Introdução à Lógica de Predicados

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  ■♥tr♦❞✉çã♦ à ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ▼❛t❡♠át✐❝❛ ❉✐s❝r❡t❛ ■

  ❘♦❞r✐❣♦ ❘✐❜❡✐r♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❆♣❧✐❝❛❞❛s ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❋❡❞❡r❛❧ ❞❡ ❖✉r♦ Pr❡t♦ ✶✵ ❞❡ ❞❡③❡♠❜r♦ ❞❡ ✷✵✶✷

  ▼♦t✐✈❛çã♦ ✖ ✭■✮ ❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦✿

  ❚♦❞♦ ❤♦♠❡♠ é ♠♦rt❛❧✳ ❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠✳ ▲♦❣♦✱ ❙ó❝r❛t❡s é ♠♦rt❛❧✳

  ❆♣ós ♣❡♥s❛r ✉♠ ♣♦✉❝♦✳✳✳

  ❱❡♠♦s ❝❧❛r❛♠❡♥t❡ q✉❡ ❡st❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦✳ P♦rt❛♥t♦✱ ✈❛♠♦s r❡♣r❡s❡♥tá✲❧♦ ✉s❛♥❞♦ ❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧✳

  ▼♦t✐✈❛çã♦ ✖ ✭■■✮ P♦ré♠ ❛♦ t❡♥t❛r r❡♣r❡s❡♥t❛r✳✳✳

  P❡r❝❡❜❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ❝♦♥❡❝t✐✈♦✦ ■st♦ é✱ q✉❡ ♦ ❛r❣✉♠❡♥t♦ é ❢♦r♠❛❞♦ ❛♣❡♥❛s ♣♦r ♣r♦♣♦s✐çõ❡s ❛tô♠✐❝❛s✳ ❙❡♥❞♦ ❛ss✐♠✱ t❡♠♦s q✉❡ ♦ ❛r❣✉♠❡♥t♦ s❡r✐❛ r❡♣r❡s❡♥t❛❞♦ ❝♦♠♦✿

  ❋ór♠✳ Pr♦♣✳ ❆ ❚♦❞♦ ❤♦♠❡♠ é ♠♦rt❛❧ ❇ ❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠ ❈ ❙ó❝r❛t❡s é ♠♦rt❛❧

  ▼♦t✐✈❛çã♦ ✖ ✭■■■✮ ❖ s❡q✉❡♥t❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❛♦ ❛r❣✉♠❡♥t♦ é✿

  ❆, ❇ ⊢ ❈

  ▼❛s✳✳✳

  ➱ ❢á❝✐❧ ♥♦t❛r q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ❞❡❞✉③✐r ❈ ❛ ♣❛rt✐r ❞❡ ❆ ❡ ❇✳ ▼❛s ♦ ❛r❣✉♠❡♥t♦ ♣❛r❡❝✐❛ ❝♦rr❡t♦✳✳✳

  ▼♦t✐✈❛çã♦ ✖ ✭■❱✮ ❖ Pr♦❜❧❡♠❛ é q✉❡✳✳✳

  ❆ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧ ♥ã♦ é ❝❛♣❛③ ❞❡ r❡♣r❡s❡♥t❛r ❛✜r♠❛çõ❡s ❝♦♥t❡♥❞♦ ✧t♦❞♦s✧✱ ✧♣❛r❛ t♦❞♦s✧✱ ✧❛❧❣✉♥s✧✱ ✧s♦♠❡♥t❡ ✉♠✧✳✳✳ ❚❛✐s ❛✜r♠❛t✐✈❛s ♦❝♦rr❡♠ ❝♦♠ ❢r❡q✉ê♥❝✐❛ ❡♠ ♠❛t❡♠át✐❝❛✿

  ❚♦❞♦ ♥ú♠❡r♦ ♣❛r é ❞✐✈✐sí✈❡❧ ♣♦r ✷✳ ❚♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♠❛✐♦r q✉❡ ✶ ♦✉ é ♣r✐♠♦ ♦✉ ♣r♦❞✉t♦ ❞❡ ♣r✐♠♦s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■✮ ❆ ▲✐♥❣✉❛❣❡♠ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  ❆ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ❝♦♥s✐st❡ ❞❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧ ❛✉♠❡♥t❛❞❛ ❝♦♠ q✉❛♥t✐✜❝❛❞♦r❡s✱ ♣r❡❞✐❝❛❞♦s ❡ ✈❛r✐á✈❡✐s✳ ❚♦❞❛s ❛s r❡❣r❛s ❞❡ ❞❡❞✉çã♦ ♥❛t✉r❛❧ ❡ r❛❝✐♦❝í♥✐♦ ❛❧❣é❜r✐❝♦ q✉❡ ✈✐♠♦s✱ ❝♦♥t✐♥✉❛♠ s❡♥❞♦ ✈á❧✐❞❛s ♣❛r❛ ❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■■✮ Pr❡❞✐❝❛❞♦s

  ❯♠ ♣r❡❞✐❝❛❞♦ é ✉♠❛ ❛✜r♠❛çã♦ ❞❡ q✉❡ ✉♠ ♦❜❥❡t♦ ① ♣♦ss✉✐ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ♣r♦♣r✐❡❞❛❞❡✳ ❚❛✐s ❛✜r♠❛t✐✈❛s ❞❡✈❡♠ s❡r ✈❡r❞❛❞❡✐r❛s ♦✉ ❢❛❧s❛s✳ ❆✜r♠❛t✐✈❛s ♣♦❞❡♠ ❡♥✈♦❧✈❡r ❡❧❡♠❡♥t♦s ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♦✉ ✈❛r✐á✈❡✐s q✉❡ ♣♦❞❡♠ ❛ss✉♠✐r ✈❛❧♦r❡s ❞❡st❡ ❝♦♥❥✉♥t♦✳ ❊①❡♠♣❧♦s✿

  ✺ é ✉♠ ♥ú♠❡r♦ ♣❛r✳ ① ❃ ②✳ ❏♦sé ❣♦st❛ ❞❡ ▼❛r✐❛✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■■■✮ ❯♥✐✈❡rs♦ ❞❡ ❉✐s❝✉rs♦

  ❉❛♠♦s ♦ ♥♦♠❡ ❞❡ ❯♥✐✈❡rs♦ ❞❡ ❉✐s❝✉rs♦ ❛♦ ❝♦♥❥✉♥t♦ q✉❡ é r❡❢❡r✐❞♦ ♣♦r ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ❊①❡♠♣❧♦✿

  ❆ ❛✜r♠❛t✐✈❛✿ ✧❚♦❞♦ ♥ú♠❡r♦ ♣❛r é ❞✐✈✐sí✈❡❧ ♣♦r ✷✧✳ P♦ss✉✐ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■❱✮ ▼❛✐s s♦❜r❡ Pr❡❞✐❝❛❞♦s

  ◆♦r♠❛❧♠❡♥t❡ r❡♣r❡s❡♥t❛♠♦s ♣r❡❞✐❝❛❞♦s ❞❡ ♠❛♥❡✐r❛ ❝♦♥❝✐s❛✳ ❊①❡♠♣❧♦s✿

  ❏♦sé ❣♦st❛ ❞❡ ▼❛r✐❛✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❢❛t♦ ♣♦r✿ ❣♦st❛✭❏♦sé✱ ▼❛r✐❛✮✳ ❖♥❞❡ ❣♦st❛ é ✉♠ ♣r❡❞✐❝❛❞♦ ❞❡ ❞♦✐s ♣❛râ♠❡tr♦s✱ ♦♥❞❡ ❝❛❞❛ ✉♠ ❞❡st❡s é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ q✉❡ ❝♦♥té♠ ♣❡ss♦❛s✳ ① é ✉♠ ❛❞♦❧❡s❝❡♥t❡✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❢❛t♦ ♣♦r ❛❞♦❧❡s❝❡♥t❡✭①✮✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱✮ ▼❛✐s s♦❜r❡ Pr❡❞✐❝❛❞♦s

  ❊①❡♠♣❧♦s✿

  ♣❛r✭✷✮ ❢ (①, ✷, ❣(②)) Pr❡❞✐❝❛❞♦s q✉❡ ♥ã♦ ♣♦ss✉❡♠ ♣❛râ♠❡tr♦s sã♦ ✈❛r✐á✈❡✐s ♣r♦♣♦s✐❝✐♦♥❛✐s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■✮ ◗✉❛♥t✐✜❝❛❞♦r❡s

  ❆ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ♣♦ss✉✐ ❞♦✐s q✉❛♥t✐✜❝❛❞♦r❡s✿

  ◗✉❛♥t✐✜❝❛❞♦r ✉♥✐✈❡rs❛❧✿ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ∀ ◗✉❛♥t✐✜❝❛❞♦r ❡①✐st❡♥❝✐❛❧✿ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ sí♠❜♦❧♦ ∃

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■■✮ ◗✉❛♥t✐✜❝❛❞♦r ❯♥✐✈❡rs❛❧

  ❙❡ ❋ (①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✱ ❝♦♥t❡♥❞♦ ❛ ✈❛r✐á✈❡❧ ①✱ ❡♥tã♦ ∀①.❋ (①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❖ s✐❣♥✐✜❝❛❞♦ ❞❡ ∀①.❋ (①) é✿ P❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ ① ♣❡rt❡♥❝❡♥t❡ ❛♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦✱ ❛ ❢ór♠✉❧❛ ❋ (①) ❞❡✈❡ s❡r ✈❡r❞❛❞❡✐r❛✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❱■■■✮ ❊①❡♠♣❧♦s

  ❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭N✮✳

  ∀

♥.♥ ≥ ✵✿ ❡st❛ ❢ór♠✉❧❛ é ✈❡r❞❛❞❡✐r❛✳

  ❈♦♥s✐❞❡r❡ ❛❣♦r❛✱ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡r❡s ❤✉♠❛♥♦s✳

  ∀ ①.❣♦st❛(①, ▼❛r✐❛)

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭■❳✮ ◗✉❛♥t✐✜❝❛❞♦r ❊①✐st❡♥❝✐❛❧

  ❙❡ ❋ (①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✱ ❝♦♥t❡♥❞♦ ❛ ✈❛r✐á✈❡❧ ①✱ ❡♥tã♦ ∃①.❋ (①) é ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❖ s✐❣♥✐✜❝❛❞♦ ❞❡ ∃①.❋ (①) é✿ ❊①✐st❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❡❧❡♠❡♥t♦ ① ♣❡rt❡♥❝❡♥t❡ ❛♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦✱ q✉❡ ❢❛ç❛ ❛ ❢ór♠✉❧❛ ❋ (①) s❡r ✈❡r❞❛❞❡✐r❛✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳✮ ❊①❡♠♣❧♦s

  ❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭N✮✳

  ∃ ♥.♥ ≤ ✵✿ ❡st❛ ❢ór♠✉❧❛ é ❢❛❧s❛✳

  ❈♦♥s✐❞❡r❡ ❛❣♦r❛✱ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s s❡r❡s ❤✉♠❛♥♦s✳

  ∃ ①.❣♦st❛(①, ▼❛r✐❛)

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■✮ ❙✐❣♥✐✜❝❛❞♦ ❞♦s ◗✉❛♥t✐✜❝❛❞♦r❡s

  , , ..., }

  ❙❡❥❛ ❯ = {✉ ✶ ✉ ✷ ✉ ♥ ✱ ✉♠ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ t❛❧ q✉❡ |❯| = ♥ ∧ ♥ 6= ∞✳ ❊♥tã♦ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ∀①.♣(①) é✿

  

) ∧ ) ∧ ... ∧ )

♣(✉ ✶ ♣(✉ ✷ ♣(✉ ♥

  ❊ ♦ s✐❣♥✐✜❝❛❞♦ ❞❡ ∃①.♣(①) é✿

  

) ∨ ) ∨ ... ∨ )

♣(✉ ✶ ♣(✉ ✷ ♣(✉ ♥

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■■✮ ❊①❡♠♣❧♦s

  ❈♦♥s✐❞❡r❡ ❯ = {✷, ✸, ✹} ❡ ♣❛r(①) ❡ ✐♠♣❛r(①) ❞♦✐s ♣r❡❞✐❝❛❞♦s s♦❜r❡ ❯✳ ❊♥tã♦ ❛s ❢ór♠✉❧❛s ∀①.♣❛r(①) ∨ ✐♠♣❛r(①) ❡ ∃

  ①.¬✐♠♣❛r(①) ∧ ① < ✶ ♣♦❞❡♠ s❡r ❡①♣❛♥❞✐❞❛s ❝♦♠♦✿

  ( ♣❛r(✷) ∨ ✐♠♣❛r(✷)) ∧ (♣❛r(✸) ∨ ✐♠♣❛r(✸)) ∧ ( ♣❛r(✹) ∨ ✐♠♣❛r(✹)) (¬ ✐♠♣❛r(✷) ∧ ✷ < ✶) ∨ (¬✐♠♣❛r(✸) ∧ ✸ < ✶) ∨ (¬✐♠♣❛r(✹) ∧ ✹ < ✶)

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■■■✮ ❊s❝♦♣♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r❡s

  P❛r❛ ❡♥t❡♥❞❡r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❡s❝♦♣♦ ❞❡ q✉❛♥t✐✜❝❛❞♦r❡s é ♥❡❝❡ssár✐♦ ❝♦♥❤❡❝❡r ♦s ❝♦♥❝❡✐t♦s ❞❡ ✈❛r✐á✈❡✐s ❧✐✈r❡s ❡ ❧✐❣❛❞❛s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳■❱✮ ❱❛r✐á✈❡✐s ❧✐✈r❡s ❡ ❧✐❣❛❞❛s

  ❯♠❛ ✈❛r✐á✈❡❧ é ❞✐t❛ s❡r ❧✐✈r❡ ❡♠ ✉♠❛ ❢ór♠✉❧❛ s❡ ❡st❛ ♥ã♦ ♣♦ss✉✐ ♥❡♥❤✉♠ q✉❛♥t✐✜❝❛❞♦r✳

  ❊①❡♠♣❧♦✿ ∀①.∃②.♣(①, ②, ③)✳ ◆❡st❛ ❢ór♠✉❧❛ ③ é ✉♠❛ ✈❛r✐á✈❡❧ ❧✐✈r❡✳

  ❯♠❛ ✈❛r✐á✈❡❧ é ❞✐t❛ s❡r ❧✐❣❛❞❛ s❡ ❡st❛ ❡stá ❛ss♦❝✐❛❞❛ ❛ ✉♠ q✉❛♥t✐✜❝❛❞♦r✳

  ❊①❡♠♣❧♦✿ ∀①.∃②.♣(①, ②, ③)✳ ◆❡st❛ ❢ór♠✉❧❛ ❛s ✈❛r✐á✈❡✐s ① ❡ ② sã♦ ❧✐❣❛❞❛s✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱✮ ❊s❝♦♣♦ ❞❡ ✉♠ ◗✉❛♥t✐✜❝❛❞♦r

  ❙❡❥❛ α ✉♠❛ ❢ór♠✉❧❛ q✉❛❧q✉❡r ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❊♥tã♦ ♦ ❡s❝♦♣♦ ❞❡ ∀ ♥❛ ❢ór♠✉❧❛ ∀

  ①.α é t♦❞❛ ❛ ❢ór♠✉❧❛ α✳ ❚♦❞❛ ♦❝♦rrê♥❝✐❛ ❞❡ ① ❡♠ α s❡rá ❧✐❣❛❞❛ ❛♦ q✉❛♥t✐✜❝❛❞♦r ∀✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ❡s❝♦♣♦ ♣❛r❛ ♦ q✉❛♥t✐✜❝❛❞♦r ∃ é ✐❞ê♥t✐❝❛✳

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱■✮ ❚r❛❞✉③✐♥❞♦ P❛r❛ ❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  Pr♦❝❡ss♦ s✐♠♣❧❡s ❡ s✐♠✐❧❛r ❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧✳ ▼❛s ❞❡✈❡♠♦s ❧❡✈❛r ❡♠ ❝♦♥t❛ ♣r❡❞✐❝❛❞♦s ❡ q✉❛♥t✐✜❝❛❞♦r❡s✳ ❊①❡♠♣❧♦✿ ❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣❡ss♦❛s ❡ ♦❜❥❡t♦s✳

  ❚♦❞♦ ♠✉♥❞♦ ❣♦st❛ ❞❡ ❛❧❣✉é♠✳ ∀①.∃②.❣♦st❛(①, ②) ❚♦❞♦s ❣♦st❛♠ ❞❡ q✉❡♠ ❣♦st❛ ❞❡ s✉❛ ♠ã❡✳ ∀

①.∀②.❣♦st❛(①, ♠❛❡(②)) → ❣♦st❛(②, ①)

❆❧✉♥♦s ♥ã♦ ❣♦st❛♠ ❞❡ ❢❛③❡r ♣r♦✈❛s✳ ∀

  

①.❛❧✉♥♦(①) → ¬❣♦st❛(①, ❢❛③❡r(Pr♦✈❛))

  ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s ✖ ✭❳❱■■✮ ❊①❡r❝í❝✐♦

  ❘❡♣r❡s❡♥t❡ ❛s s❡❣✉✐♥t❡s ❢r❛s❡s✿ ❙❡ ❛♠❛♥❤ã ❝❤♦✈❡r✱ ❡♥tã♦ t♦❞♦s ❞❡✈❡♠ ❧❡✈❛r ❣✉❛r❞❛✲❝❤✉✈❛s✳ ❚♦❞♦s q✉❡ ❣♦st❛♠ ❞❡ ❈❛r❧♦s✱ ♥ã♦ ❣♦st❛♠ ❞❡ s❡✉s ♣r✐♠♦s✳ ◆✐♥❣✉é♠ ❢♦✐ ❛ ❢❡st❛ ❞❡ ▼❛♥♦❡❧✳ ❚♦❞♦s ♦s ❣r❡❣♦s sã♦ ❡✉r♦♣❡✉s✱ ♠❛s ♥❡♠ t♦❞♦ ❡✉r♦♣❡✉ é ❣r❡❣♦✳ ❚♦❞♦s ❣♦st❛♠ ❞❛ ♠ã❡ ❞❡ ▼❛r✐❛✳

  ❙✐♥t❛①❡ ✖ ✭■✮ ❋ór♠✉❧❛s ❜❡♠ ❢♦r♠❛❞❛s

  ❆♥t❡s ❞❡ ❞❡s❝r❡✈❡r♠♦s ❝♦♠♦ é ❛ s❡♠â♥t✐❝❛ ❞❡ ❢ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✱ ❞❡✈❡♠♦s ❞❡✜♥✐r ♣r✐♠❡✐r❛♠❡♥t❡✱ ♦ q✉❡ é ✉♠❛ ❢ór♠✉❧❛ s✐♥t❛t✐❝❛♠❡♥t❡ ❝♦rr❡t❛✳

  ❙✐♥t❛①❡ ✖ ✭■■✮ ❋ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s

  ❋ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ♣♦❞❡♠ ❞❡♥♦t❛r✿

  ❊❧❡♠❡♥t♦s ❞♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ❡♠ q✉❡stã♦✳ ❱❛❧♦r❡s ❧ó❣✐❝♦s

  ❙❡♥❞♦ ❛ss✐♠✱ ❛ s✐♥t❛①❡ s❡rá ❞✐✈✐❞❛ ❡♠ ❞✉❛s ❝❛t❡❣♦r✐❛s✿

  ❚❡r♠♦s q✉❡ ❞❡♥♦t❛♠ ❡❧❡♠❡♥t♦s✳ ❋ór♠✉❧❛s q✉❡ ❞❡♥♦t❛♠ ✈❛❧♦r❡s ❧ó❣✐❝♦s✳

  ❙✐♥t❛①❡ ✖ ✭■■■✮ ❚❡r♠♦s

  ❖ ❝♦♥❥✉♥t♦ ❞❡ t❡r♠♦s T é ❞❡✜♥✐❞♦ ❝♦♠♦✿

  

❛✮ ❙❡❥❛ V ♦ ❝♦♥❥✉♥t♦ ❝♦♥t❡♥❞♦ t♦❞❛s ❛s ✈❛r✐á✈❡✐s

  ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❊♥tã♦✱ T ⊆ V✳

  ❜✮ ❙❡ ❝ é ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ✉♠

  ❡❧❡♠❡♥t♦ ❞♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦✱ ❡♥tã♦ ❝ ∈ T

  ❝✮ ❙❡❥❛ ❢ ✉♠❛ ❢✉♥çã♦ ♥✲ár✐❛ s♦❜r❡ t❡r♠♦s ❡

  , , ..., ∈ T t t t ✭♥ ≥ ✵✮ ❡♥tã♦

  ✶ ✷ ♥

  , , ..., ) ∈ T ❢ (t t t ✳

  ✶ ✷ ♥

  ❙✐♥t❛①❡ ✖ ✭■❱✮ ❚❡r♠♦s

  ❈♦♥s✐❞❡r❛♥❞♦ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ❝✐❞❛❞❡s ❡ ❡st❛❞♦s✱ t❡♠♦s q✉❡✿

  ❇❡❧♦ ❍♦r✐③♦♥t❡ é ✉♠ ❝♦♥st❛♥t❡ r❡♣r❡s❡♥t❛♥❞♦ ❛ ❝❛♣✐t❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s ❙❡ ❝❛♣✐t❛❧ é ✉♠❛ ❢✉♥çã♦ q✉❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ ❝♦♥st❛♥t❡ q✉❡ r❡♣r❡s❡♥t❛ ✉♠ ❡st❛❞♦ r❡t♦r♥❛ ❛ r❡s♣❡❝t✐✈❛ ❝❛♣✐t❛❧ ❞❡st❡✱ t❡♠♦s q✉❡ ❝❛♣✐t❛❧✭▼✐♥❛s ●❡r❛✐s✮ ❞❡♥♦t❛ ♦ ♠❡s♠♦ ❡❧❡♠❡♥t♦ q✉❡ ❛ ❝♦♥st❛♥t❡ ❇❡❧♦ ❍♦r✐③♦♥t❡✳

  ❙✐♥t❛①❡ ✖ ✭❱✮ ❋ór♠✉❧❛s

  ❖ ❝♦♥❥✉♥t♦ F ❞❡ ❢ór♠✉❧❛s é ❞❡✜♥✐❞♦ ❝♦♠♦✿ ⊥, ⊤ ∈ F

  ❛✮

  , , ...,

  ❜✮ ❙❡ ♣ é ✉♠ ♣r❡❞✐❝❛❞♦ ♥✲ár✐♦ ❡ t t t sã♦ ✶ ✷ ♥

  

✶ ✷ ♥

❝✮ ❙❡ α ∈ F ❡♥tã♦ ¬α ∈ F ❞✮ ❙❡ α, β ∈ F ❡♥tã♦ α ◦ β ∈ F✱ ♦♥❞❡

  , , ..., ) ∈ F t❡r♠♦s✱ ❡♥tã♦ ♣(t t t ✳

  ◦ ∈ {→, ∨, ∧, ↔} ✳

  

❡✮ ❙❡ ① ∈ V ❡ α ∈ F ❡♥tã♦ ∀①.α ∈ F ❡ ∃①.α ∈ F

  ❙✐♥t❛①❡ ✖ ✭❱■✮ Pr❡❝❡❞ê♥❝✐❛

  ❙❡❣✉✐r❡♠♦s ❛s ♠❡s♠❛s ❝♦♥✈❡♥çõ❡s ❞❡ ♣r❡❝❡❞ê♥❝✐❛ q✉❡ ❛❞♦t❛♠♦s ♣❛r❛ ❛ ❧ó❣✐❝❛ ♣r♦♣♦s✐❝✐♦♥❛❧✳ ❆❞✐❝✐♦♥❛❧♠❡♥t❡✱ ❝♦♥s❡r❛r❡♠♦s q✉❡ q✉❛♥t✐✜❝❛❞♦r❡s ✭∃, ∀✮ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♣r❡❝❡❞ê♥❝✐❛ ❞❡ ¬✳

  ❙❡♠â♥t✐❝❛ ✖ ✭■✮ ❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  P❛r❛ ❞❛r s❡♠â♥t✐❝❛ ❛ ✉♠❛ ❢ór♠✉❧❛ ❞❡✈❡♠♦s ✐♥t❡r♣r❡tá✲❧❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ U✳ ▼❛s s♦♠❡♥t❡ U é s✉✜❝✐❡♥t❡❄

  ❙❡❥❛ U = N✳ ❈♦♠ ✐ss♦ é ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r s❡ ∃ ✐.∀❥.▼(✐, ❥) é ✈❡r❞❛❞❡✐r❛❄ ◆ã♦✦ ◗✉❛❧ ♦ s✐❣♥✐✜❝❛❞♦ ❞♦ ♣r❡❞✐❝❛❞♦ ▼❄

  ❈♦♠♦ ❞❡t❡r♠✐♥❛r s❡ ▼(✐, ❥) é ✈❡r❞❛❞❡✐r♦❄

  ❙❡♠â♥t✐❝❛ ✖ ✭■■✮ ❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  ❙♦♠❡♥t❡ ♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ✐♥t❡r♣r❡t❛r ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❉❡✈❡ ❤❛✈❡r ❛❞✐❝✐♦♥❛♠❡♥t❡ ♠❛♥❡✐r❛s ❞❡ ✐♥t❡r♣r❡t❛r✿

  ❚❡r♠♦s ❢✉♥❝✐♦♥❛✐s✳ Pr❡❞✐❝❛❞♦s✳

  ❙❡♠â♥t✐❝❛ ✖ ✭■■■✮ ❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  ❚❡r♠♦s ❢✉♥❝✐♦♥❛✐s ♣♦❞❡♠ s❡r ♣❡♥s❛❞♦s ❝♦♠♦ ❢✉♥çõ❡s ♥✲ár✐❛s ❡♥tr❡ t❡r♠♦s✳

  P❛r❛ ❝❛❞❛ ❢✉♥çã♦ ❢ : T → T ✱ ❞❡✈❡ ❡①✐st✐r : U → U ❢ q✉❡ ❛ss♦❝✐❛ ❝❛❞❛ ❝♦♥st❛♥t❡ ❡♠ T ❛

s✉❛ r❡s♣❡❝t✐✈❛ ✐♥t❡r♣r❡t❛çã♦ ❡♠ U✳

  ❙❡♠â♥t✐❝❛ ✖ ✭■❱✮ ❙❡♠â♥t✐❝❛ ❞❛ ▲ó❣✐❝❛ ❞❡ Pr❡❞✐❝❛❞♦s

  Pr❡❞✐❝❛❞♦s ♣♦❞❡♠ s❡r ♣❡♥s❛❞♦s ❝♦♠♦ ♥✲✉♣❧❛s ❞❡ t❡r♠♦s✳

  P❛r❛ ❝❛❞❛ ♣r❡❞✐❝❛❞♦ ♣ : T × T ✱ ❞❡✈❡ ❡①✐st✐r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣❛r❡s ❞❡ t❡r♠♦s P✳ ❉✐③❡♠♦s q✉❡ ✶ ✷ ✶ ✷

, ) , ) ∈

♣(t t é ✈❡r❞❛❞❡✐r♦ s❡ (t t P✳

  ❙❡♠â♥t✐❝❛ ✖ ✭❱✮ ❱♦❧t❛♥❞♦ ❛♦ ❊①❡♠♣❧♦ ❛♥t❡r✐♦r✳✳✳

  ❙❡❥❛ U = N✱ ❡ ❝♦♥s✐❞❡r❡ q✉❡ ♣❛r❛ ①, ② ∈ N ✱ ▼(①, ②) é ✈❡r❞❛❞❡✐r♦ s❡ ① ≥ ②✳ ❆ ❢ór♠✉❧❛ ∃✐.∀❥.▼(✐, ❥) é ✈❡r❞❛❞❡✐r❛❄

  ◆ã♦✦ ❆ ❢ór♠✉❧❛ r❡♣r❡s❡♥t❛ ❛ s❡❣✉✐♥t❡ s❡♥t❡♥ç❛✿ ✏❡①✐st❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ q✉❡ é ♠❛✐♦r q✉❡ t♦❞♦s ♦s ♦✉tr♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✑✳

  ▼❛s ❡ ❛ ❢ór♠✉❧❛ ∀❥.∃✐.▼(✐, ❥)❄

  ❙❡♠â♥t✐❝❛ ✖ ✭❱■✮ ❋✐♥❛❧✐③❛♥❞♦✳✳✳

  ■♥t❡r♣r❡t❛♠♦s ❢ór♠✉❧❛s ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s ✉t✐❧✐③❛♥❞♦✿

  ❯♠ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ❯♠ ❝♦♥❥✉♥t♦ ❞❡ ❞❡✜♥✐çõ❡s ❞❡ ♣r❡❞✐❝❛❞♦s ❡ sí♠❜♦❧♦s ❢✉♥❝✐♦♥❛✐s✳

  ❙❡♠â♥t✐❝❛ ✖ ✭❱■■✮ ❊①❡r❝í❝✐♦

  ❈♦♥s✐❞❡r❡ ❝♦♠♦ ✉♥✐✈❡rs♦ ❞❡ ❞✐s❝✉rs♦ ♦ ❝♦♥❥✉♥t♦ N ❡ ♦s s❡❣✉✐♥t❡s ♣r❡❞✐❝❛❞♦s ❡ sí♠❜♦❧♦s ❢✉♥❝✐♦♥❛✐s✿

  ♣❛r(①)✿ ✈❡r❞❛❞❡✐r♦ s❡ ① é ♣❛r✳ ✐♠♣❛r(①)✿ ✈❡r❞❛❞❡✐r♦ s❡ ① é í♠♣❛r✳ s : N → N✿ ❢✉♥çã♦ s✉❝❡ss♦r✳

  ◗✉❛❧ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛s s❡❣✉✐♥t❡s ❢ór♠✉❧❛s❄ ♥.♣❛r(♥) ∨ ✐♠♣❛r(♥)✳ ∀ ♥.✐♠♣❛r(♥) → ♣❛r(s(♥))✳ ¬∃

  ♥.♣❛r(♥) ∧ ✐♠♣❛r(♥)✳

  ❙❡♠â♥t✐❝❛ ✖ ✭❱■■■✮ ❊①❡r❝í❝✐♦

  ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❢r❛s❡✿ ✏❚♦❞♦s ♦s ✉♥✐❝ór♥✐♦s ♣♦ss✉❡♠ ✺ ♣❛t❛s✑✳

  ❘❡♣r❡s❡♥t❡ ❡st❛ ❢r❛s❡ ✉t✐❧✐③❛♥❞♦ ✉♠❛ ❢ór♠✉❧❛ ❞❛ ❧ó❣✐❝❛ ❞❡ ♣r❡❞✐❝❛❞♦s✳ ❊♥❝♦♥tr❡ ✉♥✐✈❡rs♦s ❞❡ ❞✐s❝✉rs♦ ❡ ✐♥t❡r♣r❡t❛çõ❡s ❞❡ ♣r❡❞✐❝❛❞♦s ❡ sí♠❜♦❧♦s ❢✉♥❝✐♦♥❛✐s ♣❛r❛ s✉❛ ❢ór♠✉❧❛✳ ❊♠ s❡✉ ♠♦❞❡❧♦ ❞❡ ✐♥t❡r♣r❡t❛çã♦✱ s✉❛ ❢ór♠✉❧❛ é ✈❡r❞❛❞❡✐r❛❄

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