œ ✶ ✐❢ ✇ ( α ) ï¿œ ✇ ( β ) ✇ ( α → β ) = ✶ − ( ✇ ( α ) − ✇ ( β )) ♩t❀❡r✇✐s❡✳

  59. ❙❛② ❋♩r♠ ❛r❡ ❧♊❣✐❝❛❧❧② ❡q✉✐✈❛❧❡♥t ✐❢ ✳ ❲r✐t❡ ✳ ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ MV-algebras ❈✳ ❈✳ ❈❀❛♥❣ ✐♥ ❘♊♠❡✱ ✶✟✻✟✳

  60. ✥ ▲✉❊❛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 ✳

  61. ❋♩r♠ ❚❀❡ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡ ✵ ✐s ❛♥ ▌❱✲❛❧❣❡❜r❛✳ ❵▌❱✲❛❧❣❡❜r❛✬ ✐s s❀♊rt ❢♊r ❵▌❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬ ✱ ❭❢♊r ❧❛❝❩ ♊❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❀❛♥❣✱ ✶✟✜✻✮ ✳ ▌❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❊❛s✐❡✇✐❝③ ❧♊❣✐❝ ❂ ❇♊♊❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♊❣✐❝ ❆❜str❛❝t❧②✿ â–Œ ✵ ✐s ❛♥ ▌❱✲❛❧❣❡❜r❛ ✐❢ â–Œ ✵ ✐s ❛ ❝♩♠♠✉t❛t✐✈❡ ♠♊♥♊✐❞✱ ① ✱ ✶ ✵ ✐s ❛❜s♩r❜✐♥❣ ❢♊r ① ✭ 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 ] ≡

  62. ❵▌❱✲❛❧❣❡❜r❛✬ ✐s s❀♊rt ❢♊r ❵▌❛♥②✲❱❛❧✉❡❞ ❆❧❣❡❜r❛✬ ✱ ❭❢♊r ❧❛❝❩ ♊❢ ❛ ❜❡tt❡r ♥❛♠❡✳✧ ✭❈✳❈✳ ❈❀❛♥❣✱ ✶✟✜✻✮ ✳ ▌❱✲❛❧❣❡❜r❛s ✿ ✥ ▲✉❊❛s✐❡✇✐❝③ ❧♊❣✐❝ ❂ ❇♊♊❧❡❛♥ ❛❧❣❡❜r❛s ✿ ❈❧❛ss✐❝❛❧ ❧♊❣✐❝ ❆❜str❛❝t❧②✿ â–Œ ✵ ✐s ❛♥ ▌❱✲❛❧❣❡❜r❛ ✐❢ â–Œ ✵ ✐s ❛ ❝♩♠♠✉t❛t✐✈❡ ♠♊♥♊✐❞✱ ① ✱ ✶ ✵ ✐s ❛❜s♩r❜✐♥❣ ❢♊r ① ✭ 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❛✳

  63. ❆❜str❛❝t❧②✿ â–Œ ✵ ✐s ❛♥ ▌❱✲❛❧❣❡❜r❛ ✐❢ â–Œ ✵ ✐s ❛ ❝♩♠♠✉t❛t✐✈❡ ♠♊♥♊✐❞✱ ① ✱ ✶ ✵ ✐s ❛❜s♩r❜✐♥❣ ❢♊r ① ✭ 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✐❝❛❧ ❧♊❣✐❝

  64. ✥ ▲✉❊❛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✿ ( â–Œ , ⊕ , ¬ , ✵ ) ✐s ❛♥ ▌❱✲❛❧❣❡❜r❛ ✐❢ ( â–Œ , ⊕ , ✵ ) ✐s ❛ ❝♩♠♠✉t❛t✐✈❡ ♠♊♥♊✐❞✱ ¬¬ ① = ① ✱ ✶ := ¬ ✵ ✐s ❛❜s♩r❜✐♥❣ ❢♊r ⊕ ✭ ① ⊕ ✶ = ✶✮✱ ❛♥❞✱ ❝❀❛r❛❝t❡r✐st✐❝❛❧❧②✱ ¬ ( ¬ ① ⊕ ② ) ⊕ ② = ¬ ( ¬ ② ⊕ ① ) ⊕ ① ✭✯✮

  65. ❚❀✉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

  66. ▌❡❡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

  67. ❇♊♊❧❡❛♥ ❛❧❣❡❜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 )

  68. ✥ ▲✉❊❛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 = ✶ ✳

  69. ❚❀❡ ✈❛r✐❡t② ♊❢ ▌❱✲❛❧❣❡❜r❛s ✐s ❣❡♥❡r❛t❡❞ ❜② ✵ ✶ ✳ ❈✳❈✳ ❈❀❛♥❣✱ ❚r❛♥s✳ ♊❢ t❀❡ ❆▌❙ ✱ ✶✟✺✟✳ ❚❀✐s ♠❡❛♥s✿ ❚❀❡ ❝❧❛ss ♊❢ ▌❱✲❛❧❣❡❜r❛s ❝♊✐♥❝✐❞❡s ✇✐t❀ ❍❙P ✵ ✶ â‘€ ❛♥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 ♊❢ [ ✵ , ✶ ] ✳

  70. ❚❀✐s ♠❡❛♥s✿ ❚❀❡ ❝❧❛ss ♊❢ ▌❱✲❛❧❣❡❜r❛s ❝♊✐♥❝✐❞❡s ✇✐t❀ ❍❙P ✵ ✶ â‘€ ❛♥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❀❡ ❆▌❙ ✱ ✶✟✺✟✳

  71. ✥ ▲✉❊❛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 ([ ✵ , ✶ ]) â‘€ ❛♥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✲♠♊❞❡❧ ✐♥ [ ✵ , ✶ ] ✳

  72. ❍❡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 ï¿œ → ✵ ❛♥❞ ① ï¿œ → ✶ â‘€ t❀❡ ❇♊♊❧❡❛♥✱ ♩r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♊♥s✳

  73. ✵ ✶ ✷ ✱ ❚❀❡ ❡✈❛❧✉❛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 ï¿œ → ✵ ❛♥❞ ① ï¿œ → ✶ â‘€ t❀❡ ❇♊♊❧❡❛♥✱ ♩r ❝❧❛ss✐❝❛❧✱ ❡✈❛❧✉❛t✐♊♥s✳ ❍❡r❡ ✐s ❛ ✷✲✈❛r✐❛❜❧❡ ❣❡♥❡r❛❧✐s❛t✐♊♥ ♊❢ t❀❡ t❡rt✐✉♠ ♥♊♥ ❞❛t✉r t❡r♠✿ ① √ ¬ ① √ ② √ ¬ ② = ✶ ✭ ⋆⋆ ✮

  74. ❚❀❡ ❜♊✉♥❞❛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 ï¿œ → ✵ ❛♥❞ ① ï¿œ → ✶ â‘€ 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❡✿

  75. ✥ ▲✉❊❛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 ï¿œ → ✵ ❛♥❞ ① ï¿œ → ✶ â‘€ 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❡✳

  76. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❳ √ ¬ ❳ = ✶ ✭ ⋆ ✮ ❚❀❡ ❜♊✉♥❞❛r② ♊❢ t❀❡ ✉♥✐t ✐♥t❡r✈❛❧✳ ❳ √ ¬ ❳ √ ❚ √ ¬ ❚ = ✶ ✭ ⋆⋆ ✮ ❚❀❡ ❜♊✉♥❞❛r② ♊❢ t❀❡ ✉♥✐t sq✉❛r❡✳

  77. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❚❀❡ t✇✐st❡❞ ❝✉❜✐❝ ✿ V ( { ② − ① ✷ , ③ − ① ✾ } ) → ( t , t ✷ , t ✾ ) ✳✮ ✭P❛r❛♠❡tr✐s❛t✐♊♥✿ t ￜ−

  78. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ Rational polyhedra ▲❡♊♥❛r❞♊✬s ❚r✉♥❝❛t❡❞ ■❝♩s❛❀❡❞r♊♥ ✭ ■❧❧✉str❛t✐♊♥ ❢♊r ▲✉❝❛ P❛❝✐♊❧✐✬s ❚❀❡ ❉✐✈✐♥❡ Pr♊♣♊rt✐♊♥✱ ✶✺✵✟✳ ✮

  79. ♥ ✇✐t❀ P ❛ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ❝♊♥✈ ❋ ❀ ♥ ✇✐t❀ ❛ r❛t✐♊♥❛❧ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ❝♊♥✈ ❋ ✳ P ♥ ✱ ✇r✐tt❡♥ ❝♊♥✈ P ✱ ✐s t❀❡ ❚❀❡ ❝♊♥✈❡① ❀✉❧❧ ♊❢ ❛ s❡t P ❝♊❧❧❡❝t✐♊♥ ♊❢ ❛❧❧ ❝♊♥✈❡① ❝♊♠❜✐♥❛t✐♊♥s ♊❢ ❡❧❡♠❡♥ts ♊❢ P ✿ ♠ ♠ ❝♊♥✈ P P ❛♥❞ ✵ ✇✐t❀ ✶ r ✐ ✈ ✐ ✈ ✐ r ✐ r ✐ ✐ ✶ ✐ ✶ ❙✉❝❀ ❛ s❡t ✐s ❝♊♥✈❡① ✐❢ P ❝♊♥✈ P ✳ ❚❀❡ s❡t P ✐s ❝❛❧❧❡❞✿ ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❲❡ ❝♊♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▌❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❀♊s❡ ♊❢ t❀❡ ❢♊r♠ F ♥ /Ξ ✱ ✇✐t❀ Ξ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♊♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❀❡ ❛ss✉♠♣t✐♊♥ ♊♥ Ξ ✐s ❢❛r ❢r♩♠ ✐♠♠❛t❡r✐❛❧✿ t❀❡r❡ ✐s ♥♊ ❍✐❧❜❡rt✬s ❇❛s✐s ❚❀❡♊r❡♠ ❢♊r ▌❱✲❛❧❣❡❜r❛s✳

  80. ♥ ✇✐t❀ P ❛ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ❝♊♥✈ ❋ ❀ ♥ ✇✐t❀ ❛ r❛t✐♊♥❛❧ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ❝♊♥✈ ❋ ✳ P ❚❀❡ s❡t P ✐s ❝❛❧❧❡❞✿ ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❲❡ ❝♊♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▌❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❀♊s❡ ♊❢ t❀❡ ❢♊r♠ F ♥ /Ξ ✱ ✇✐t❀ Ξ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♊♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❀❡ ❛ss✉♠♣t✐♊♥ ♊♥ Ξ ✐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 ✿ ï¿œ ♠ ï¿œ ♠ ï¿œ ï¿œ ❝♊♥✈ P = r ✐ ✈ ✐ | ✈ ✐ ∈ P ❛♥❞ ✵ ï¿œ r ✐ ∈ R ✇✐t❀ r ✐ = ✶ . ✐ = ✶ ✐ = ✶ ❙✉❝❀ ❛ s❡t ✐s ❝♊♥✈❡① ✐❢ P = ❝♊♥✈ P ✳

  81. ♥ ✇✐t❀ ❛ r❛t✐♊♥❛❧ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ❝♊♥✈ ❋ ✳ P ▲✉❊❛s✐❡✇✐❝③ ✥ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❲❡ ❝♊♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▌❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❀♊s❡ ♊❢ t❀❡ ❢♊r♠ F ♥ /Ξ ✱ ✇✐t❀ Ξ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♊♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❀❡ ❛ss✉♠♣t✐♊♥ ♊♥ Ξ ✐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 ✿ ï¿œ ♠ ï¿œ ♠ ï¿œ ï¿œ ❝♊♥✈ P = r ✐ ✈ ✐ | ✈ ✐ ∈ P ❛♥❞ ✵ ï¿œ r ✐ ∈ R ✇✐t❀ r ✐ = ✶ . ✐ = ✶ ✐ = ✶ ❙✉❝❀ ❛ s❡t ✐s ❝♊♥✈❡① ✐❢ P = ❝♊♥✈ P ✳ ❚❀❡ s❡t P ✐s ❝❛❧❧❡❞✿ ❛ ♣♊❧②t♊♣❡✱ ✐❢ t❀❡r❡ ✐s ❛ ☞♥✐t❡ ❋ ⊆ R ♥ ✇✐t❀ P = ❝♊♥✈ ❋ ❀

  82. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❲❡ ❝♊♥s✐❞❡r ☞♥✐t❡❧② ♣r❡s❡♥t❡❞ ▌❱✲❛❧❣❡❜r❛s✱ ✐✳❡✳ t❀♊s❡ ♊❢ t❀❡ ❢♊r♠ F ♥ /Ξ ✱ ✇✐t❀ Ξ ❛ ☞♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❝♊♥❣r✉❡♥❝❡ ✭✐❞❡❛❧✮✳ ❚❀❡ ❛ss✉♠♣t✐♊♥ ♊♥ Ξ ✐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 ✿ ï¿œ ♠ ï¿œ ♠ ï¿œ ï¿œ ❝♊♥✈ P = r ✐ ✈ ✐ | ✈ ✐ ∈ P ❛♥❞ ✵ ï¿œ 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 = ❝♊♥✈ ❋ ✳

  83. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❆ ♣♊❧②t♊♣❡ ✐♥ R ✷ ✳

  84. ✥ ▲✉❊❛s✐❡✇✐❝③ ❈❀❛♥❣ P♊❧②❀❡❞r❛ ❊♣✐❧♊❣✉❡ ❆ ♣♊❧②t♊♣❡ ✐♥ R ✷ ✭ ❛ s✐♠♣❧❡①✮ ✳

  85. ✥ ▲✉❊❛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✳

  86. ❊❛❝❀ ▲ ✐ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ♣♊❧②♥♊♠✐❛❧ ✇✐t❀ ✐♥t❡❣❡r ❝♊❡✍❝✐❡♥ts✳ ♠ ❜❡t✇❡❡♥ ♣♊❧②❀❡❞r❛ ❛❧✇❛②s ✐s ♊❢ t❀❡ ♥ ❆ ♠❛♣ ❋ P ◗ ❢♊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♊♠❡ ✶ ï¿œ ✐ ① ï¿œ ♠ ✳ ❆ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♊♥ [ ✵ , ✶ ] → R ✳

  87. ♠ ❜❡t✇❡❡♥ ♣♊❧②❀❡❞r❛ ❛❧✇❛②s ✐s ♊❢ t❀❡ ♥ ❆ ♠❛♣ ❋ P ◗ ❢♊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♊♠❡ ✶ ï¿œ ✐ ① ï¿œ ♠ ✳ 2 ❊❛❝❀ ▲ ✐ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❧✐♥❡❛r ♣♊❧②♥♊♠✐❛❧ ✇✐t❀ ✐♥t❡❣❡r ❝♊❡✍❝✐❡♥ts✳ ❆ ♣✐❡❝❡✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♊♥ [ ✵ , ✶ ] → R ✳

  88. ✥ ▲✉❊❛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♊♠❡ ✶ ï¿œ ✐ ① ï¿œ ♠ ✳ 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✳

  89. ❚❀❡ ❝❛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 ❧❛♥❣✉❛❣❡✳

  90. ✥ ▲✉❊❛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 ♊♣ MV ❢♣ Q

  91. ♥ ✱ t❀❡ ❋r♩♠ r❛t✐♊♥❛❧ ♣♊❧②❀❡❞r❛ t♩ ▌❱✲❛❧❣❡❜r❛s ✿ ●✐✈❡♥ P ❝♊❧❧❡❝t✐♊♥ ♊❢ ❛❧❧ ✲♠❛♣s P ✵ ✶ ✐s ❛ ✭☞♥✐t❡❧② P ♣r❡s❡♥t❛❜❧❡✮ ▌❱✲❛❧❣❡❜r❛ ✉♥❞❡r t❀❡ ♣♊✐♥t✇✐s❡ ♊♣❡r❛t✐♊♥ ✐♥❀❡r✐t❡❞ ❢r♩♠ ✵ ✶ ✳ ❊①❛♠♣❧❡ ✳ ■❢ ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♩ ✵ ✐♥ ❛♥② ① ✶ ① ♥ ▌❱✲❛❧❣❡❜r❛✱ t❀❡♥ ✐t ❣❡♥❡r❛t❡s t❀❡ tr✐✈✐❛❧ ✐❞❡❛❧ ✵ ✳ ■♥ t❀✐s ❝❛s❡✱ ✵ ✶ ♥ ✳ ❍❡♥❝❡ t❀❡ ❞✉❛❧s ♊❢ ❢r❡❡ ♥ ✱ ❛♥❞ ♥ ❛❧❣❡❜r❛s ❛r❡ t❀❡ ✉♥✐t ❝✉❜❡s✳ ✵ ✶ ♥ ❀♊♠❡♊♠♊r♣❀✐❝ t♩ t❀❡ ♠❛①✐♠❛❧ ❘❡♠❛r❊ ✳ ❚❀❡ s✉❜s♣❛❝❡ 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 ♥ / ï¿œ τ ( ① ✶ , . . . , ① ♥ ) ï¿œ ✱ t❀❡ ❛ss♩❝✐❛t❡❞ r❛t✐♊♥❛❧ ♣♊❧②❀❡❞r♊♥ V ( τ ) ✐s t❀❡ s❡t ♊❢ ♥ ✲t✉♣❧❡s ( r ✶ , . . . , r ♥ ) ∈ [ ✵ , ✶ ] ♥ s✉❝❀ t❀❛t τ ( r ✶ , . . . , r ♥ ) = ✵ ✐♥ [ ✵ , ✶ ] ✳

  92. ❊①❛♠♣❧❡ ✳ ■❢ ✐s ✐❞❡♥t✐❝❛❧❧② ❡q✉❛❧ t♩ ✵ ✐♥ ❛♥② ① ✶ ① ♥ ▌❱✲❛❧❣❡❜r❛✱ t❀❡♥ ✐t ❣❡♥❡r❛t❡s t❀❡ tr✐✈✐❛❧ ✐❞❡❛❧ ✵ ✳ ■♥ t❀✐s ❝❛s❡✱ ✵ ✶ ♥ ✳ ❍❡♥❝❡ t❀❡ ❞✉❛❧s ♊❢ ❢r❡❡ ♥ ✱ ❛♥❞ ♥ ❛❧❣❡❜r❛s ❛r❡ t❀❡ ✉♥✐t ❝✉❜❡s✳ ✵ ✶ ♥ ❀♊♠❡♊♠♊r♣❀✐❝ t♩ t❀❡ ♠❛①✐♠❛❧ ❘❡♠❛r❊ ✳ ❚❀❡ s✉❜s♣❛❝❡ 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 ♥ / ï¿œ τ ( ① ✶ , . . . , ① ♥ ) ï¿œ ✱ 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♩♠ [ ✵ , ✶ ] ✳

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