Apostila de Lógica para Programação e Sistemas

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❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✸
✶✳✶ ▲ó❣✐❝❛ ❋♦r♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✷ ❉❡❞✉çã♦ ❡ ■♥❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✸ ▲ó❣✐❝❛ ❈❧áss✐❝❛ ❡ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✹ Pr♦♣♦s✐çõ❡s ❡ Pr❡❞✐❝❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✺ Pr✐♥❝í♣✐♦s ❞❛ ▲ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧ ✶✵
✷✳✶ ▲✐♥❣✉❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷✳✶✳✶ Pr♦♣♦s✐çõ❡s ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵
✷✳✶✳✷ Pr♦♣♦s✐çõ❡s ❈♦♠♣♦st❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✷✳✶✳✸ ❋ór♠✉❧❛s ❇❡♠ ❋♦r♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✷ ❙❡♠â♥t✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✷✳✷✳✶ ❖♣❡r❛çõ❡s ❧ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷✳✷ ❚❛❜❡❧❛✲✈❡r❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✸ ❊q✉✐✈❛❧ê♥❝✐❛ ▲ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✷✳✹ ❋♦r♠❛s ♥♦r♠❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✷✳✺ ■♥❢❡rê♥❝✐❛ ▲ó❣✐❝❛ ✭❈♦♥s❡q✉ê♥❝✐❛ ▲ó❣✐❝❛✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✸ ❉❡❞✉çã♦ ♥♦ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧ ✸✸
✸✳✶ ❆r❣✉♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✸✳✷ ❉❡❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✸✳✸ ❙✐♠♣❧✐✜❝❛çã♦ ❞❛ ❈♦♥❝❧✉sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✸✳✹ ■♥✈❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✶
❙❯▼➪❘■❖ ✷
✹ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s ✺✵
✹✳✶ Pr❡❞✐❝❛❞♦s ❡ ❚❡r♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
✹✳✷ ❖♣❡r❛çõ❡s ▲ó❣✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
✹✳✸ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼
✹✳✹ ❋ór♠✉❧❛s ❇❡♠ ❋♦r♠❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺
✺ ❉❡❞✉çã♦ ♥♦ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s ✻✼
✺✳✶ ❊❧✐♠✐♥❛çã♦ ❡ ■♥s❡rçã♦ ❞❡ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼
✺✳✶✳✶ ■♥st❛♥❝✐❛çã♦ ❯♥✐✈❡rs❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽
✺✳✶✳✷ ●❡♥❡r❛❧✐③❛çã♦ ❯♥✐✈❡rs❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾
✺✳✶✳✸ ●❡♥❡r❛❧✐③❛çã♦ ❊①✐st❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵
✺✳✶✳✹ ■♥st❛♥❝✐❛çã♦ ❊①✐st❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶
✺✳✷ ❈✉✐❞❛❞♦s ♥❛ ❛♣❧✐❝❛çã♦ ❞❛s r❡❣r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷
✻ ▼❡❝❛♥✐③❛çã♦ ❞❛ ▲ó❣✐❝❛ ❈❧áss✐❝❛ ✼✺
✻✳✶ ❘❡s♦❧✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✻✳✶✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❝❧❛✉s❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻
✻✳✶✳✷ ❯♥✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼
✻✳✷ ❘❡s♦❧✉çã♦ ♣♦r ❘❡❢✉t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵
❆ ❊q✉✐✈❛❧ê♥❝✐❛s ❡ ■♥❢❡rê♥❝✐❛s ❢✉♥❞❛♠❡♥t❛✐s ✽✹
❆✳✶ ▲❡✐s ❞❡ ❊q✉✐✈❛❧ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹
❆✳✷ ❘❡❣r❛s ❞❡ ■♥❢❡rê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺
❆✳✸ ❊q✉✐✈❛❧ê♥❝✐❛s ❡ ■♥❢❡rê♥❝✐❛s ❝♦♠ ◗✉❛♥t✐✜❝❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
✶✳✶ ▲ó❣✐❝❛ ❋♦r♠❛❧
❊♠❜♦r❛ ❡①✐st❛♠ ♠✉✐t❛s ❞❡✜♥✐çõ❡s ♣❛r❛ ♦ ❝❛♠♣♦ ❞❡ ❡st✉❞♦ ❞❛ ❧ó❣✐❝❛✱ ❡ss❛s ❞❡✜♥✐çõ❡s
♥ã♦ ❞✐❢❡r❡♠ ❡ss❡♥❝✐❛❧♠❡♥t❡ ✉♠❛s ❞❛s ♦✉tr❛s❀ ❤á ✉♠ ❝❡rt♦ ❝♦♥s❡♥s♦ ❡♥tr❡ ♦s ❛✉✲
t♦r❡s ❞❡ q✉❡ ❛ ▲ó❣✐❝❛ t❡♠✱ ♣♦r ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✱ ❛s ❧❡✐s ❣❡r❛✐s ❞♦ ♣❡♥s❛♠❡♥t♦✱ ❡
❛s ❢♦r♠❛s ❞❡ ❛♣❧✐❝❛r ❡ss❛s ❧❡✐s ❝♦rr❡t❛♠❡♥t❡ ♥❛ ✐♥✈❡st✐❣❛çã♦ ❞❛ ✈❡r❞❛❞❡✳
❊♠❜♦r❛ t❡♥❤❛♠ s✐❞♦ ❡♥❝♦♥tr❛❞♦s ♥❛ ❮♥❞✐❛✱ t❡①t♦s s♦❜r❡ ❡ss❡ ❛ss✉♥t♦✱ ❡s❝r✐t♦s
❡♠ é♣♦❝❛s r❡♠♦t❛s✱ é tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ❛❝❡✐t♦ q✉❡ ❛ ▲ó❣✐❝❛ t❡♥❤❛ ♥❛s❝✐❞♦ ♥❛ ●ré✲
❝✐❛ ❆♥t✐❣❛✱ ♣♦r ✈♦❧t❛ ❞♦ sé❝✉❧♦ ■❱ ❛♥t❡s ❞❡ ❈r✐st♦✳ ❖s ♣r✐♠❡✐r♦s tr❛❜❛❧❤♦s s♦❜r❡
▲ó❣✐❝❛ sã♦ ❞❡✈✐❞♦s ❛ P❛r♠ê♥✐❞❡s✱ ❩❡♥ã♦✱ ❡ ❛♦ ❣r✉♣♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✏s♦✜s✲
t❛s✑✱ ♠❛s ♦ ✈❡r❞❛❞❡✐r♦ ❝r✐❛❞♦r ❞❛ ▲ó❣✐❝❛ é✱ s❡♠ ❞ú✈✐❞❛✱ ❆r✐stót❡❧❡s✱ ♣♦✐s ❢♦✐
❡❧❡ q✉❡♠ s✐st❡♠❛t✐③♦✉ ❡ ♦r❣❛♥✐③♦✉ ❡ss❡ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡❧❡✈❛♥❞♦✲♦ à ❝❛t❡❣♦r✐❛ ❞❡
❝✐ê♥❝✐❛✳ ❊♠ s✉❛ ♦❜r❛ ❝❤❛♠❛❞❛ ❖r❣❛♥✉♠ ✭q✉❡✱ ❡♠ tr❛❞✉çã♦ ❧✐✈r❡✱ s✐❣♥✐✜❝❛ ✏❢❡rr❛✲
♠❡♥t❛✑✮ ❆r✐stót❡❧❡s ❡st❛❜❡❧❡❝❡✉ ♣r✐♥❝í♣✐♦s tã♦ ❣❡r❛✐s ❡ tã♦ só❧✐❞♦s q✉❡ ❞♦♠✐♥♦✉
♦ ♣❡♥s❛♠❡♥t♦ ♦❝✐❞❡♥t❛❧ ❞✉r❛♥t❡ ❞♦✐s ♠✐❧ ❛♥♦s✱ ❡ ❛té ❤♦❥❡ sã♦ ❝♦♥s✐❞❡r❛❞♦s ✈á❧✐❞♦s✳
❆r✐stót❡❧❡s t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛ ❜✉s❝❛ ❞❛ ✈❡r❞❛❞❡✱ ❡✱ ♣❛r❛ ✐ss♦✱ ♣r♦❝✉r❛✈❛
❝❛r❛❝t❡r✐③❛r ♦s ✐♥str✉♠❡♥t♦s ❞❡ q✉❡ s❡ s❡r✈✐❛ ❛ r❛③ã♦✱ ♥❡ss❛ ❜✉s❝❛✳ ❊♠ ♦✉tr❛s
♣❛❧❛✈r❛s✱ ❆r✐stót❡❧❡s s❡ ♣r❡♦❝✉♣❛✈❛ ❝♦♠ ❛s ❢♦r♠❛s ❞❡ r❛❝✐♦❝í♥✐♦ q✉❡✱ ❛ ♣❛rt✐r
❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❝♦♥s✐❞❡r❛❞♦s ✈❡r❞❛❞❡✐r♦s✱ ♣❡r♠✐t✐❛♠ ♦❜t❡r ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✳
❈❛❜❡r✐❛✱ ♣♦✐s✱ à ▲ó❣✐❝❛✱ ❛ ❢♦r♠✉❧❛çã♦ ❞❡ ❧❡✐s ❣❡r❛✐s ❞❡ ❡♥❝❛❞❡❛♠❡♥t♦s ❞❡ ❝♦♥❝❡✐t♦s
❡ ❥✉í③♦s q✉❡ ❧❡✈❛r✐❛♠ à ❞❡s❝♦❜❡rt❛ ❞❡ ♥♦✈❛s ✈❡r❞❛❞❡s✳
✸
✹❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖
❊ss❛ ❢♦r♠❛ ❞❡ ❡♥❝❛❞❡❛♠❡♥t♦ é ❝❤❛♠❛❞♦✱ ❡♠ ▲ó❣✐❝❛✱ ❞❡ ❛r❣✉♠❡♥t♦✱ ❡♥q✉❛♥t♦
❛s ❛✜r♠❛çõ❡s ❡♥✈♦❧✈✐❞❛s sã♦ ❝❤❛♠❛❞❛s ♣r♦♣♦s✐çõ❡s ❀ ✉♠ ❛r❣✉♠❡♥t♦ é✱ ♣♦✐s✱ ✉♠
❝♦♥❥✉♥t♦ ❞❡ ♣r♦♣♦s✐çõ❡s t❛❧ q✉❡ s❡ ❛✜r♠❡ q✉❡ ✉♠❛ ❞❡❧❛s é ❞❡r✐✈❛❞❛ ❞❛s ❞❡♠❛✐s❀
✉s✉❛❧♠❡♥t❡✱ ❛ ♣r♦♣♦s✐çã♦ ❞❡r✐✈❛❞❛ é ❝❤❛♠❛❞❛ ❝♦♥❝❧✉sã♦✱ ❡ ❛s ❞❡♠❛✐s✱ ♣r❡♠✐ss❛s✳
❊♠ ✉♠ ❛r❣✉♠❡♥t♦ ✈á❧✐❞♦✱ ❛s ♣r❡♠✐ss❛s sã♦ ❝♦♥s✐❞❡r❛❞❛s ♣r♦✈❛s ❡✈✐❞❡♥t❡s ❞❛ ✈❡r✲
❞❛❞❡ ❞❛ ❝♦♥❝❧✉sã♦✳
❊✐s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛r❣✉♠❡♥t♦✿
❙❡ ❡✉ ❣❛♥❤❛r ♥❛ ❧♦t❡r✐❛✱ s❡r❡✐ r✐❝♦
❊✉ ❣❛♥❤❡✐ ♥❛ ❧♦t❡r✐❛
▲♦❣♦✱ s♦✉ r✐❝♦
❈♦♠♦ ❛ ❝♦♥❝❧✉sã♦ ✏s♦✉ r✐❝♦✑ é ✉♠❛ ❞❡❝♦rrê♥❝✐❛ ❧ó❣✐❝❛ ❞❛s ❞✉❛s ♣r❡♠✐ss❛s✱ ❡ss❡
❛r❣✉♠❡♥t♦ é ❝♦♥s✐❞❡r❛❞♦ ✈á❧✐❞♦✳
➱ ♣r❡❝✐s♦ ❞❡✐①❛r ❝❧❛r♦ q✉❡ ❛ ▲ó❣✐❝❛ s❡ ♣r❡♦❝✉♣❛ ❝♦♠ ♦ r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♥tr❡
❛s ♣r❡♠✐ss❛s ❡ ❛ ❝♦♥❝❧✉sã♦✱ ❝♦♠ ❛ ❡str✉t✉r❛ ❡ ❛ ❢♦r♠❛ ❞♦ r❛❝✐♦❝í♥✐♦✱ ❡ ♥ã♦ ❝♦♠
s❡✉ ❝♦♥t❡ú❞♦✱ ✐st♦ é✱ ❝♦♠ ❛s ♣r♦♣♦s✐çõ❡s t♦♠❛❞❛s ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡✳ ❊♠ ♦✉tr❛s
♣❛❧❛✈r❛s✱ ♥ã♦ é ♦❜❥❡t♦ ❞❛ ▲ó❣✐❝❛ s❛❜❡r s❡ q✉❡♠ ❣❛♥❤❛ ♥❛ ▲♦t❡r✐❛ ✜❝❛ r✐❝♦ ♦✉
♥ã♦✱ ♦✉ s❡ ❡✉ ❣❛♥❤❡✐ ♦✉ ♥ã♦ ♥❛ ▲♦t❡r✐❛✳ ❖ ♦❜❥❡t♦ ❞❛ ▲ó❣✐❝❛ é ❞❡t❡r♠✐♥❛r s❡ ❛
❝♦♥❝❧✉sã♦ é ♦✉ ♥ã♦ ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❧ó❣✐❝❛ ❞❛s ♣r❡♠✐ss❛s✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ♣♦r
q✉❡ ♦ ♦❜❥❡t♦ ❞❛ ▲ó❣✐❝❛ é ❛ ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ♦ r❛❝✐♦❝í♥✐♦ ❡stá ❡str✉t✉r❛❞♦✱ ❛ ▲ó❣✐❝❛
❝♦st✉♠❛ r❡❝❡❜❡r ♦ ♥♦♠❡ ❞❡ ▲ó❣✐❝❛ ❋♦r♠❛❧ ✳
❆ ✈❛❧✐❞❛❞❡ ❞♦ ❛r❣✉♠❡♥t♦ ❡stá ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞❛ à ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ❡❧❡ s❡
❛♣r❡s❡♥t❛✱ ❝♦♠♦ ♣♦❞❡ s❡r ♠♦str❛❞♦ ♣❡❧♦ ❡♥✉♥❝✐❛❞♦ ❛❜❛✐①♦✱
❙❡ ❡✉ ❣❛♥❤❛r ♥❛ ❧♦t❡r✐❛✱ s❡r❡✐ r✐❝♦
◆ã♦ ❣❛♥❤❡✐ ♥❛ ❧♦t❡r✐❛
▲♦❣♦✱ ♥ã♦ s♦✉ r✐❝♦
q✉❡✱ ❡♠❜♦r❛ s❡❥❛ s❡♠❡❧❤❛♥t❡ ❛♦ ❛♥t❡r✐♦r✱ t❡♠ ♦✉tr❛ ❢♦r♠❛✱ ❡✱ ♥❡ss❛ ❢♦r♠❛✱ ❛ ❝♦♥✲
❝❧✉sã♦ ♥ã♦ s❡ s❡❣✉❡ ❧♦❣✐❝❛♠❡♥t❡ ❞❛s ♣r❡♠✐ss❛s✱ ❡✱ ♣♦rt❛♥t♦✱ ♥ã♦ é ✉♠ ❛r❣✉♠❡♥t♦
✈á❧✐❞♦
✶
✳
✶
◆❛ ▲ó❣✐❝❛ ❋♦r♠❛❧✱ ♦ t❡♠♣♦ ❝r♦♥♦❧ó❣✐❝♦ ♥ã♦ é ❝♦♥s✐❞❡r❛❞♦✱ ✈❡r❡♠♦s ✐ss♦ ♠❛✐s à ❢r❡♥t❡ ♥♦s
Pr✐♥❝í♣✐♦s ❞❛ ▲ó❣✐❝❛ ❈❧áss✐❝❛
❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✺
✶✳✷ ❉❡❞✉çã♦ ❡ ■♥❞✉çã♦
❆ ▲ó❣✐❝❛ ❞✐s♣õ❡ ❞❡ ❞✉❛s ❢❡rr❛♠❡♥t❛s ♣r✐♥❝✐♣❛✐s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞❛s ♣❡❧♦
♣❡♥s❛♠❡♥t♦ ♥❛ ❜✉s❝❛ ❞❡ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s✿ ❛ ❞❡❞✉çã♦ ❡ ❛ ✐♥❞✉çã♦✱ q✉❡ ❞ã♦
♦r✐❣❡♠ ❛ ❞♦✐s t✐♣♦s ❞❡ ❛r❣✉♠❡♥t♦s✱ ❞❡❞✉t✐✈♦s ❡ ✐♥❞✉t✐✈♦s✳ ❖s ❛r❣✉♠❡♥t♦s ❞❡❞✉t✐✈♦s
♣r❡t❡♥❞❡♠ q✉❡ s✉❛s ♣r❡♠✐ss❛s ❢♦r♥❡ç❛♠ ✉♠❛ ♣r♦✈❛ ❝♦♥❝❧✉s✐✈❛ ❞❛ ✈❡r❛❝✐❞❛❞❡ ❞❛
❝♦♥❝❧✉sã♦✳ ❯♠ ❛r❣✉♠❡♥t♦ ❞❡❞✉t✐✈♦ é ✈á❧✐❞♦ q✉❛♥❞♦ s✉❛s ♣r❡♠✐ss❛s✱ s❡ ✈❡r❞❛❞❡✐r❛s✱
❢♦r♥❡❝❡♠ ♣r♦✈❛s ❝♦♥✈✐♥❝❡♥t❡s ♣❛r❛ s✉❛ ❝♦♥❝❧✉sã♦✱ ✐st♦ é✱ q✉❛♥❞♦ ❢♦r ✐♠♣♦ssí✈❡❧ q✉❡
❛s ♣r❡♠✐ss❛s s❡❥❛♠ ✈❡r❞❛❞❡✐r❛s ❡ ❛ ❝♦♥❝❧✉sã♦ ❢❛❧s❛❀ ❝❛s♦ ❝♦♥trár✐♦✱ ♦ ❛r❣✉♠❡♥t♦
❞❡❞✉t✐✈♦ é ❞✐t♦ ✐♥✈á❧✐❞♦
✷
✳ ❖s ❞♦✐s ❛r❣✉♠❡♥t♦s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ sã♦ ❞♦ t✐♣♦
❞❡❞✉t✐✈♦✱ ♦ ♣r✐♠❡✐r♦ ✈á❧✐❞♦ ❡ ♦ s❡❣✉♥❞♦ ✐♥✈á❧✐❞♦✳
❖s ❛r❣✉♠❡♥t♦s ✐♥❞✉t✐✈♦s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ♥ã♦ ♣r❡t❡♥❞❡♠ q✉❡ s✉❛s ♣r❡♠✐ss❛s
❢♦r♥❡ç❛♠ ♣r♦✈❛s ❝❛❜❛✐s ❞❛ ✈❡r❛❝✐❞❛❞❡ ❞❛ ❝♦♥❝❧✉sã♦✱ ♠❛s ❛♣❡♥❛s q✉❡ ❢♦r♥❡ç❛♠
✐♥❞✐❝❛çõ❡s ❞❡ss❛ ✈❡r❛❝✐❞❛❞❡✳ ❱❡❥❛ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛r❣✉♠❡♥t♦ ✐♥❞✉t✐✈♦✿
❏♦❣✉❡✐ ✉♠❛ ♣❡❞r❛ ♥♦ ❧❛❣♦✱ ❡ ❛ ♣❡❞r❛ ❛❢✉♥❞♦✉
❏♦❣✉❡✐ ♦✉tr❛ ♣❡❞r❛ ♥♦ ❧❛❣♦ ❡ ❡❧❛ t❛♠❜é♠ ❛❢✉♥❞♦✉
❏♦❣✉❡✐ ♠❛✐s ✉♠❛ ♣❡❞r❛ ♥♦ ❧❛❣♦✱ ❡ t❛♠❜é♠ ❡st❛ ❛❢✉♥❞♦✉
▲♦❣♦✱ s❡ ❡✉ ❥♦❣❛r ✉♠❛ ♦✉tr❛ ♣❡❞r❛ ♥♦ ❧❛❣♦✱ ❡❧❛ ✈❛✐ ❛❢✉♥❞❛r
❖s t❡r♠♦s ✏✈á❧✐❞♦s✑ ❡ ✏✐♥✈á❧✐❞♦s✑ ♥ã♦ s❡ ❛♣❧✐❝❛♠ ❛♦s ❛r❣✉♠❡♥t♦s ✐♥❞✉t✐✈♦s❀ ❡❧❡s
❝♦st✉♠❛♠ s❡r ❛✈❛❧✐❛❞♦s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♠❛✐♦r ♦✉ ♠❡♥♦r ♣♦ss✐❜✐❧✐❞❛❞❡ ❝♦♠ q✉❡
s✉❛s ❝♦♥❝❧✉sõ❡s s❡❥❛♠ ❡st❛❜❡❧❡❝✐❞❛s✳
❈♦st✉♠❛✲s❡ ❞✐③❡r q✉❡ ♦s ❛r❣✉♠❡♥t♦s
✐♥❞✉t✐✈♦s ♣❛rt❡♠ ❞♦ ♣❛rt✐❝✉❧❛r ♣❛r❛ ♦
❣❡r❛❧✱ ✐st♦ é✱ ❛ ♣❛rt✐r ❞❡ ♦❜s❡r✈❛çõ❡s ♣❛rt✐❝✉❧❛r❡s✱ ♣r♦❝✉r❛ ❡st❛❜❡❧❡❝❡r r❡❣r❛s ❣❡r❛✐s✱
q✉❡✱ ♥♦ ❝❛s♦ ❞❛s ❝✐ê♥❝✐❛s ♥❛t✉r❛✐s✱ ❞❡✈❡♠ s❡r ♣r♦✈❛❞❛s ♣♦r ♦✉tr♦s ♠❡✐♦s❀ ♦s ❛r❣✉✲
♠❡♥t♦s ❞❡❞✉t✐✈♦s✱ ♣♦r s❡✉ ❧❛❞♦✱ ♣❛rt❡♠ ❞❡ r❡❣r❛s ❣❡r❛✐s ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❛ ✈❡r❛❝✐✲
❞❛❞❡ ❞❡ ❛❝♦♥t❡❝✐♠❡♥t♦s ♣❛rt✐❝✉❧❛r❡s✳ ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝✐ê♥❝✐❛ t❡♠ ❞❡♣❡♥❞✐❞♦✱
❡♠ ❣r❛♥❞❡ ♣❛rt❡✱ ❞❛ ❤❛❜✐❧✐❞❛❞❡ ❡♠ ❝♦♠❜✐♥❛r ♦s ❞♦✐s t✐♣♦s ❞❡ r❛❝✐♦❝í♥✐♦✳
✷
❖s ❛r❣✉♠❡♥t♦s ✐♥✈á❧✐❞♦s sã♦ t❛♠❜é♠ ❝❤❛♠❛❞♦s s♦✜s♠❛s ♦✉ ❢❛❧á❝✐❛s✳ ❖ q✉❡ ❞✐❢❡r❡♥❝✐❛ ♦
s♦✜s♠❛ ❞❛ ❢❛❧á❝✐❛✱ é q✉❡✱ ❡♠❜♦r❛ ❛♠❜♦s s❡❥❛♠ ❜❛s✐❝❛♠❡♥t❡ r❛❝✐♦❝í♥✐♦s ❡rr❛❞♦s✱ ❛ ❢❛❧á❝✐❛ é ✐♥✈♦❧✲
✉♥tár✐❛✳ ❆♦ ♣❛ss♦ q✉❡ ♦ s♦✜s♠❛ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐♥❞✉③✐r ❛ ❛✉❞✐ê♥❝✐❛ ❛♦ ❡♥❣❛♥♦✱ ♦ r❛❝✐♦❝í♥✐♦
❢❛❧❛❝✐♦s♦ ❞❡❝♦rr❡ ❞❡ ✉♠❛ ❢❛❧❤❛ ❞❡ q✉❡♠ ❛r❣✉♠❡♥t❛✳
❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✻
✶✳✸ ▲ó❣✐❝❛ ❈❧áss✐❝❛ ❡ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛
❖s ❛r❣✉♠❡♥t♦s ❢♦r♠✉❧❛❞♦s ❡♠ ✉♠❛ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✱ ❝♦♠♦ ♦ ✐♥❣❧ês ♦✉ ♣♦rt✉❣✉ês✱
sã♦✱ ♠✉✐t❛s ✈❡③❡s✱ ❞❡ ❞✐❢í❝✐❧ ❛✈❛❧✐❛çã♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣♦r ❝❛✉s❛ ❞❛ ❛♠❜✐❣✉✐❞❛❞❡
✐♥❡r❡♥t❡ às ❧✐♥❣✉❛❣❡♥s ♥❛t✉r❛✐s✱ ❡ ❞❛s ❝♦♥str✉çõ❡s às ✈❡③❡s ✈❛❣❛s ♦✉ ❝♦♥❢✉s❛s ❞♦s
t❡r♠♦s✳ ❊♠ ✈✐rt✉❞❡ ❞❡ss❡s ❢❛t♦s✱ ❛ ♣❛rt✐r ❞♦s tr❛❜❛❧❤♦s ❞❡ ●❡♦r❣❡ ❇♦♦❧❡✱ ❡♠
♠❡❛❞♦s ❞♦ sé❝✉❧♦ ❳■❳✱ ❢♦r❛♠ s❡♥❞♦ ✉t✐❧✐③❛❞♦s ❝❛❞❛ ✈❡③ ♠❛✐s sí♠❜♦❧♦s ❞❡ ♦r✐❣❡♠
♠❛t❡♠át✐❝❛ ♣❛r❛ ❡①♣r❡ss❛r ♦s ❡♥✉♥❝✐❛❞♦s ❡ r❛❝✐♦❝í♥✐♦s ❞❛ ▲ó❣✐❝❛✳ ❆ ▲ó❣✐❝❛ ❛♣r❡✲
s❡♥t❛❞❛ ❞❡ss❛ ❢♦r♠❛ é ❝❤❛♠❛❞❛ ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛ ♦✉ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✱ ❡♥q✉❛♥t♦
❛ ▲ó❣✐❝❛ ❜❛s❡❛❞❛ ❡♠ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧ é ❝❤❛♠❛❞❛ ▲ó❣✐❝❛ ❈❧áss✐❝❛✳
➚ ♠❡❞✐❞❛ q✉❡ ❛ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛ ❞❡s❡♥✈♦❧✈❡ s✉❛ ♣ró♣r✐❛ ❧✐♥❣✉❛❣❡♠ té❝♥✐❝❛✱
✈❡♠ s❡ t♦r♥❛♥❞♦ ✉♠ ✐♥str✉♠❡♥t♦ ❝❛❞❛ ✈❡③ ♠❛✐s ♣♦❞❡r♦s♦ ♣❛r❛ ❛ ❛♥á❧✐s❡ ❡ ❛ ❞❡❞✉çã♦
❞♦s ❛r❣✉♠❡♥t♦s✳ ❆ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ s✐♠❜♦❧♦❣✐❛ ♠❛t❡♠át✐❝❛ ❛❥✉❞❛ ❛ ❡①♣♦r✱ ❝♦♠
♠❛✐♦r ❝❧❛r❡③❛✱ ❛s ❡str✉t✉r❛s ❧ó❣✐❝❛s ❞❛s ♣r♦♣♦s✐çõ❡s ❡ ❞♦s ❛r❣✉♠❡♥t♦s✱ q✉❡ ♣♦❞❡♠
♥ã♦ ✜❝❛r s✉✜❝✐❡♥t❡♠❡♥t❡ ❝❧❛r❛s s❡ ❡①♣r❡ss❛s ❡♠ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✳
❯♠❛ ♦✉tr❛ ✈❛♥t❛❣❡♠ ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ s✐♠❜ó❧✐❝❛ ♣❛r❛ ❛ ▲ó❣✐❝❛
é ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✉t✐❧✐③❛çã♦ ❞❡ r❡❝✉rs♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♥♦ tr❛t❛♠❡♥t♦ ❞❡ ❡♥✉♥✲
❝✐❛❞♦s ❡ ❛r❣✉♠❡♥t♦s❀ ♦s ❝♦♠♣✉t❛❞♦r❡s ❞✐❣✐t❛✐s s❡ ♠♦str❛♠ ❜❛st❛♥t❡ ❛❞❡q✉❛❞♦s
à ♠❛♥✐♣✉❧❛çã♦ ❞❡ sí♠❜♦❧♦s✱ ❡♥q✉❛♥t♦ ❛♣r❡s❡♥t❛♠ ❡①tr❡♠❛ ❞✐✜❝✉❧❞❛❞❡ ♥♦ tr❛t❛✲
♠❡♥t♦ ❞❡ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✳ ❊♠ ✶✾✻✺✱ ✉♠ ♣❡sq✉✐s❛❞♦r ❝❤❛♠❛❞♦ ❘♦❜✐♥s♦♥ ❞❡✲
s❡♥✈♦❧✈❡✉ ✉♠ ♣r♦❝❡❞✐♠❡♥t♦ ❝♦♠♣✉t❛❝✐♦♥❛❧ ♣❛r❛ ❛ ❞❡❞✉çã♦✱ ❝❤❛♠❛❞♦ ❘❡s♦❧✉çã♦✱
❡✈✐❞❡♥❝✐❛♥❞♦ ❛s ✈❛♥t❛❣❡♥s ❞❛ ✉t✐❧✐③❛çã♦ ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ s✐♠❜ó❧✐❝❛ ♣❛r❛ ❛ ▲ó❣✐❝❛✳
✶✳✹ Pr♦♣♦s✐çõ❡s ❡ Pr❡❞✐❝❛❞♦s
▼✉✐t❛s ❞❛s ✐❞❡✐❛s ❡♥✈♦❧✈✐❞❛s ♥♦s ❛r❣✉♠❡♥t♦s ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞❛s ❛tr❛✈és ❞❡
♣r♦♣♦s✐çõ❡s ✭t❛♠❜é♠ ❝❤❛♠❛❞♦s ❡♥✉♥❝✐❛❞♦s ♦✉ s❡♥t❡♥ç❛s✮ q✉❡ s❡ r❡❢❡r❡♠ ❛ ✉♠
♦❜❥❡t♦❀ ♣♦r ❡①❡♠♣❧♦✱ ✏❡✉ ❣❛♥❤❡✐ ♥❛ ❧♦t❡r✐❛✑✱ ✏❏♦sé ❛t✐r♦✉ ✉♠❛ ♣❡❞r❛ ♥♦ ❧❛❣♦✑✱
✏❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠✑✳ ❚❛✐s ♣r♦♣♦s✐çõ❡s sã♦ ❝❤❛♠❛❞❛s s✐♥❣✉❧❛r❡s ✳
❊①✐st❡♠ ♦✉tr❛s ♣r♦♣♦s✐çõ❡s✱ ♥♦ ❡♥t❛♥t♦✱ q✉❡ ❢❛③❡♠ r❡❢❡rê♥❝✐❛ ❛ ❝♦♥❥✉♥t♦s ❞❡
♦❜❥❡t♦s❀ ♣♦r ❡①❡♠♣❧♦✱ ✏t♦❞♦s ♦s ❤♦♠❡♥s sã♦ ♠♦rt❛✐s✑✱ ✏❛❧❣✉♥s ❛str♦♥❛✉t❛s ❢♦r❛♠
à ▲✉❛✑✱ ✏♥❡♠ t♦❞♦s ♦s ❣❛t♦s ❝❛ç❛♠ r❛t♦s✑✳
❖s t❡r♠♦s ✏❤♦♠❡♥s✑✱ ✏❛str♦♥❛✉t❛s✑ ❡ ✏❣❛t♦s✑ sã♦ ❝♦♥❝❡✐t♦s❀ ♥ã♦ s❡ r❡❢❡r❡♠
✼❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖
❛ ♥❡♥❤✉♠ ❤♦♠❡♠✱ ❛str♦♥❛✉t❛ ♦✉ ❣❛t♦ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♠❛s s✐♠ ❛♦ ❝♦♥❥✉♥t♦ ❞❡
♣r♦♣r✐❡❞❛❞❡s q✉❡ ❢❛③ ❝♦♠ q✉❡ ✉♠ ♦❜❥❡t♦ ❡st❡❥❛ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛ ♦✉ ❡♠ ♦✉tr❛✳
❚❛✐s ♣r♦♣r✐❡❞❛❞❡s sã♦ ❝❤❛♠❛❞❛s ♣r❡❞✐❝❛❞♦s ✳
❈♦♠♦ ❛ ▲ó❣✐❝❛ q✉❡ tr❛t❛ ❛♣❡♥❛s ❞❛s ♣r♦♣♦s✐çõ❡s s✐♥❣✉❧❛r❡s é ♠❛✐s s✐♠♣❧❡s q✉❡
❛ q✉❡ tr❛t❛ t❛♠❜é♠ ❝♦♠ ❝♦♥❥✉♥t♦s ❞❡ ♦❜❥❡t♦s✱ ♦s ❛✉t♦r❡s ♣r❡❢❡r✐r❛♠ s❡♣❛r❛r ♦
❡st✉❞♦ ❞❛ ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛ ❡♠ ❞✉❛s ♣❛rt❡s✿ ♦ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧ ✱ ♦✉ ▲ó❣✐❝❛
❙❡♥t❡♥❝✐❛❧✱ q✉❡ s❡ ♦❝✉♣❛ ❞❛s ♣r♦♣♦s✐çõ❡s s✐♥❣✉❧❛r❡s✱ ❡ ♦ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s✱ ♦✉
▲ó❣✐❝❛ ❞♦s Pr❡❞✐❝❛❞♦s✱ q✉❡ tr❛t❛ ❞♦s ❝♦♥❥✉♥t♦s ❞❡ ♦❜❥❡t♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳
P❛r❛ tr❛t❛r ❝♦♠ ♦❜❥❡t♦s ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ♦ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s ❛♣r❡s❡♥t❛
❞♦✐s ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✱ ❛ ✈❛r✐á✈❡❧ ✱ ♣❛r❛ s❡ r❡❢❡r✐r ❛ ✉♠ ♦❜❥❡t♦ ❣❡♥ér✐❝♦ ❞❡ ✉♠❛
❝❛t❡❣♦r✐❛✱ ❡ ♦s q✉❛♥t✐✜❝❛❞♦r❡s ✱ ❡①♣r❡ssõ❡s ❝♦♠♦ ✏♣❛r❛ t♦❞♦✑ ❡ ✏❡①✐st❡ ❛❧❣✉♠✑ ♣❛r❛
s❡ r❡❢❡r✐r❡♠ à q✉❛♥t✐❞❛❞❡ ❞❡ ♦❜❥❡t♦s q✉❡ ♣❛rt✐❧❤❛♠ ♦ ♠❡s♠♦ ♣r❡❞✐❝❛❞♦❀ ❛ss✐♠ ❛
♣r♦♣♦s✐çã♦ ✏t♦❞♦s ♦s ❤♦♠❡♥s sã♦ ♠♦rt❛✐s✑ ❛ss✉♠❡ ❛ ❢♦r♠❛ ✏♣❛r❛ t♦❞♦ x✱ s❡ x é ✉♠
❤♦♠❡♠✱ ❡♥tã♦ x é ♠♦rt❛❧✑ ❡ ❛s ♣r♦♣♦s✐çõ❡s ✏❛❧❣✉♥s ❛str♦♥❛✉t❛s ❢♦r❛♠ à ▲✉❛✑ ❡
✏♥❡♠ t♦❞♦s ♦s ❣❛t♦s ❝❛ç❛♠ r❛t♦s✑ ❛ss✉♠❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ ❛s ❢♦r♠❛s ✏❡①✐st❡ ✉♠
x t❛❧ q✉❡ x é ✉♠ ❛str♦♥❛✉t❛ ❡ x ❢♦✐ à ▲✉❛✑ ❡ ✏❡①✐st❡ ✉♠ x t❛❧ q✉❡ x é ✉♠ ❣❛t♦ ❡ x
♥ã♦ ❝❛ç❛ r❛t♦s✑✳
◗✉❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❡ q✉❛♥t✐✜❝❛❞♦r❡s s❡ r❡❢❡r❡♠ ❛♣❡♥❛s ❛♦s ♦❜❥❡t♦s✱ ♦ ❈á❧❝✉❧♦
❞❡ Pr❡❞✐❝❛❞♦s t❛♠❜é♠ é ❝❤❛♠❛❞♦ ▲ó❣✐❝❛ ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠❀ ♠❛s ♣♦❞❡♠♦s ♣❡♥s❛r
❡♠ ✉♠❛ s✐t✉❛çã♦ ♥❛ q✉❛❧ ❛s ✈❛r✐á✈❡✐s ❡ q✉❛♥t✐✜❝❛❞♦r❡s s❡ r❡✜r❛♠ t❛♠❜é♠ ❛♦s
♣r❡❞✐❝❛❞♦s❀ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ♦ ❡♥✉♥❝✐❛❞♦ ✏❡①✐st❡ ✉♠ ♣r❡❞✐❝❛❞♦ q✉❡ t♦❞❛s
❛s ♣❡ss♦❛s ♣♦ss✉❡♠✑✱ q✉❡ ♣♦❞❡ s❡r ❡①♣r❡ss❛ ♣♦r ✏❡①✐st❡ ✉♠ P t❛❧ q✉❡ P é ✉♠
♣r❡❞✐❝❛❞♦ ❡ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ x✱ s❡ x é ✉♠❛ ♣❡ss♦❛✱ x ♣♦ss✉✐ P ✑
◗✉❛♥❞♦ ❛s ✈❛r✐á✈❡✐s ❡ q✉❛♥t✐✜❝❛❞♦r❡s s❡ r❡❢❡r❡♠ t❛♠❜é♠ ❛♦s ♣r❡❞✐❝❛❞♦s✱ ❝♦♠♦
♥❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ t❡♠♦s ♦ q✉❡ ❝❤❛♠❛♠♦s ▲ó❣✐❝❛ ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠✳ ❯♠ ❡①❡♠✲
♣❧♦ ✐♠♣♦rt❛♥t❡ ❞❛ ▲ó❣✐❝❛ ❞❡ ❙❡❣✉♥❞❛ ❖r❞❡♠ é ♦ Pr✐♥❝✐♣✐♦ ❞❡ ■♥❞✉çã♦ ▼❛t❡♠át✐❝❛✿
✏s❡ ♦ ♥✉♠❡r♦ ✶ t✐✈❡r ✉♠ ♣r❡❞✐❝❛❞♦✱ ❡ ♦ ❢❛t♦ ❞❡ n ♣♦ss✉✐r ❡ss❡ ♣r❡❞✐❝❛❞♦ ✐♠♣❧✐❝❛
❡♠ q✉❡ n + 1 t❛♠❜é♠ ♦ ♣♦ss✉❛✱ ❡♥tã♦ ♦ ♣r❡❞✐❝❛❞♦ s❡ ❛♣❧✐❝❛ ❛ t♦❞♦s ♦s ♥ú♠❡r♦s
♥❛t✉r❛✐s✑ ✳
❖s ♣r❡❞✐❝❛❞♦s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ sã♦✱ ♣♦✐s✱ ❛q✉❡❧❡s q✉❡ s❡ ❛♣❧✐❝❛♠ ❛ ✐♥❞✐✈í❞✉♦s❀
❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ sã♦ ❛q✉❡❧❡s q✉❡ s❡ ❛♣❧✐❝❛♠ ❛ ✐♥❞✐✈í❞✉♦s ❡ ❛♦s ♣r❡❞✐❝❛❞♦s ❞❡
♣r✐♠❡✐r❛ ♦r❞❡♠✳ ❆ ❣❡♥❡r❛❧✐③❛çã♦ ♣♦❞❡ ♣r♦ss❡❣✉✐r✱ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ♣r❡❞✐❝❛❞♦s ❞❡
t❡r❝❡✐r❛ ♦r❞❡♠✱ ❞❡ q✉❛rt❛ ♦r❞❡♠✱ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✱ ❝❛❞❛ ✉♠ ❞❡❧❡s ❛♣❧✐❝❛♥❞♦✲s❡
❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖ ✽
❛♦s ✐♥❞✐✈í❞✉♦s ❡ ❛♦s ♣r❡❞✐❝❛❞♦s ❞❛s ♦r❞❡♥s ❛♥t❡r✐♦r❡s✳
❊♠ ♥♦ss♦ ❝✉rs♦ ✈❛♠♦s ♥♦s ❧✐♠✐t❛r ❛♦ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧ ❡ ❛♦ ❈á❧❝✉❧♦ ❞❡
Pr❡❞✐❝❛❞♦s ❞❡ Pr✐♠❡✐r❛ ❖r❞❡♠✳
✶✳✺ Pr✐♥❝í♣✐♦s ❞❛ ▲ó❣✐❝❛
❆ ▲ó❣✐❝❛ ❋♦r♠❛❧ r❡♣♦✉s❛ s♦❜r❡ três ♣r✐♥❝í♣✐♦s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ♣❡r♠✐t❡♠ t♦❞♦
s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ♣♦st❡r✐♦r✱ ❡ q✉❡ ❞ã♦ ✈❛❧✐❞❛❞❡ ❛ t♦❞♦s ♦s ❛t♦s ❞♦ ♣❡♥s❛♠❡♥t♦
❡ ❞♦ r❛❝✐♦❝í♥✐♦✳ ❙ã♦ ❡❧❡s✿
Pr✐♥❝✐♣✐♦ ❞❛ ■❞❡♥t✐❞❛❞❡
❖ q✉❡ é✱ é❀ ♦✉ s❡❥❛✱ t♦❞♦ ♦❜❥❡t♦ é ✐❞ê♥t✐❝♦ ❛ s✐ ♣ró♣r✐♦✳ ■ss♦ ♥ã♦ é ✉♠ s✐♠♣❧❡s ❥♦❣♦
❞❡ ♣❛❧❛✈r❛s❀ ♥❛ ✈❡r❞❛❞❡✱ é ♣♦ssí✈❡❧ ❞❡❢❡♥❞❡r ❛ ♥♦çã♦ ♦♣♦st❛✱ ❞❡ q✉❡ ❛ r❡❛❧✐❞❛❞❡ é
✢✉✐❞❛✱ ❞❡ q✉❡ ♥❛❞❛ ♣❡r♠❛♥❡❝❡ ✐❣✉❛❧ ❛ s✐ ♣ró♣r✐♦✱ ❡ q✉❡ q✉❛❧q✉❡r r❛❝✐♦❝í♥✐♦ s♦❜r❡
♦❜❥❡t♦s é ✉♠❛ ✜❝çã♦✳
Pr✐♥❝í♣✐♦ ❞❛ ◆ã♦✲❈♦♥tr❛❞✐çã♦
❯♠ ♦❜❥❡t♦ ♥ã♦ ♣♦❞❡✱ s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ s❡r ❡ ♥ã♦ s❡r✳ ❖✉ s❡❥❛✱ ♥ã♦ é ♣♦ssí✈❡❧
❛✜r♠❛r ❡ ♥❡❣❛r ♦ ♠❡s♠♦ ♣r❡❞✐❝❛❞♦ ♣❛r❛ ♦ ♠❡s♠♦ ♦❜❥❡t♦ ❛♦ ♠❡s♠♦ t❡♠♣♦❀ ♦✉
❛✐♥❞❛✱ ❞❡ ❞✉❛s ❛✜r♠❛çõ❡s ❝♦♥tr❛❞✐tór✐❛s✱ ✉♠❛ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❢❛❧s❛✳
Pr✐♥❝í♣✐♦ ❞♦ ❚❡r❝❡✐r♦ ❊①❝❧✉í❞♦
❚♦❞♦ ♦❜❥❡t♦ é ♦✉ ♥ã♦ é✳ ❖✉ s❡❥❛✱ ✉♠❛ ❞❛❞❛ ❛✜r♠❛çã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✈❡r❞❛❞❡✐r❛
♦✉ ❢❛❧s❛✱ ♥ã♦ ❡①✐st✐♥❞♦ ✉♠❛ t❡r❝❡✐r❛ ♦♣çã♦✳
❙♦❜r❡ ❡ss❡s ♣r✐♥❝í♣✐♦s r❡♣♦✉s❛ t♦❞♦ ♦ ❛r❝❛❜♦✉ç♦ ❞❛ ▲ó❣✐❝❛ ❈❧áss✐❝❛✳ ❆ ♥❡❣❛çã♦
❞❡ ✉♠ ♦✉ ♠❛✐s ❞❡ss❡s ♣r✐♥❝í♣✐♦s ❞á ♦r✐❣❡♠ ❛ ♦✉tr❛s ❧ó❣✐❝❛s✱ ❝❤❛♠❛❞❛s ❣❡♥❡r✐❝❛✲
♠❡♥t❡ ❞❡ ▲ó❣✐❝❛s ◆ã♦✲❈❧áss✐❝❛s ✱ ❝✉❥❛s ♣r✐♥❝✐♣❛✐s ✈❡rt❡♥t❡s sã♦✿
❼ ❛s ▲ó❣✐❝❛s ▼♦❞❛✐s✱ q✉❡ ✐♥❝❧✉❡♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣♦ss✐❜✐❧✐❞❛❞❡ ❡ ❞❡ ♥❡❝❡ss✐❞❛❞❡✱
❡ ♥❛s q✉❛✐s ♦ ✈❡r❜♦ ♣♦❞❡ s❡r ♠♦❞✐✜❝❛❞♦ ♣♦r ✉♠ ❛❞✈ér❜✐♦ ❞❡ ♠♦❞♦✱ ❝♦♠♦ ♥♦s
❡♥✉♥❝✐❛❞♦s ✏t❛❧✈❡③ ❝❤♦✈❛ ❛♠❛♥❤ã✑✱ ❡ ✏❝❡rt❛♠❡♥t❡ ❏♦ã♦ s❛✐✉✑❀
✾❈❆P❮❚❯▲❖ ✶✳ ■◆❚❘❖❉❯➬➹❖
❼ ❛s ▲ó❣✐❝❛s P❧✉r✐✈❛❧❡♥t❡s✱ ♥❛s q✉❛✐s ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈❡r❛❝✐❞❛❞❡ ❞❡✐①❛ ❞❡ s❡r
❜✐♥ár✐♦ ✭✈❡r❞❛❞❡✐r♦ ❡ ❢❛❧s♦✮ ♣❛r❛ ❛ss✉♠✐r ♦✉tr♦s ✈❛❧♦r❡s✱ ❝♦♠♦ ♥❛s ▲ó❣✐❝❛s
❚r✐✈❛❧❡♥t❡s✱ ♥❛s q✉❛✐s ❛s ♣r♦♣♦s✐çõ❡s ♣♦❞❡♠ s❡r ✈❡r❞❛❞❡✐r❛s✱ ❢❛❧s❛s ❡ ♥❡✉tr❛s✱
♥❛s ▲ó❣✐❝❛s ◆❡❜✉❧♦s❛s✱ ❡♠ q✉❡ ❡①✐st❡♠ ❣r❛❞❛çõ❡s ❞❡ ✈❡r❛❝✐❞❛❞❡ ❡ ♥❛s q✉❛✐s
✉♠❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ♠❛✐s ✈❡r❞❛❞❡✐r❛ q✉❡ ♦✉tr❛✱ ❡ ♥❛s ❧ó❣✐❝❛s ♣r♦❜❛❜✐❧ís✲
t✐❝❛s✱ ♥❛s q✉❛✐s ❡①✐st❡ ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞❡ q✉❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ♣♦ss❛ s❡r
✈❡r❞❛❞❡✐r❛❀
❼ ❛s ▲ó❣✐❝❛s ❋r❛❝❛s✱ ❝♦♠♦ ❛ ■♥t✉✐❝✐♦♥✐st❛✱ q✉❡ ♥ã♦ ❛❝❡✐t❛ ♦ Pr✐♥❝✐♣✐♦ ❞♦ ❚❡r❝❡✐r♦
❊①❝❧✉í❞♦✱ ❡✱ ♣❛r❛ ❛ q✉❛❧✱ ❛ ❞✉♣❧❛ ♥❡❣❛çã♦ ♥ã♦ ❡q✉✐✈❛❧❡ à ❛✜r♠❛çã♦✱ ♦ q✉❡
♣♦❞❡ s❡r ❡①❡♠♣❧✐✜❝❛❞♦ ♣❡❧♦ ❡♥✉♥❝✐❛❞♦ ✏♥ã♦ t❡♥❤♦ ♥❛❞❛✑ ♦♥❞❡ ♦ t❡r♠♦ ✏♥ã♦✑
❛♦ ✐♥✈és ❞❡ s❡ ❝♦♥tr❛♣♦r ❛♦ t❡r♠♦ ✏♥❛❞❛✑ ♦ r❡❢♦rç❛✳
◆♦s ú❧t✐♠♦s ❛♥♦s ❛s ▲ó❣✐❝❛s ◆ã♦✲❈❧áss✐❝❛s tê♠ s♦❢r✐❞♦ ❡♥♦r♠❡ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱
❡♠ ✈✐rt✉❞❡✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ❞❡ ♥♦✈❛s ❛♣❧✐❝❛çõ❡s✱ ❛❧❣✉♠❛s ❞❛s q✉❛✐s ♥❛ ár❡❛ ❝♦♠✲
♣✉t❛❝✐♦♥❛❧✱ ❝♦♠♦ ❛ ▲ó❣✐❝❛ ◆❡❜✉❧♦s❛ ♦✉ ▲ó❣✐❝❛ ❋✉③③②✳
❈❛♣ít✉❧♦ ✷
❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧
✷✳✶ ▲✐♥❣✉❛❣❡♠
✷✳✶✳✶ Pr♦♣♦s✐çõ❡s ❙✐♠♣❧❡s
❖ ♣r✐♠❡✐r♦ ♣❛ss♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ ✉♠❛ ❧✐♥❣✉❛❣❡♠ s✐♠❜ó❧✐❝❛✱ ♠❛✐s ❛❞❡q✉❛❞❛ à
❢♦r♠✉❧❛çã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❛ ▲ó❣✐❝❛✱ é ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ q✉❡ ❝❤❛♠❛♠♦s ♣r♦♣♦s✐çã♦
s✐♠♣❧❡s ♦✉ ❛tô♠✐❝❛✳ ❊♠ ❧✐♥❤❛s ❣❡r❛✐s✱ ✉♠❛ ♣r♦♣♦s✐çã♦ s✐♠♣❧❡s ✭♦✉ ❡♥✉♥❝✐❛❞♦✱ ♦✉
s❡♥t❡♥ç❛✮✱ é ✉♠❛ ❞❡❝❧❛r❛çã♦ q✉❡ ❡①♣r✐♠❡ ✉♠ ♣❡♥s❛♠❡♥t♦ ❝♦♠ s❡♥t✐❞♦ ❝♦♠♣❧❡t♦✳
❙ã♦ ❡①❡♠♣❧♦s ❞❡ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s✿
❆ ▲✉❛ é ✉♠ s❛té❧✐t❡ ❞❛ ❚❡rr❛✳
❙ó❝r❛t❡s é ✉♠ ❤♦♠❡♠✳
❊✉ ❡st✉❞♦ ▲ó❣✐❝❛✳
❚♦❞♦s ♦s ❤♦♠❡♥s sã♦ ♠♦rt❛✐s✳
❊♠ ❣❡r❛❧✱ ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s sã♦ ❝♦♥st✐t✉í❞❛s ♣♦r ✉♠ s✉❥❡✐t♦✱ ✉♠ ✈❡r❜♦✱ ❡ s❡✉s
❝♦♠♣❧❡♠❡♥t♦s✳ ❊♠ ❧✐♥❣✉❛❣❡♠ s✐♠❜ó❧✐❝❛✱ ❝♦st✉♠❛♠♦s r❡♣r❡s❡♥t❛r ❛s ♣r♦♣♦s✐çõ❡s
s✐♠♣❧❡s ♣❡❧❛s ❧❡tr❛s p✱ q✱ r✱ s✱ t ❡t❝✱ ❝❤❛♠❛❞❛s ❧❡tr❛s s❡♥t❡♥❝✐❛✐s ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s
❢❛③❡r ❛s s❡❣✉✐♥t❡s r❡♣r❡s❡♥t❛çõ❡s✿
p✿ ❆ ▲✉❛ é ♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛
q✿ P❡❞r♦ ➪❧✈❛r❡s ❈❛❜r❛❧ ❞❡s❝♦❜r✐✉ ♦ ❇r❛s✐❧
✶✵
✶✶❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
r✿ ❉❛♥t❡ ❡s❝r❡✈❡✉ ❖s ▲✉sí❛❞❛s
s✿ ❖ ❇r❛s✐❧ é ✉♠❛ ♠♦♥❛rq✉✐❛
✷✳✶✳✷ Pr♦♣♦s✐çõ❡s ❈♦♠♣♦st❛s
❆s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦st❛s ♦✉ ♠♦❧❡❝✉❧❛r❡s sã♦ ♦❜t✐❞❛s ❝♦♠❜✐♥❛♥❞♦ ♣r♦♣♦s✐çõ❡s s✐♠✲
♣❧❡s ❛tr❛✈és ❞❡ ❝❡rt♦s t❡r♠♦s ❝❤❛♠❛❞♦s ❝♦♥❡❝t✐✈♦s ✳ ❆ ▲ó❣✐❝❛ ❞✐s♣õ❡ ❞❡ ❝✐♥❝♦
❝♦♥❡❝t✐✈♦s✿
❼ ❡
❼ ♦✉
❼ ♥ã♦
❼ s❡ ✳✳✳ ❡♥tã♦
❼ s❡ ❡ s♦♠❡♥t❡ s❡
❯t✐❧✐③❛♥❞♦ ❡ss❡s ❝♦♥❡❝t✐✈♦s ♣♦❞❡♠♦s ❝♦♥str✉✐r✱ ♣♦r ❡①❡♠♣❧♦✱ ❛s s❡❣✉✐♥t❡s ♣r♦♣♦s✐çõ❡s
❝♦♠♣♦st❛s✿
❆ ▲✉❛ ♥ã♦ é ♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛✳
❏♦ã♦ é ♠❛❣r♦ ❡ ❏♦sé é ❛❧t♦✳
▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✱ ❏♦ã♦ ❢♦✐ ❛♦ t❡❛tr♦ ❡ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✳
▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ♦✉ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✳
▼❛r✐❛ ❢♦✐ à ♣r❛✐❛ ♦✉ ❛♦ ♠❡r❝❛❞♦✳
❙❡ ❛ ❝❤✉✈❛ ❝♦♥t✐♥✉❛r ❛ ❝❛✐r ❡♥tã♦ ♦ r✐♦ ✈❛✐ tr❛♥s❜♦r❞❛r✳
❙❡ ❏♦ã♦ ❡st✉❞❛r✱ s❡rá ❛♣r♦✈❛❞♦✳
❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❡st✉❞❛r✳
❊♠ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✱ ❛ ❛çã♦ ❞❡ ❝♦♠❜✐♥❛r ♣r♦♣♦s✐çõ❡s é ❝❤❛♠❛❞❛ ♦♣❡r❛çã♦✱ ❡ ♦s
❝♦♥❡❝t✐✈♦s sã♦ ❝❤❛♠❛❞♦s ♦♣❡r❛❞♦r❡s✱ ❡ sã♦ r❡♣r❡s❡♥t❛❞♦s ♣♦r sí♠❜♦❧♦s ❡s♣❡❝í✜❝♦s✳
❆♣r❡s❡♥t❛♠♦s ❛❜❛✐①♦ ❛s ❝✐♥❝♦ ♦♣❡r❛çõ❡s ❧ó❣✐❝❛s✱ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ❝♦♥❡❝t✐✈♦s
❡ sí♠❜♦❧♦s✿
✶✷❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❖♣❡r❛çã♦ ❈♦♥❡❝t✐✈♦ ❙í♠❜♦❧♦
◆❡❣❛çã♦ ♥ã♦ ¬
❈♦♥❥✉♥çã♦ ❡ ∧
❉✐s❥✉♥çã♦ ♦✉ ∨
❈♦♥❞✐❝✐♦♥❛❧ s❡ ✳✳✳ ❡♥tã♦ →
❇✐❝♦♥❞✐❝✐♦♥❛❧ s❡ ❡ s♦♠❡♥t❡ s❡ ↔
P❛r❛ r❡♣r❡s❡♥t❛r ❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦st❛s✱ ♣r✐♠❡✐r♦ r❡♣r❡s❡♥t❛♠♦s ❛s ♣r♦♣♦s✐çõ❡s
s✐♠♣❧❡s ❞❛ ❡①♣r❡ssã♦ ❝♦♠ ✉♠❛ ❧❡tr❛ s❡♥t❡♥❝✐❛❧ ❡ ❞❡♣♦✐s ❛s ❝♦♠❜✐♥❛♠♦s ✉t✐❧✐③❛♥❞♦
♦s sí♠❜♦❧♦s ❞♦s ❝♦♥❡❝t✐✈♦s✳ ❆ss✐♠✱ ❛s ❡①♣r❡ssõ❡s ❞♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ ♣♦❞❡♠ s❡r
r❡♣r❡s❡♥t❛❞❛s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❆ ▲✉❛ ♥ã♦ é ♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛✳
p✿ ❛ ▲✉❛ é ♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛
( p)¬
❏♦ã♦ é ♠❛❣r♦ ❡ ❏♦sé é ❛❧t♦✳
p✿ ❏♦ã♦ é ♠❛❣r♦✱ q✿ ❏♦sé é ❛❧t♦
(p ∧ q)
▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✱ ❏♦ã♦ ❢♦✐ ❛♦ t❡❛tr♦ ❡ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✳
p✿ ▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✱ q✿ ❏♦ã♦ ❢♦✐ ❛♦ t❡❛tr♦✱ r✿ ▼❛r❝❡❧♦
✜❝♦✉ ❡♠ ❝❛s❛
(p ∧ q ∧ r)
▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ♦✉ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✳
p✿ ▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✱ q✿ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛
(p ∨ q)
▼❛r✐❛ ❢♦✐ à ♣r❛✐❛ ♦✉ ❛♦ ♠❡r❝❛❞♦✳
p✿ ▼❛r✐❛ ❢♦✐ à ♣r❛✐❛✱ q✿ ▼❛r✐❛ ❢♦✐ ❛♦ ♠❡r❝❛❞♦
(p ∨ q)
✶✸❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❙❡ ❛ ❝❤✉✈❛ ❝♦♥t✐♥✉❛r ❛ ❝❛✐r ❡♥tã♦ ♦ r✐♦ ✈❛✐ tr❛♥s❜♦r❞❛r✳
p✿ ❛ ❝❤✉✈❛ ❝♦♥t✐♥✉❛ ❛ ❝❛✐r✱ q✿ ♦ r✐♦ ✈❛✐ tr❛♥s❜♦r❞❛r
(p q)→
❙❡ ❏♦ã♦ ❡st✉❞❛r✱ s❡rá ❛♣r♦✈❛❞♦✳
p✿ ❏♦ã♦ ❡st✉❞❛✱ q✿ ❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦
(p q)→
❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❡st✉❞❛r✳
p✿ ❏♦ã♦ é ❛♣r♦✈❛❞♦✱ q✿ ❏♦ã♦ ❡st✉❞❛
(p q)↔
❉❡✈✐❞♦ à r✐q✉❡③❛ ❞❛ ▲í♥❣✉❛ P♦rt✉❣✉❡s❛✱ ♣❛❧❛✈r❛s ❝♦♠ s✐❣♥✐✜❝❛❞♦s ❧✐❣❡✐r❛♠❡♥t❡
❞✐❢❡r❡♥t❡s sã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ ♠❡s♠♦ ❝♦♥❡❝t✐✈♦ ❧ó❣✐❝♦✳ ❆ s❡❣✉✐r sã♦ ♠♦str❛❞❛s
❛❧❣✉♠❛s ❡①♣r❡ssõ❡s ❝♦♠✉♥s ❡♠ ♣♦rt✉❣✉ês ❛ss♦❝✐❛❞❛s ❛ ❞✐✈❡rs❛s ♦♣❡r❛çõ❡s ❧ó❣✐❝❛s✿
◆❡❣❛çã♦ (¬A)
♥ã♦ ❆
♥ã♦ s❡ ❞á q✉❡ ❆
♥ã♦ é ❢❛t♦ q✉❡ ❆
♥ã♦ é ✈❡r❞❛❞❡ q✉❡ ❆
♥ã♦ é q✉❡ ❆
♥ã♦ s❡ t❡♠ ❆
é ❢❛❧s♦ q✉❡ ❆
♥ã♦ é ♦ ❝❛s♦ q✉❡ ❆
❆ ❡ ❇
❆✱ ♠❛s ❇
❆✱ ❡♠❜♦r❛ ❇
❆✱ ❛ss✐♠ ❝♦♠♦ ❇
❈♦♥❥✉♥çã♦ (A ∧B) ❆ ❡✱ ❛❧é♠ ❞✐ss♦✱ ❇
t❛♥t♦ ❆ ❝♦♠♦ ❇
❆ ❡ t❛♠❜é♠ ❇
♥ã♦ só ❆✱ ♠❛s t❛♠❜é♠ ❇
❆✱ ❛♣❡s❛r ❞❡ ❇
✶✹❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❉✐s❥✉♥çã♦ (A ∨B)
❆ ♦✉ ❇
♦✉ ❆ ♦✉ ❇
❆ ♦✉ ❇ ♦✉ ❛♠❜♦s
s❡ ❆✱ ❡♥tã♦ ❇
s❡ ❆✱ ❇
s❡ ❆✱ ✐st♦ s✐❣♥✐✜❝❛ q✉❡ ❇
t❡♥❞♦✲s❡ ❆✱ ❡♥tã♦ ❇
q✉❛♥❞♦ ❆✱ ❡♥tã♦ ❇
s❡♠♣r❡ q✉❡ ❆✱ ❇
❇✱ s❡♠♣r❡ q✉❡ s❡ t❡♥❤❛ ❆
❇✱ ❝♦♥t❛♥t♦ q✉❡ ❆
❇✱ s❡ ❆
❇✱ q✉❛♥❞♦ ❆
❇ ♥♦ ❝❛s♦ ❞❡ ❆
❈♦♥❞✐❝✐♦♥❛❧ (A→ B) ❆✱ só s❡ ❇
❆✱ s♦♠❡♥t❡ s❡ ❇
❆ ✐♠♣❧✐❝❛ ❇
❆ ❛❝❛rr❡t❛ ❇
❇ é ✐♠♣❧✐❝❛❞❛ ♣♦r ❆
❆✱ ❧♦❣♦ ❇
❇ s❡❣✉❡ ❞❡ ❆
❜❛st❛ ❆ ♣❛r❛ ❇
❆ é ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❇
❇ é ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❆
✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❇ é ❆
✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❆ é ❇
❆ s❡ ❡ só s❡ ❇
❆ s❡ ❡ s♦♠❡♥t❡ s❡ ❇
❆ q✉❛♥❞♦ ❡ s♦♠❡♥t❡ q✉❛♥❞♦ ❇
❇✐❝♦♥❞✐❝✐♦♥❛❧ (A↔ B)
❆ ❡q✉✐✈❛❧❡ ❛ ❇
❆ ✉♥✐❝❛♠❡♥t❡ s❡ ❇
♥ã♦ ❆ ❡①❝❡t♦ s❡ ❇
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✶✺
❆ é ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❇
✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ ❆ é ❇
◆❡st❡ ♣♦♥t♦✱ ✈❛❧❡ ❞❡st❛❝❛r ❛s ❡①♣r❡ssõ❡s ✏❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡✑ ❡ ✏❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✑✳
❆♠❜❛s sã♦ tr❛t❛❞❛s ❝♦♠♦ ❝♦♥❞✐❝✐♦♥❛✐s ❞♦ t✐♣♦ A B✱ ♦♥❞❡ ❞✐③❡♠♦s q✉❡✿ ❆ é→
✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❇ ♦✉ q✉❡ ❇ é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❆✳
❊①❡♠♣❧♦s✿
❯♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ♣❛r❛ ❞✐r✐❣✐r é t❡r ❝❛rt❡✐r❛ ❞❡ ♠♦t♦r✐st❛
p✿ ❡✉ ❞✐r✐❥♦✱ q✿ ❡✉ t❡♥❤♦ ❝❛rt❡✐r❛ ❞❡ ♠♦t♦r✐st❛
(p q)→
◆♦t❡ q✉❡✱ ♥♦ ❡♥t❛♥t♦✱ ❡①✐st❡♠ ♦✉tr❛s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s
♣❛r❛ ❞✐r✐❣✐r✱ ❝♦♠♦✿ s❛❜❡r ❞✐r✐❣✐r✱ t❡r ✉♠ ❝❛rr♦ ❡t❝✳
❯♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ✈♦t❛r é ♣♦ss✉✐r tít✉❧♦ ❞❡ ❡❧❡✐t♦r✳
p✿ ❡✉ ♣♦ss✉♦ tít✉❧♦ ❞❡ ❡❧❡✐t♦r✱ q✿ ❡✉ ✈♦t♦
(p q)→
◆❡st❡ ❝❛s♦✱ ❛ ❝♦♥❞✐çã♦ ✏♣♦ss✉✐r tít✉❧♦ ❞❡ ❡❧❡✐t♦r✑ ❜❛st❛ ✭é
s✉✜❝✐❡♥t❡✮ ♣❛r❛ ✈♦t❛r✳
◆♦ ❡♥t❛♥t♦✱ q✉❛♥❞♦ ❤♦✉✈❡r ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✱ ♦✉ s❡❥❛✱ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜✲
❝✐❡♥t❡✱ ❛ ❡①♣r❡ssã♦ s❡rá ✉♠❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧✳ P♦r ❡①❡♠♣❧♦✿
❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❡st✉❞❛r
❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦ ✉♥✐❝❛♠❡♥t❡ s❡ ❡st✉❞❛r
❏♦ã♦ ♥ã♦ s❡rá ❛♣r♦✈❛❞♦✱ ❡①❝❡t♦ s❡ ❡st✉❞❛r
❏♦ã♦ ❡st✉❞❛r é ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛ ❡ s✉✜❝✐❡♥t❡ ♣❛r❛ s❡r ❛♣r♦✈❛❞♦
t♦❞❛s ♣♦❞❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ♣♦r (p q)✱ ♦♥❞❡ p r❡♣r❡s❡♥t❛ ✏❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦✑↔
❡ q r❡♣r❡s❡♥t❛ ✏❏♦ã♦ ❡st✉❞❛✑✳
✶✻❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
✷✳✶✳✸ ❋ór♠✉❧❛s ❇❡♠ ❋♦r♠❛❞❛s
❈♦♠ ♦ ❛✉①í❧✐♦ ❞♦s ❝♦♥❡❝t✐✈♦s ♣♦❞❡♠♦s ❝♦♥str✉✐r ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦st❛s ♠❛✐s ❡❧❛❜✲
♦r❛❞❛s✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♣r♦♣♦s✐çã♦✿
❙❡ ♦ ❞❡✜❝✐t ♣❡rs✐st✐r ❡ ❛ ❛rr❡❝❛❞❛çã♦ ♥ã♦ ❛✉♠❡♥t❛r✱ ❡♥tã♦ ♦✉ ❛✉♠❡♥✲
t❛♠♦s ♦s ✐♠♣♦st♦s ♦✉ ❤❛✈❡rá ✐♥✢❛çã♦✳
❈♦♠ ❛ r❡♣r❡s❡♥t❛çã♦✿
p✿ ♦ ❞❡✜❝✐t ♣❡rs✐st❡
q✿ ❛ ❛rr❡❝❛❞❛çã♦ ❛✉♠❡♥t❛
r✿ ❛✉♠❡♥t❛♠♦s ♦s ✐♠♣♦st♦s
s✿ ❤❛✈❡rá ✐♥✢❛çã♦
❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡rá s❡r ❡s❝r✐t❛ ♥❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛✿
((p ∧ ( q)) (r ∨ s))¬ →
❯♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛ é ♣♦rt❛♥t♦ ❢♦r♠❛❞❛ ♣♦r ❝♦♥❡①õ❡s ❞❡ ♣r♦♣♦s✐çõ❡s s✐♠✲
♣❧❡s✱ ♦✉ s❡❥❛✱ ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛ é ✉♠❛ ❝❛❞❡✐❛ ❝♦♥st✐t✉í❞❛ ♣❡❧♦s sí♠❜♦❧♦s
p✱ q✱ r ❡t❝✳ ✭r❡♣r❡s❡♥t❛♥❞♦ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s✮✱ sí♠❜♦❧♦s ❞❡ ❝♦♥❡❝t✐✈♦s ❡ ♣❛rê♥✲
t❡s❡s✳ ◆♦ ❡♥t❛♥t♦✱ ♥❡♠ t♦❞❛ ❝❛❞❡✐❛ ❞❡ss❡s sí♠❜♦❧♦s r❡♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦s✐çã♦
❝♦♠♣♦st❛✳ P♦r ❡①❡♠♣❧♦✱ ❛ ❝❛❞❡✐❛✿
pq ↔) ∧ ∧ ∨ (r →
♥ã♦ t❡♠ ♥❡♥❤✉♠ s✐❣♥✐✜❝❛❞♦ ❡♠ ▲ó❣✐❝❛✳ ❚❡♠♦s ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❡ r❡❝♦♥✲
❤❡❝❡r q✉❛♥❞♦ ✉♠❛ ❝❛❞❡✐❛ ❞❡ss❡s sí♠❜♦❧♦s r❡♣r❡s❡♥t❛ r❡❛❧♠❡♥t❡ ✉♠❛ ♣r♦♣♦s✐çã♦
❝♦♠♣♦st❛✳ ❆s ♣r♦♣♦s✐çõ❡s sã♦ t❛♠❜é♠ ❝♦♥❤❡❝✐❞❛s ♣♦r ❢ór♠✉❧❛s ❜❡♠ ❢♦r♠❛❞❛s
✭❢❜❢✮ ✭♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ ❢ór♠✉❧❛s✮ ❡ ♣♦ss✉❡♠ ✉♠❛ ❧❡✐ ❞❡ ❢♦r♠❛çã♦✱ ❡♥✉♥❝✐❛❞❛
❛❜❛✐①♦✿
✶✳ Pr♦♣♦s✐çõ❡s s✐♠♣❧❡s sã♦ ❢ór♠✉❧❛s
✷✳ ❙❡ A ❡ B sã♦ ❢ór♠✉❧❛s✱ ❡♥tã♦ sã♦ t❛♠❜é♠ ❢ór♠✉❧❛s✿
(A ∧B), (A ∨ B), ( A), (A B), (A B)¬ → ↔
✶✼❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
✸✳ ◆❛❞❛ ♠❛✐s é ❢ór♠✉❧❛
❖❜s❡r✈❡ q✉❡ ♦s sí♠❜♦❧♦s A ❡ B ✐♥tr♦❞✉③✐❞♦s ♥❛ s❡çã♦ ❛♥t❡r✐♦r s❡ tr❛t❛♠ ❞❡ ✈❛r✲
✐á✈❡✐s q✉❡ ❞❡♥♦t❛♠ s❡♥t❡♥ç❛s q✉❛✐sq✉❡r ❞❛ ❧✐♥❣✉❛❣❡♠ ♣r♦♣♦s✐❝✐♦♥❛❧✳ ❖ ❧❡✐t♦r ❞❡✈❡
❡st❛r ❛t❡♥t♦ ♣❛r❛ ♦ ❢❛t♦ ❞❡ q✉❡ t❛✐s sí♠❜♦❧♦s ♥ã♦ sã♦ ♣r♦♣r✐❛♠❡♥t❡ sí♠❜♦❧♦s ❞❛
❧✐♥❣✉❛❣❡♠✱ ♠❛s s✐♠ sí♠❜♦❧♦s q✉❡ ❡stã♦ ✏❢♦r❛✑ ❞❛ ❧✐♥❣✉❛❣❡♠ ♣r♦♣♦s✐❝✐♦♥❛❧✳ ❚❛✐s
✈❛r✐á✈❡✐s ❞❡♥♦♠✐♥❛♠✲s❡✱
❝♦st✉♠❡✐r❛♠❡♥t❡✱ ♠❡t❛✲✈❛r✐á✈❡✐s ✳
❚❛♥t♦ ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♦ ❛s ❝♦♠♣♦st❛s sã♦ ❝❤❛♠❛❞❛s ❡①♣r❡ssõ❡s
♣r♦♣♦s✐❝✐♦♥❛✐s✳ ◆♦ q✉❡ s❡ s❡❣✉❡✱ ♣♦r ✉♠❛ q✉❡stã♦ ❞❡ s✐♠♣❧✐❝✐❞❛❞❡✱ ✉t✐❧✐③❛r❡✲
♠♦s ♦ t❡r♠♦ ♣r♦♣♦s✐çã♦ ♣❛r❛ ✐♥❞✐❝❛r ✉♠❛ ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧✱ ♦✉ s❡❥❛✱ t❛♥t♦
♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♦ ❝♦♠♣♦st❛s✳
❖r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛ ❞❛s ♦♣❡r❛çõ❡s
P❛r❛ ❡✈✐t❛r q✉❡ ✉♠❛ ❢ór♠✉❧❛ ✜q✉❡ ❝♦♠ ✉♠ ♥ú♠❡r♦ ❡①❝❡ss✐✈♦ ❞❡ ♣❛rê♥t❡s❡s ♣♦❞❡✲
♠♦s ♦♠✐t✐r ❛❧❣✉♥s ❞❡❧❡s✱ ❛ ✜♠ ❞❡ ❢❛❝✐❧✐t❛r ❛ ❡s❝r✐t❛ ♦✉ ❛ ❧❡✐t✉r❛ ❞❛ ❢ór♠✉❧❛✳ P❛r❛
✐ss♦ é ♥❡❝❡ssár✐♦ ❡st❛❜❡❧❡❝❡r ✉♠❛ ❤✐❡r❛rq✉✐❛ ❞❡ ♦♣❡r❛çã♦ ❞♦s ❝♦♥❡❝t✐✈♦s✳ ❚❛❧ ❤✐❡r✲
❛rq✉✐❛ ✭♦✉ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛✮ é ❛ s❡❣✉✐♥t❡✿
✶✳ ¬
✷✳ ∧,∨
✸✳ →
✹✳ ↔
❊ss❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛ ✐♥❞✐❝❛ q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ♥❡❣❛çã♦ é ❛ ♣r✐♠❡✐r❛ ❛ s❡r
❡①❡❝✉t❛❞❛❀ ❡♠ s❡❣✉✐❞❛✱ ❛s ♦♣❡r❛çõ❡s ❞❡ ❝♦♥❥✉♥çã♦ ❡ ❞✐s❥✉♥çã♦ ♥❛ ♦r❞❡♠ ❡♠ q✉❡
❡st✐✈❡r❡♠ ❞✐s♣♦st❛s❀ ❞❡♣♦✐s ❞❡✈❡ s❡r ❡①❡❝✉t❛❞❛ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✱ ❡✱
♣♦r ✜♠✱ ❛ ❞❡ ❜✐❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✳
❉❡ss❛ ❢♦r♠❛✱ ❛s ♦♣❡r❛çõ❡s ❞❛ ❡①♣r❡ssã♦ ❛❝✐♠❛✱ s❡♠ ♣❛rê♥t❡s❡s✱ s❡rã♦ ❡①❡❝✉✲
t❛❞❛s ♥❛ s❡❣✉✐♥t❡ ♦r❞❡♠✿
p q r s∧ ¬ → ∨
2 1 4 3
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✶✽
◗✉❛♥❞♦ ❢♦r ♥❡❝❡ssár✐♦ ♠♦❞✐✜❝❛r ❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r
♣❛rê♥t❡s❡s✳ ❆ss✐♠✱ ♦ ❡①❡♠♣❧♦ ✧❙❡ ▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ❡ ❏♦ã♦ ❢♦✐ ❛♦ t❡❛tr♦✱ ❡♥tã♦
▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✧ é r❡♣r❡s❡♥t❛❞♦ ♣♦r p ∧ q → r ❡♥q✉❛♥t♦ q✉❡ ❛ ❡①♣r❡ssã♦
✏▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✱ ❡✱ s❡ ❏♦ã♦ ❢♦✐ ❛♦ t❡❛tr♦✱ ❡♥tã♦ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✑ é
r❡♣r❡s❡♥t❛❞❛ ♣♦r p ∧ (q → r)
❆ ✉t✐❧✐③❛çã♦ ❞♦s ❝♦♥❡❝t✐✈♦s ∧ ❡ ∨ ♣♦❞❡ ❝❛✉s❛r ❛♠❜✐❣✉✐❞❛❞❡ ❛té ♠❡s♠♦ ❡♠
❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✳ P♦r ❡①❡♠♣❧♦ ❛ ❡①♣r❡ssã♦✿
▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ❡ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛ ♦✉ ▼❛r✐❛ ❢♦✐ à ♣r❛✐❛
r❡♣r❡s❡♥t❛❞❛ ♣♦r p ∧ q ∨ s✱ ♥ã♦ ❞❡✐①❛ ❝❧❛r♦ s❡✉ s✐❣♥✐✜❝❛❞♦❀ t❛♥t♦ ♣♦❞❡ s✐❣♥✐✜❝❛r
✏▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ❡ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✱ ♦✉ ❡♥tã♦ ▼❛r✐❛ ❢♦✐ à ♣r❛✐❛✑✱ r❡♣r❡✲
s❡♥t❛❞❛ ♣♦r (p ∧ q) ∨ s✱ ❝♦♠♦ ♣♦❞❡ s✐❣♥✐✜❝❛r ✏▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ❡ ♦✉ ▼❛r❝❡❧♦
✜❝♦✉ ❡♠ ❝❛s❛ ♦✉ ▼❛r✐❛ ❢♦✐ à ♣r❛✐❛✑✱ r❡♣r❡s❡♥t❛❞❛ ♣♦r p∧(q∨s)✱ q✉❡ sã♦ ❝❧❛r❛♠❡♥t❡
❛✜r♠❛çõ❡s ❞✐st✐♥t❛s✳
❙❡❣✉♥❞♦ ❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛ ❞❛ ▲ó❣✐❝❛✱ ❛ ❡①♣r❡ssã♦ ❞❛❞❛ ❝♦rr❡s♣♦♥❞❡ à
♣r✐♠❡✐r❛ ❢♦r♠❛ ❛♣r❡s❡♥t❛❞❛✱ ♠❛s✱ ♣❛r❛ ❡✈✐t❛r q✉❛❧q✉❡r ♠❛❧✲❡♥t❡♥❞✐❞♦✱ ❛❝♦♥s❡❧✲
❤❛♠♦s ❛ ✉t✐❧✐③❛r ♣❛rê♥t❡s❡s✱ ♥❡ss❡s ❝❛s♦s✳
❯t✐❧✐③❛♥❞♦ ♣❛rê♥t❡s❡s ❡ ❝♦♥❡❝t✐✈♦s✱ ❛s ❡①♣r❡ssõ❡s s✐♠❜ó❧✐❝❛s ♣♦❞❡♠ ❛ss✉♠✐r
❛s♣❡❝t♦s ❛✐♥❞❛ ♠❛✐s ❝♦♠♣❧❡①♦s✱ ❝♦♠♦✱ ♣♦r ❡①❡♠♣❧♦✿
(p↔ q ∨ (¬r → s)) ∧ ¬t
◗✉❛♥❞♦ ❛ ❡①♣r❡ssã♦ ♣♦ss✉✐r ♣❛rê♥t❡s❡s✱ ❞❡✈❡♠♦s ♣r✐♠❡✐r♦ ❡①❡❝✉t❛r ❛s ♦♣❡r✲
❛çõ❡s ❞♦s ♣❛rê♥t❡s❡s ♠❛✐s ✐♥t❡r♥♦s ❝♦♥t✐♥✉❛♥❞♦ ❛té ❝❤❡❣❛r ❛♦s ♠❛✐s ❡①t❡r♥♦s✳
❉❡st❛ ❢♦r♠❛✱ ❛s ♦♣❡r❛çõ❡s ❞❛ ❡①♣r❡ssã♦ ❛♥t❡r✐♦r s❡r✐❛♠ ❡①❡❝✉t❛❞❛s ♥❛ s❡❣✉✐♥t❡
♦r❞❡♠✿
(p q ( r s)) t↔ ∨ ¬ → ∧ ¬
4 3 1 2 6 5
✷✳✷ ❙❡♠â♥t✐❝❛
❆❧é♠ ❞❛s ♣r♦♣♦s✐çõ❡s✱ ❛ ▲ó❣✐❝❛ ❞✐s♣õ❡ ❞❡ ✉♠❛ ❢✉♥çã♦✱ ❝❤❛♠❛❞❛ ✈❛❧♦r ❧ó❣✐❝♦ ✭r❡♣✲
r❡s❡♥t❛❞❛ ♣♦r vl✮✱ q✉❡ ❛ss♦❝✐❛ ❛ ❝❛❞❛ ♣r♦♣♦s✐çã♦ s✐♠♣❧❡s ✉♠ ❞❡ ❞♦✐s ✈❛❧♦r❡s ❧ó❣✐❝♦s✱
✶✾❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❝❤❛♠❛❞♦s ✈❡r❞❛❞❡✐r♦ ✭r❡♣r❡s❡♥t❛❞♦ ♣♦r ❱✮ ♦✉ ❢❛❧s♦ ✭r❡♣r❡s❡♥t❛❞♦ ♣♦r ❋✮✳ ●❡r❛❧✲
♠❡♥t❡✱ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❱ ♦✉ ❋ é ❛ss♦❝✐❛❞♦ à ♣r♦♣♦s✐çã♦✱ ❡♠ ❝♦♥s♦♥â♥❝✐❛ ❝♦♠ ♦
s✐❣♥✐✜❝❛❞♦ ❞❛ ♣r♦♣♦s✐çã♦ ♥♦ ♠✉♥❞♦ r❡❛❧✱ ❡♠❜♦r❛ ✐ss♦ ♥ã♦ s❡❥❛ ❡ss❡♥❝✐❛❧✳
❈♦♠ ❡ss❡ s❡♥t✐❞♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s✿
❆ ▲✉❛ é ♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛✳
P❡❞r♦ ➪❧✈❛r❡s ❈❛❜r❛❧ ❞❡s❝♦❜r✐✉ ♦ ❇r❛s✐❧✳
sã♦ ✈❡r❞❛❞❡✐r❛s✱ ✐st♦ é ❛ss✉♠❡♠ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❱✱ ❡ q✉❡ ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s✿
❉❛♥t❡ ❡s❝r❡✈❡✉ ❖s ▲✉sí❛❞❛s✳
❖ ❇r❛s✐❧ é ✉♠❛ ♠♦♥❛rq✉✐❛✳
sã♦ ❝❧❛r❛♠❡♥t❡ ❢❛❧s❛s✱ ❡ ♣♦rt❛♥t♦ ❛ss✉♠❡♠ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❋✳
❖ ♦❜❥❡t✐✈♦ ❞❛ ▲ó❣✐❝❛✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ é ✈❡r✐✜❝❛r s❡ ❛s ♣r♦♣♦s✐çõ❡s sã♦ ✈❡r✲
❞❛❞❡✐r❛s ♦✉ ❢❛❧s❛s❀ ❛♦ ✐♥✈és ❞✐ss♦✱ ♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦ ❞❛ ▲ó❣✐❝❛ é ❡①❛♠✐♥❛r ♦
r❡❧❛❝✐♦♥❛♠❡♥t♦ ❡♥tr❡ ❛s ♣r♦♣♦s✐çõ❡s✱ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞♦s s❡✉s ✈❛❧♦r❡s ❧ó❣✐❝♦s✳ ❉✐t♦
❞❡ ♦✉tr❛ ❢♦r♠❛✱ ❛ ▲ó❣✐❝❛ ♥ã♦ s❡ ✐♥t❡r❡ss❛ ♣❡❧♦ s✐❣♥✐✜❝❛❞♦ ❞❛s ♣r♦♣♦s✐çõ❡s✱ ♠❛s
❛♣❡♥❛s ♣♦r s✉❛ ❢♦r♠❛❀ ♥♦ q✉❡ ❝♦♥❝❡r♥❡ à ▲ó❣✐❝❛✱ ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♦ ✏❆ ▲✉❛ é
♦ s❛té❧✐t❡ ❞❛ ❚❡rr❛✑ ♣♦❞❡ s❡r tr❛t❛❞❛ ❝♦♠♦ ✏❛ ♣r♦♣♦s✐çã♦ p✑✱ ♥ã♦ s❡♥❞♦ ♥❡❝❡ssár✐♦
♥❡♥❤✉♠❛ r❡❢❡rê♥❝✐❛ ❛ ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡ ❛str♦♥♦♠✐❛✳
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s Pr✐♥❝í♣✐♦s ❞❛ ▲ó❣✐❝❛✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡✿
❼ ❚♦❞❛ ♣r♦♣♦s✐çã♦ ✈❡r❞❛❞❡✐r❛ é s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛✱ ♥ã♦ ♣♦❞❡♥❞♦ s❡r ♦r❛ ✈❡r✲
❞❛❞❡✐r❛ ♦r❛ ❢❛❧s❛✳ ✭■❞❡♥t✐❞❛❞❡✮
❼ ◆❡♥❤✉♠❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r❛ ❡ ❢❛❧s❛ s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ✭◆ã♦✲
❝♦♥tr❛❞✐çã♦✮
❼ ❚♦❞❛ ♣r♦♣♦s✐çã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✱ ♥ã♦ ❡①✐st✐♥❞♦ ♦✉tr❛
♣♦ss✐❜✐❧✐❞❛❞❡✳ ✭❚❡r❝❡✐r♦ ❡①❝❧✉í❞♦✮
♣♦❞❡♠♦s ❡s❝r❡✈❡r✿ vl(p) = V✱ vl(q) = V✱ vl(r) = F✱ vl(s) = F
❈♦♠♦ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛✱ ❡♠
❢✉♥çã♦ ❞♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s q✉❡ ❛ ❝♦♠♣õ❡❄
P❛r❛ r❡s♣♦♥❞❡r ❛ ❡ss❛ ♣❡r❣✉♥t❛✱ t❡♠♦s q✉❡ ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s✱ ✐st♦ é✱ ❞❛r ♦
r❡s✉❧t❛❞♦ ❞❛ ♦♣❡r❛çã♦ ♣❛r❛ ❝❛❞❛ ♣♦ssí✈❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ❞♦s ♦♣❡r❛♥❞♦s✳
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✷✵
✷✳✷✳✶ ❖♣❡r❛çõ❡s ❧ó❣✐❝❛s
◆❡❣❛çã♦
❙❡ A é ✉♠❛ ♣r♦♣♦s✐çã♦✱ ❛ ❡①♣r❡ssã♦ ¬A é ❝❤❛♠❛❞❛ ♥❡❣❛çã♦ ❞❡ A✳ ❈❧❛r❛♠❡♥t❡✱ ❛
♥❡❣❛çã♦ ✐♥✈❡rt❡ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❡ ✉♠❛ ❡①♣r❡ssã♦❀ s❡ A ❢♦r ✈❡r❞❛❞❡✐r❛✱ A é ❢❛❧s❛✱¬
❡♥q✉❛♥t♦ q✉❡ s❡ A ❢♦r ❢❛❧s❛✱ A é ✈❡r❞❛❞❡✐r❛✳ ❆ t❛❜❡❧❛ ❞❛ ♦♣❡r❛çã♦ ❞❡ ♥❡❣❛çã♦ é¬
♠✉✐t♦ s✐♠♣❧❡s✱ ❡ é ❛♣r❡s❡♥t❛❞❛ ❛❜❛✐①♦✱ ♦♥❞❡ A é ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❛❧q✉❡r✳
A (¬A)
❱ ❋
❋ ❱
❊♥tã♦✱ s❡ ❛ ❡①♣r❡ssã♦ ✏▼❛r✐❛ ❢♦✐ ❛♦ ❝✐♥❡♠❛✑ é r❡♣r❡s❡♥t❛❞❛ ♣♦r p✱ ❡①♣r❡ssõ❡s
❝♦♠♦ ✏▼❛r✐❛ ♥ã♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛✑ ♦✉ ✏➱ ❢❛❧s♦ q✉❡ ▼❛r✐❛ t❡♥❤❛ ✐❞♦ ❛♦ ❝✐♥❡♠❛✑ ✜❝❛♠
r❡♣r❡s❡♥t❛❞❛s ♣♦r ( p)✳¬
❈♦♥❥✉♥çã♦
❙❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s✱ ❛ ❡①♣r❡ssã♦ (A ∧ B) é ❝❤❛♠❛❞❛ ❝♦♥❥✉♥çã♦ ❞❡ A ❡ B✱
❡ ❛s ♣r♦♣♦s✐çõ❡s A ❡ B sã♦ ❝❤❛♠❛❞❛s ❢❛t♦r❡s ❞❛ ❡①♣r❡ssã♦✳ ❙❡ ❝♦♥❤❡❝❡r♠♦s ♦s
✈❛❧♦r❡s ❧ó❣✐❝♦s ❞♦s ❢❛t♦r❡s ❞❡ ✉♠❛ ❝♦♥❥✉♥çã♦✱ ♦ q✉❡ ♣♦❞❡♠♦s ❞✐③❡r ❞♦ ✈❛❧♦r ❧ó❣✐❝♦
❞❛ ❝♦♥❥✉♥çã♦❄
❆ ❡①♣r❡ssã♦ ✏❏♦ã♦ é ♠❛❣r♦ ❡ ❏♦sé é ❛❧t♦✑ s❡rá ✈❡r❞❛❞❡✐r❛ ✉♥✐❝❛♠❡♥t❡ ♥♦ ❝❛s♦
❡♠ q✉❡ ♦s ❞♦✐s ❢❛t♦r❡s ❢♦r❡♠ ✈❡r❞❛❞❡✐r♦s✱ ✐st♦ é✱ s❡ ❏♦ã♦ ❢♦r ♠❛❣r♦ ❡ ❏♦sé ❢♦r ❛❧t♦❀
s❡ ✉♠ ❞♦s ❞♦✐s ❢❛t♦r❡s ✭♦✉ ❛♠❜♦s✮ ❢♦r ❢❛❧s♦✱ ❛ ❝♦♥❥✉♥çã♦ é ❢❛❧s❛✳
❖ ✈❛❧♦r ❧ó❣✐❝♦ ❞♦ r❡s✉❧t❛❞♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❥✉♥çã♦ ♣♦❞❡ s❡r ❛♣r❡s❡♥t❛❞♦
❛tr❛✈és ❞❛ t❛❜❡❧❛ ❛❜❛✐①♦✱ ♦♥❞❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s q✉❛✐sq✉❡r✿
A B (A ∧B)
❱ ❱ ❱
❋ ❱ ❋
❱ ❋ ❋
❋ ❋ ❋
✷✶❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❉✐s❥✉♥çã♦
➚s ✈❡③❡s✱ ❛ ❧í♥❣✉❛ ♣♦rt✉❣✉❡s❛ ❡♥❝❡rr❛ ❛❧❣✉♠❛ ❛♠❜✐❣✉✐❞❛❞❡ ♥♦ ✉s♦ ❞♦ ❝♦♥❡❝t✐✈♦
✏♦✉✑❀ ❛ ✉t✐❧✐③❛çã♦ ❞❡ ✏♦✉✑ ❡♥tr❡ ❞♦✐s ❢❛t♦s ✐♥❞✐❝❛ q✉❡ ✉♠ ❞❡❧❡s é ✈❡r❞❛❞❡✐r♦✱
♠❛s ♣♦❞❡ ♥ã♦ ❞❡✐①❛r ❝❧❛r♦ s❡ ❛♠❜♦s ♦ sã♦❀ ♥♦r♠❛❧♠❡♥t❡✱ ♥❛ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✱
♣r♦❝✉r❛✲s❡ r❡s♦❧✈❡r ❛ ❛♠❜✐❣✉✐❞❛❞❡ ✉t✐❧✐③❛♥❞♦✲s❡ ♦ ❝♦♥t❡①t♦✳ P♦r ❡①❡♠♣❧♦✱ ♥❛ ❢r❛s❡
✏▼❛r✐❛ ❢♦✐ à ♣r❛✐❛ ♦✉ ❛♦ ♠❡r❝❛❞♦✑ ♣❛r❡❝❡ q✉❡ ❛♣❡♥❛s ✉♠ ❞♦s ❢❛t♦s é ✈❡r❞❛❞❡✐r♦✱
♣♦✐s é ❞✐❢í❝✐❧ ❛❧❣✉é♠ ✐r à ♣r❛✐❛ ❡ ❛♦ ♠❡r❝❛❞♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡❀ ♥♦ ❡♥t❛♥t♦✱ s❡ ♥ã♦
❤♦✉✈❡r ❡①✐❣ê♥❝✐❛ s❡ s✐♠✉❧t❛♥❡✐❞❛❞❡✱ ♣♦❞❡ ♦❝♦rr❡r q✉❡ ▼❛r✐❛ t❡♥❤❛ ✐❞♦ à ♣r❛✐❛ ❡
❞❡♣♦✐s ❛♦ ♠❡r❝❛❞♦✱ ❡ ❛♠❜♦s ♦s ❢❛t♦s sã♦ ✈❡r❞❛❞❡✐r♦s✳
❖ ♦✉tr♦ ❡①❡♠♣❧♦✱ ✏▼ár✐♦ ❢♦✐ ❛♦ ❝✐♥❡♠❛ ♦✉ ▼❛r❝❡❧♦ ✜❝♦✉ ❡♠ ❝❛s❛✑ é ❛✐♥❞❛ ♣✐♦r✱
♣♦✐s ♥ã♦ ❤á ♥❡♥❤✉♠❛ ✐♥❞✐❝❛çã♦ s❡ ❛♣❡♥❛s ✉♠ ♦✉ ♦s ❞♦✐s ❢❛t♦s ♦❝♦rr❡r❛♠✳ ❈♦♠♦
♥❛ ▲ó❣✐❝❛ ♥ã♦ sã♦ ♣❡r♠✐t✐❞❛s ❛♠❜✐❣✉✐❞❛❞❡s✱ ❢♦✐ ♥❡❝❡ssár✐♦ ❞❡✜♥✐r ❞♦✐s ❝♦♥❡❝t✐✈♦s
♣❛r❛ ♦ t❡r♠♦ ✏♦✉✑✿
❼ ♦ ♦✉✲✐♥❝❧✉s✐✈♦✱ ♦♥❞❡ s❡ ♣❡r♠✐t❡ q✉❡ ✉♠ ❞♦s ❢❛t♦s ♦✉ ❛♠❜♦s ♦❝♦rr❛♠✱ ❡
❼ ♦ ♦✉✲❡①❝❧✉s✐✈♦ ♦♥❞❡ ✉♠ ❡ ❛♣❡♥❛s ✉♠ ❞♦s ❢❛t♦s ♦❝♦rr❡♠✳
❙❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s✱ ❛ ❡①♣r❡ssã♦ (A ∨ B) é ❝❤❛♠❛❞❛ ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛
❞❡ A ❡ B❀ ♣♦r s❡✉ t✉r♥♦✱ ❛ ❞✐s❥✉♥çã♦ ❡①❝❧✉s✐✈❛ ❞❛s ❡①♣r❡ssõ❡s A ❡ B é ✐♥❞✐❝❛❞❛
♣♦r (A ⊻ B)❀ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ❛s ♣r♦♣♦s✐çõ❡s A ❡ B sã♦ ❝❤❛♠❛❞❛s ♣❛r❝❡❧❛s ❞❛
❡①♣r❡ssã♦✳
❊♠ q✉❡ ❝♦♥❞✐çõ❡s ❛ ❡①♣r❡ssã♦ ✏▼❛r✐❛ ❢♦✐ à ♣r❛✐❛ ♦✉ ❛♦ ♠❡r❝❛❞♦✑ é ✈❡r❞❛❞❡✐r❛❄
◆♦ ❝♦♥❝❡✐t♦ ✐♥❝❧✉s✐✈♦ ❞♦ ❝♦♥❡❝t✐✈♦ ✏♦✉✑ ❜❛st❛ q✉❡ ▼❛r✐❛ t❡♥❤❛ ✐❞♦ ♣❡❧♦ ♠❡♥♦s ✉♠
❞♦s ❧✉❣❛r❡s❀ ♦✉ s❡❥❛✱ ♣❛r❛ q✉❡ ✉♠❛ ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛ s❡❥❛ ✈❡r❞❛❞❡✐r❛✱ ❜❛st❛ q✉❡
✉♠❛ ❞❛s ♣❛r❝❡❧❛s ✭♦✉ ❛♠❜❛s✮ ♦ s❡❥❛❀ ✉♥✐❝❛♠❡♥t❡ s❡ ❛♠❜❛s ❛s ♣❛r❝❡❧❛s ❢♦r❡♠ ❢❛❧s❛s✱
❛ ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛ ♦ s❡rá✳
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ s❡ tr❛t❛r ❞❡ ✉♠❛ ❞✐s❥✉♥çã♦ ❡①❝❧✉s✐✈❛✱ ❛ ❡①♣r❡ssã♦ só s❡rá
✈❡r❞❛❞❡✐r❛ s❡ ▼❛r✐❛ t✐✈❡r ✐❞♦ ❛ ✉♠ ❞♦s ❧✉❣❛r❡s✱ ♠❛s ♥ã♦ ❛♦ ♦✉tr♦✳ ❆ ❞✐s❥✉♥çã♦
❡①❝❧✉s✐✈❛ s❡rá ✈❡r❞❛❞❡✐r❛ s❡ ✉♠❛ ❞❛s ♣❛r❝❡❧❛s ❢♦r ✈❡r❞❛❞❡✐r❛ ❡ ❛
♦✉tr❛ ❢❛❧s❛❀ s❡
❛♠❜❛s ❛s ♣❛r❝❡❧❛s t✐✈❡r❡♠ ♦ ♠❡s♠♦ ✈❛❧♦r ❧ó❣✐❝♦✱ ❛ ❞✐s❥✉♥çã♦ ❡①❝❧✉s✐✈❛ s❡rá ❢❛❧s❛✳
❊♠ ♥♦ss♦ t❡①t♦✱ tr❛t❛r❡♠♦s ✉♥✐❝❛♠❡♥t❡ ❞❛ ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛❀ ✐st♦ é✱ ♦ t❡r♠♦
❞✐s❥✉♥çã♦ s❡ r❡❢❡r✐rá à ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛❀ q✉❛♥❞♦ s❡ tr❛t❛r ❞❛ ❞✐s❥✉♥çã♦ ❡①❝❧✉s✐✈❛✱
✐ss♦ s❡rá ❡①♣r❡ss❛♠❡♥t❡ ❝✐t❛❞♦✳
✷✷❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❆ t❛❜❡❧❛ ❛❜❛✐①♦ ❛♣r❡s❡♥t❛ ♦s r❡s✉❧t❛❞♦s ❞❛s ♦♣❡r❛çõ❡s ❞❡ ❞✐s❥✉♥çã♦ ✐♥❝❧✉s✐✈❛ ❡
❡①❝❧✉s✐✈❛✱ ♦♥❞❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s q✉❛✐sq✉❡r✳
A B (A ∨B) (A ⊻B)
❱ ❱ ❱ ❋
❋ ❱ ❱ ❱
❱ ❋ ❱ ❱
❋ ❋ ❋ ❋
◆❛ ❢♦r♠❛ t❡①t✉❛❧✱ às ✈❡③❡s ❡s❝r❡✈❡♠♦s ✉♠ ✏♦✉✑ ❛♥t❡s ❞❛ ❢r❛s❡✱ às ✈❡③❡s ♦♠✐t✐♠♦s
♣❛rt❡ ❞❛ ❡①♣r❡ssã♦✱ ♠❛s ✐ss♦ ♥ã♦ ❛❧t❡r❛ ❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛❀ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❡①♣r❡ssã♦
✏❖✉ ▼❛r✐❛ ❢♦✐ ❛♦ t❡❛tr♦ ♦✉ ❢♦✐ ❛♦ ❝✐♥❡♠❛✑ ✜❝❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r (p ∨ q)✱ ♦♥❞❡ p
r❡♣r❡s❡♥t❛ ✏▼❛r✐❛ ❢♦✐ ❛♦ t❡❛tr♦✑ ❡ q r❡♣r❡s❡♥t❛ ✏▼❛r✐❛ ❢♦✐ ❛♦ ❝✐♥❡♠❛✑ ✳
❈♦♥❞✐❝✐♦♥❛♠❡♥t♦
❙❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s✱ ❛ ❡①♣r❡ssã♦ (A B) é ❝❤❛♠❛❞❛ ❝♦♥❞✐❝✐♦♥❛❧ ♦✉ ✐♠♣❧✐✲→
❝❛çã♦ ❞❡ A ❡ B❀ ❛ ♣r♦♣♦s✐çã♦ A é ❝❤❛♠❛❞❛ ❛♥t❡❝❡❞❡♥t❡✱ ❡ ❛ ♣r♦♣♦s✐çã♦ B ❝♦♥s❡✲
q✉❡♥t❡ ❞❛ ❝♦♥❞✐❝✐♦♥❛❧✳ ❆ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ ✐♥❞✐❝❛ q✉❡ ♦ ❛❝♦♥t❡❝✐♠❡♥t♦
❞❡ A é ✉♠❛ ❝♦♥❞✐çã♦ ♣❛r❛ q✉❡ B ❛❝♦♥t❡ç❛✳ ❈♦♠♦ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ♦ ✈❛❧♦r
❧ó❣✐❝♦ ❞❛ ♣r♦♣♦s✐çã♦ ❝♦♥❞✐❝✐♦♥❛❞❛✱ ❝♦♥❤❡❝✐❞♦s ♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞♦ ❛♥t❡❝❡❞❡♥t❡ ❡
❞♦ ❝♦♥s❡q✉❡♥t❡❄
❈♦♥s✐❞❡r❡ ❛ ❡①♣r❡ssã♦ ✏❙❡ ❛ ❝❤✉✈❛ ❝♦♥t✐♥✉❛r ❛ ❝❛✐r ❡♥tã♦ ♦ r✐♦ ✈❛✐ tr❛♥s❜♦r❞❛r✑✳
❙✉♣♦♥❤❛ q✉❡ ❛♠❜❛s ❛s ❝♦✐s❛s ❛❝♦♥t❡ç❛♠✱ ✐st♦ é✱ q✉❡ ❛ ❝❤✉✈❛ t❡♥❤❛ ❝♦♥t✐♥✉❛❞♦ ❛
❝❛✐r✱ ❡ ♦ r✐♦ t❡♥❤❛ tr❛♥s❜♦r❞❛❞♦❀ ♥❡ss❡ ❝❛s♦✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ✈❡r❞❛❞❡✐r❛✳ ❙✉♣♦♥❤❛✱
♣♦r ♦✉tr♦ ❧❛❞♦✱ q✉❡ ❛ ❝❤✉✈❛ t❡♥❤❛ ❝♦♥t✐♥✉❛❞♦ ❛ ❝❛✐r✱ ♠❛s q✉❡ ♦ r✐♦ ♥ã♦ t❡♥❤❛
tr❛♥s❜♦r❞❛❞♦❀ ♥❡ss❡ ❝❛s♦✱ A ♥ã♦ ❢♦✐ ❝♦♥❞✐çã♦ ♣❛r❛ B✱ ✐st♦ é✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ❢❛❧s❛✳
❋✐♥❛❧♠❡♥t❡✱ ❝♦♥s✐❞❡r❡ q✉❡ ❛ ❝❤✉✈❛ ♥ã♦ t❡♥❤❛ ❝♦♥t✐♥✉❛❞♦ ❛ ❝❛✐r❀ ♥❡ss❡ ❝❛s♦✱ ✐♥✲
❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ q✉❡ t❡♥❤❛ ❛❝♦♥t❡❝✐❞♦ ❝♦♠ ♦ r✐♦✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ❝♦♥s✐❞❡r❛❞❛
✈❡r❞❛❞❡✐r❛✳
P♦r q✉❡ ❡ss❡ ❢❛t♦ ♦❝♦rr❡❄ P♦r q✉❡ ♠♦t✐✈♦✱ ❛ ▲ó❣✐❝❛ ❝♦♥s✐❞❡r❛ q✉❡ s❡ ♦ ❛♥✲
t❡❝❡❞❡♥t❡ ❢♦r ❢❛❧s♦✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ✈❡r❞❛❞❡✐r❛✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞♦
❝♦♥s❡q✉❡♥t❡❄
✷✸❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❊①✐st❡♠ ✈ár✐♦s ♠♦t✐✈♦s ♣❛r❛ ✐ss♦✱ ❡ ✈❛♠♦s ❛q✉✐ ❛♣r❡s❡♥t❛r ♦ ♠❛✐s s✐♠♣❧❡s✳
◗✉❛♥❞♦ ♦ ❛♥t❡❝❡❞❡♥t❡ ❢♦r ❢❛❧s♦✱ t❡♠♦s q✉❛tr♦ ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛
❝♦♥❞✐❝✐♦♥❛❧✿
P♦ss✐❜✐❧✐❞❛❞❡s
❛♥t❡❝❡❞❡♥t❡ ❝♦♥s❡q✉❡♥t❡ ✶➟ ✷➟ ✸➟ ✹➟
❋ ❱ ❱ ❱ ❋ ❋
❋ ❋ ❱ ❋ ❱ ❋
❙❡ ❛ ▲ó❣✐❝❛ ❛❞♦t❛ss❡ ❛ s❡❣✉♥❞❛ ♣♦ss✐❜✐❧✐❞❛❞❡✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ ❛ss✉♠✐r✐❛ ♦s ♠❡s✲
♠♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞♦ ❝♦♥s❡qü❡♥t❡✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦ ❛♥t❡❝❡❞❡♥t❡✱ ♦ q✉❡ ♥ã♦
♣❛r❡❝❡ r❛③♦á✈❡❧❀ s❡ ❛ss✉♠✐ss❡ ❛ t❡r❝❡✐r❛✱ ♦ ❛♥t❡❝❡❞❡♥t❡ ❡ ♦ ❝♦♥s❡qü❡♥t❡ ♣♦❞❡r✐❛♠
s❡r ♣❡r♠✉t❛❞♦s✱ s❡♠ ♠♦❞✐✜❝❛r ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛ ❝♦♥❞✐❝✐♦♥❛❧✱ ♦ q✉❡ t❛♠❜é♠ ♥ã♦
♣❛r❡❝❡ s❡r r❛③♦á✈❡❧ ✭s❡ ♦ r✐♦ tr❛♥s❜♦r❞❛r✱ ❛ ❝❤✉✈❛ ✈❛✐ ❝♦♥t✐♥✉❛r ❝❛✐♥❞♦✮✳
❋✐♥❛❧♠❡♥t❡✱ s❡ ❛ q✉❛rt❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❢♦ss❡ ❛❞♦t❛❞❛✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ ♥ã♦ s❡ ❞✐s✲
t✐♥❣✉✐r✐❛ ❞❛ ❝♦♥❥✉♥çã♦❀ r❡st❛ ❡♥tã♦ ❛ ♣r✐♠❡✐r❛ ♣♦ss✐❜✐❧✐❞❛❞❡✱ q✉❡ é ❛ ❛❞♦t❛❞❛ ♣❡❧❛
▲ó❣✐❝❛✳
❈♦♠♦ ❥á ❞✐ss❡♠♦s✱ ❛ ▲ó❣✐❝❛ ♥ã♦ s❡ ♣r❡♦❝✉♣❛ ❝♦♠ ♦s s✐❣♥✐✜❝❛❞♦s ❞❛s ❡①♣r❡ssõ❡s✱
♠❛s s♦♠❡♥t❡ ❝♦♠ s✉❛ ❢♦r♠❛✳ P♦❞❡♠♦s ❡♥tã♦ ❝♦♥s✐❞❡r❛r q✉❡ ❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐✲
❝✐♦♥❛❧ ❢♦✐ ❞❡✜♥✐❞❛ ❞❛ ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ❢♦✐ ❛♣r❡s❡♥t❛❞❛✱ s❡♠ ❛ ♣r❡♦❝✉♣❛çã♦ ❞❡ q✉❡
♦ ❛♥t❡❝❡❞❡♥t❡ s❡❥❛ ✏❝❛✉s❛✑ ❞♦ ❝♦♥s❡q✉❡♥t❡✳ ◆❡ss❡ s❡♥t✐❞♦ ❛s ❝♦♥❞✐❝✐♦♥❛✐s✿
❙❡ ❉❛♥t❡ ❡s❝r❡✈❡✉ ❖s ❙❡rtõ❡s ❡♥tã♦ ❈❛❜r❛❧ ❞❡s❝♦❜r✐✉ ♦ ❇r❛s✐❧
❙❡ ❉❛♥t❡ ❡s❝r❡✈❡✉ ❉♦♠ ❈❛s♠✉rr♦ ❡♥tã♦ ❱❛s❝♦ ❞❛ ●❛♠❛ ❞❡s❝♦❜r✐✉ ♦
❇r❛s✐❧
sã♦ ❛♠❜❛s ✈❡r❞❛❞❡✐r❛s✱ ♣♦✐s ❡♠ ❛♠❜❛s ♦ ❛♥t❡❝❡❞❡♥t❡ é ❢❛❧s♦✳
❆ t❛❜❡❧❛ q✉❡ ✐♥❞✐❝❛ ♦ r❡s✉❧t❛❞♦ ❞❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ é ❛♣r❡s❡♥t❛❞❛
❛❜❛✐①♦✱ ♦♥❞❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s q✉❛✐sq✉❡r✳
A B (A→ B)
❱ ❱ ❱
❋ ❱ ❱
❱ ❋ ❋
❋ ❋ ❱
✷✹❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❇✐❝♦♥❞✐❝✐♦♥❛♠❡♥t♦
❙❡ A ❡ B sã♦ ♣r♦♣♦s✐çõ❡s✱ ❛ ❡①♣r❡ssã♦ (A B) é ❝❤❛♠❛❞❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧ ♦✉↔
❜✐✐♠♣❧✐❝❛çã♦ ❞❡ A ❡ B✳ ❉✐③❡♠♦s q✉❡ ❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧ é ✈❡r❞❛❞❡✐r❛ q✉❛♥❞♦ ❛♠❜♦s ♦s
t❡r♠♦s sã♦ ✈❡r❞❛❞❡✐r♦s ♦✉ ❛♠❜♦s sã♦ ❢❛❧s♦s❀ q✉❛♥❞♦ ✉♠ é ❢❛❧s♦ ❡ ♦✉tr♦ é ✈❡r❞❛❞❡✐r♦✱
❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧ é ❢❛❧s❛✳
◆❛ ❡①♣r❡ssã♦ ✏❏♦ã♦ s❡rá ❛♣r♦✈❛❞♦ s❡ ❡ s♦♠❡♥t❡ s❡ ❡❧❡ ❡st✉❞❛r✑✱ ♦ ❝♦♥❡❝t✐✈♦
✏s❡ ❡ s♦♠❡♥t❡ s❡✑ ✐♥❞✐❝❛ q✉❡ s❡ ❏♦ã♦ ❡st✉❞❛r s❡rá ❛♣r♦✈❛❞♦✱ ❡ q✉❡ ❡ss❛ é ❛ ú♥✐❝❛
♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ❏♦ã♦ s❡r ❛♣r♦✈❛❞♦✱ ✐st♦ é✱ s❡ ❏♦ã♦ ♥ã♦ ❡st✉❞❛r✱ ♥ã♦ s❡rá ❛♣r♦✈❛❞♦✳
❖s ❞♦✐s ❛❝♦♥t❡❝✐♠❡♥t♦s s❡rã♦ ❛♠❜♦s ✈❡r❞❛❞❡✐r♦s ♦✉ ❛♠❜♦s ❢❛❧s♦s✱ ♥ã♦ ❡①✐st✐♥❞♦
♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ t❡r❝❡✐r❛ ♦♣çã♦✳
❆ t❛❜❡❧❛ ❞♦ ❜✐❝♦♥❞✐❝✐♦♥❛♠❡♥t♦ é ❛♣r❡s❡♥t❛❞❛ ❛❜❛✐①♦✱ ♦♥❞❡A ❡B sã♦ ♣r♦♣♦s✐çõ❡s
q✉❛✐sq✉❡r✳
A B (A↔ B)
❱ ❱ ❱
❋ ❱ ❋
❱ ❋ ❋
❋ ❋ ❱
✷✳✷✳✷ ❚❛❜❡❧❛✲✈❡r❞❛❞❡
P❛r❛ s✐♠♣❧✐✜❝❛r ❛ ❞❡t❡r♠✐♥❛çã♦ ❞♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❡ ✉♠❛ ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧✱
♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠❛ ♣❡q✉❡♥❛ t❛❜❡❧❛✱ ♥❛ q✉❛❧ ❞✐s♣♦♠♦s ❡♠ ❝♦❧✉♥❛s ♦s ✈❛❧♦r❡s
❧ó❣✐❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦♥❡♥t❡s✱ ❡✱ ❛ s❡❣✉✐r✱ ♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞❛s ♦♣❡r❛çõ❡s✱
♥❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛
✶
✳ P♦r ❡①❡♠♣❧♦✱ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❛ ❡①♣r❡ssã♦
p ∧ ¬q → r ∨ s✱ ♥❛ q✉❛❧ p ❡ r sã♦ ❢❛❧s❛s✱ ❡ q ❡ s ✈❡r❞❛❞❡✐r❛s✿
p q r s ¬q p ∧ ¬q r ∨ s p ∧ ¬q → r ∨ s
❋ ❱ ❋ ❱ ❋ ❋ ❱ ❱
▼✉✐t❛s ✈❡③❡s✱ ♥♦ ❡♥t❛♥t♦✱ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ❧ó❣✐✲
❝♦s q✉❡ ❛ ❡①♣r❡ssã♦ ♣♦❞❡ ❛ss✉♠✐r✱ ♣❛r❛ q✉❛✐sq✉❡r ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s
s✐♠♣❧❡s ❝♦♠♣♦♥❡♥t❡s✳
✶
P❛r❛ q✉❡ ❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛ ✜q✉❡ ✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞❛✱ ♣♦❞❡♠♦s ❝♦♥✈❡♥❝✐♦♥❛r q✉❡
❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♣♦♥❡♥t❡s ✜q✉❡♠ ❞✐s♣♦st❛s ❡♠ ♦r❞❡♠ ❛❧❢❛❜ét✐❝❛✳
✷✺❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❱❡❥❛♠♦s ✉♠ ❡①❡♠♣❧♦✳ ❈♦♥s✐❞❡r❡ ❛ ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧✿
p ∨ q → p ∧ q
◆♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ❝♦♥str✉í♠♦s ✉♠❛ ♣❡q✉❡♥❛ t❛❜❡❧❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦ ✈❛❧♦r
❧ó❣✐❝♦ ❞❛ ❡①♣r❡ssã♦✱ ❛ ♣❛rt✐r ❞♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞♦s ❝♦♠♣♦♥❡♥t❡s❀ ❛❣♦r❛✱ ✈❛♠♦s
❛♠♣❧✐❛r ❛q✉❡❧❛ t❛❜❡❧❛✱ ♣❛r❛ ✐♥❝❧✉✐r ❝❛❞❛ ❝♦♠❜✐♥❛çã♦ ❞♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞♦s ❝♦♠✲
♣♦♥❡♥t❡s✳
◆❛ ❡①♣r❡ssã♦✱ ❡①✐st❡♠ ❛♣❡♥❛s ❞✉❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♣♦♥❡♥t❡s✱ p ❡ q❀
❝♦♠♦ ❝❛❞❛ ✉♠❛ ♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛✱ t❡♠♦s ❡♥tã♦ q✉❛tr♦ ♣♦ss✐❜✐❧✐❞❛❞❡s✿ p
❡ q ❛♠❜❛s ✈❡r❞❛❞❡✐r❛s✱ p ✈❡r❞❛❞❡✐r❛ ❡ q ❢❛❧s❛✱ p ❢❛❧s❛ ❡ q ✈❡r❞❛❞❡✐r❛✱ ♦✉✱ ✜♥❛❧♠❡♥t❡✱
p ❡ q ❛♠❜❛s ❢❛❧s❛s✳
❚❡♥❞♦ ♦❜t✐❞♦ t❛♠❜é♠ ❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛ ❞❛s ♦♣❡r❛çõ❡s✱ ♥♦ss❛ t❛❜❡❧❛
❛ss✉♠❡ ❛ ❢♦r♠❛✿
p q p ∨ q p ∧ q p ∨ q → p ∧ q
❱ ❱ ❱ ❱ ❱
❋ ❱ ❱ ❋ ❋
❱ ❋ ❱ ❋ ❋
❋ ❋ ❋ ❋ ❱
❯♠❛ t❛❜❡❧❛ ❝♦♠♦ ❡ss❛✱ ♥❛ q✉❛❧ sã♦ ❛♣r❡s❡♥t❛❞♦s t♦❞♦s ♦s ✈❛❧♦r❡s ✈❡r❞❛❞❡
♣♦ssí✈❡✐s ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛✱ ♣❛r❛ ❝❛❞❛ ❝♦♠❜✐♥❛çã♦ ❞♦s ✈❛❧♦r❡s ✈❡r❞❛❞❡
❞❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦♥❡♥t❡s✱ é ❝❤❛♠❛❞❛ t❛❜❡❧❛✲✈❡r❞❛❞❡ ❞❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛✳
❈❛❞❛ ❧✐♥❤❛ ❞❛ t❛❜❡❧❛ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠❛ ♣♦ssí✈❡❧ ❝♦♠❜✐♥❛çã♦ ❞♦s ✈❛❧♦r❡s ❧ó❣✐✲
❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦♥❡♥t❡s❀ ❝♦♠♦ sã♦ ❞♦✐s ♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s✱ ❡①✐st❡♠✱ ♣❛r❛
n ❝♦♠♣♦♥❡♥t❡s✱ 2n ❝♦♠❜✐♥❛çõ❡s ♣♦ssí✈❡✐s✳ P♦rt❛♥t♦✱ ❛ t❛❜❡❧❛✲✈❡r❞❛❞❡ ❞❡ ✉♠❛
❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧ t❡♠ 2n ❧✐♥❤❛s✱ ❛❧❡♠ ❞♦ ❝❛❜❡ç❛❧❤♦✳
❆ ❛tr✐❜✉✐çã♦ ❞❡ ✈❛❧♦r❡s ❧ó❣✐❝♦s ❛♦s ❝♦♠♣♦♥❡♥t❡s s✐♠♣❧❡s ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦
❝♦♠♣♦st❛ é ❝❤❛♠❛❞❛ ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❞❡ss❛ ♣r♦♣♦s✐çã♦✳ ❆ss✐♠✱ ✉♠❛ ♣r♦♣♦s✐çã♦
❝♦♠ n ❝♦♠♣♦♥❡♥t❡s s✐♠♣❧❡s ❞✐st✐♥t♦s ❛❞♠✐t✐rá 2n ✐♥t❡r♣r❡t❛çõ❡s✳
❖❜s❡r✈❡ q✉❡ ❛ t❛❜❡❧❛✲✈❡r❞❛❞❡ ♣♦ss✉✐ ❞♦✐s t✐♣♦s ❞❡ ❝♦❧✉♥❛s✿ ❝♦❧✉♥❛s ♣❛r❛ ❛s
♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♣♦♥❡♥t❡s ✭♦♥❞❡ sã♦ ❞✐str✐❜✉í❞♦s ♦s ✈❛❧♦r❡s ❱ ❡ ❋ ❞❡ ❢♦r♠❛
❛ ✐♥❝❧✉✐r ❝❛❞❛ ♣♦ssí✈❡❧ ❝♦♠❜✐♥❛çã♦✮ ❡ ❝♦❧✉♥❛s ♣❛r❛ ❛s ♦♣❡r❛çõ❡s ✭♦♥❞❡ ♦s ✈❛❧♦r❡s
✷✻❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❱ ❡ ❋ sã♦ ♦❜t✐❞♦s ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❛s ♦♣❡r❛çõ❡s✮❀ ❛ss✐♠✱ s❡ ❛ ❡①♣r❡ssã♦ ♣♦ss✉✐ n
❝♦♠♣♦♥❡♥t❡s ❡ m ♦♣❡r❛çõ❡s✱ ❛ t❛❜❡❧❛ t❡rá m+ n ❝♦❧✉♥❛s✳
P❛r❛ ❞❡t❡r♠✐♥❛r ✉♥✐❝❛♠❡♥t❡ ❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ❡st❛❜❡❧❡❝❡r ❝❡rt❛s ❝♦♥✲
✈❡♥çõ❡s ♣❛r❛ s✉❛ ❝♦♥str✉çã♦✿
✶✳ P❛r❛ ❛s ❝♦❧✉♥❛s✿
✶✳✶ ❉✐s♣♦r ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❝♦♠♣♦♥❡♥t❡s ❡♠ ♦r❞❡♠ ❛❧❢❛❜ét✐❝❛✳
✶✳✷ ❉✐s♣♦r ❛s ♦♣❡r❛çõ❡s ♥❛ ♦r❞❡♠ ❞❡ ♣r❡❝❡❞ê♥❝✐❛ ❞❡t❡r♠✐♥❛❞❛✳
✷✳ P❛r❛ ❛s ❧✐♥❤❛s✿
✷✳✶ ❆❧t❡r♥❛r ❱ ❡ ❋ ♣❛r❛ ❛ ❝♦❧✉♥❛ ❞❛ ♣r✐♠❡✐r❛ ❝♦♠♣♦♥❡♥t❡ s✐♠♣❧❡s✳
✷✳✷ ❆❧t❡r♥❛r ❱ ❱ ❡ ❋ ❋ ♣❛r❛ ❛ ❝♦❧✉♥❛ ❞❛ s❡❣✉♥❞❛ ❝♦♠♣♦♥❡♥t❡ s✐♠♣❧❡s✳
✷✳✸ ❆❧t❡r♥❛r ❱ ❱ ❱ ❱ ❡ ❋ ❋ ❋ ❋ ♣❛r❛ ❛ ❝♦❧✉♥❛ ❞❛ t❡r❝❡✐r❛ ❝♦♠♣♦♥❡♥t❡
s✐♠♣❧❡s✳
✷✳✹ Pr♦ss❡❣✉✐r ❞❡ss❛ ❢♦r♠❛✱ s❡ ❤♦✉✈❡r ♠❛✐s ❝♦♠♣♦♥❡♥t❡s s✐♠♣❧❡s✱ s❡♠♣r❡
❞♦❜r❛♥❞♦ ♦ ♥✉♠❡r♦ ❞❡ ❱✬s ❡ ❋✬s ♣❛r❛ ❝❛❞❛ ❝♦❧✉♥❛
à ❞✐r❡✐t❛✳
P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ❝♦♥s✐❞❡r❡ ❛ ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧✿
(p→ q) ∨ ¬((p↔ r)→ ¬r)
❆ ♣r❡❝❡❞ê♥❝✐❛ ❞❛s ♦♣❡r❛çõ❡s é ❞❛❞❛ ♣♦r✿
(p q) ((p r) r)→ ∨ ¬ ↔ → ¬
1 6 5 2 4 3
❆ t❛❜❡❧❛✲✈❡r❞❛❞❡ ❛ss✉♠❡ ♦ ❛s♣❡❝t♦✿
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✷✼
p q r p→ q p↔ r ¬r (p↔ r)→ ¬r ¬((p↔ r)→ ¬r) (p→ q) ∨ ¬((p↔ r)→ ¬r)
❱ ❱ ❱ ❱ ❱ ❋ ❋ ❱ ❱
❋ ❱ ❱ ❱ ❋ ❋ ❱ ❋ ❱
❱ ❋ ❱ ❋ ❱ ❋ ❋ ❱ ❱
❋ ❋ ❱ ❱ ❋ ❋ ❱ ❋ ❱
❱ ❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱
❋ ❱ ❋ ❱ ❱ ❱ ❱ ❋ ❱
❱ ❋ ❋ ❋ ❋ ❱ ❱ ❋ ❋
❋ ❋ ❋ ❱ ❱ ❱ ❱ ❋ ❱
❚❛✉t♦❧♦❣✐❛✱ ❈♦♥tr❛❞✐çã♦ ❡ ❈♦♥t✐♥❣ê♥❝✐❛
❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s q✉❡ ❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦st❛s ❛ss✉♠❡♠✱ ❡♠ s✉❛s
♣♦ssí✈❡✐s ✐♥t❡r♣r❡t❛çõ❡s✱ ❡❧❛s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ♥♦s s❡❣✉✐♥t❡s t✐♣♦s✿
✶✳ ❙❡ ❛ ❡①♣r❡ssã♦ ❛ss✉♠❡ s❡♠♣r❡ ♦ ✈❛❧♦r ❱✱ ❡♠ q✉❛❧q✉❡r ✐♥t❡r♣r❡t❛çã♦✱ é ❝❤❛♠❛❞❛
✉♠❛ t❛✉t♦❧♦❣✐❛✱ ♦✉ ✉♠❛ ❡①♣r❡ssã♦ ✈á❧✐❞❛ ❀ ♣♦r ❡①❡♠♣❧♦✿ (p ∨ ¬p)
✶✳✶ ❙❡ ❛ ❡①♣r❡ssã♦ ❛ss✉♠❡ ♦ ✈❛❧♦r ❱ ❡♠ ❛❧❣✉♠❛ ✐♥t❡r♣r❡t❛çã♦✱ é ❞✐t❛ s❛t✲
✐s❢❛tí✈❡❧✱ ♦✉ ❝♦♥s✐st❡♥t❡ ❀ ❡✈✐❞❡♥t❡♠❡♥t❡✱ ❛s t❛✉t♦❧♦❣✐❛s sã♦ ❡①❡♠♣❧♦s ❞❡
❡①♣r❡ssõ❡s s❛t✐s❢❛tí✈❡✐s✳
✷✳ ❙❡ ❛ ❡①♣r❡ssã♦ ❛ss✉♠❡ s❡♠♣r❡ ♦ ✈❛❧♦r ❋✱ ❡♠ q✉❛❧q✉❡r ✐♥t❡r♣r❡t❛çã♦✱ é ❝❤❛♠❛❞❛
✉♠❛ ❝♦♥tr❛❞✐çã♦✱ ♦✉ ✉♠❛ ❡①♣r❡ssã♦ ✐♥s❛t✐s❢❛tí✈❡❧✱ ♦✉ ✐♥❝♦♥s✐st❡♥t❡ ❀ ♣♦r ❡①✲
❡♠♣❧♦✿ (p ∧ ¬p)
✷✳✶ ❙❡ ❛ ❡①♣r❡ssã♦ ❛ss✉♠❡ ♦ ✈❛❧♦r ❋ ❡♠ ❛❧❣✉♠❛ ✐♥t❡r♣r❡t❛çã♦✱ é ❝❤❛♠❛❞❛
✉♠❛ ❡①♣r❡ssã♦ ✐♥✈á❧✐❞❛ ❀ ❝❧❛r❛♠❡♥t❡✱ ❛s ❝♦♥tr❛❞✐çõ❡s sã♦✱ t❛♠❜é♠✱ ❡①✲
♣r❡ssõ❡s ✐♥✈á❧✐❞❛s✳
✸✳ ❙❡ ❛ ❡①♣r❡ssã♦ é s❛t✐s❢❛tí✈❡❧ ❡ ✐♥✈á❧✐❞❛✱ ♦✉ s❡❥❛✱ ❛ss✉♠❡ ♦ ✈❛❧♦r ❱ ❡♠ ♣❡❧♦
♠❡♥♦s ✉♠❛ ✐♥t❡r♣r❡t❛çã♦ ❡ ♦ ✈❛❧♦r ❋ ❡♠ ♦✉tr❛ ✐♥t❡r♣r❡t❛çã♦✱ é ❝❤❛♠❛❞❛ ❞❡
❝♦♥t✐♥❣ê♥❝✐❛ ❀ ♣♦r ❡①❡♠♣❧♦✿ (p p)→ ¬
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✷✽
✷✳✸ ❊q✉✐✈❛❧ê♥❝✐❛ ▲ó❣✐❝❛
❯♠❛ ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧ ❞❛ ❢♦r♠❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧ A B q✉❡ é✱ t❛♠❜é♠✱ ✉♠❛↔
t❛✉t♦❧♦❣✐❛✱ é ❝❤❛♠❛❞❛ ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛ ✭♦✉ ❡q✉✐✈❛❧ê♥❝✐❛ ❧ó❣✐❝❛✮✳ ❆s ♣r♦♣♦s✐çõ❡s
A ❡ B sã♦ ❞✐t❛s ❡q✉✐✈❛❧❡♥t❡s✱ ❡ ❡s❝r❡✈❡♠♦s A ≡ B✳
P♦r ❡①❡♠♣❧♦✱ ❛ ❡①♣r❡ssã♦ (p → q) ↔ (¬p ∨ q) é ✉♠❛ ❡q✉✐✈❛❧ê♥❝✐❛✳ ❱❡❥❛ s✉❛
t❛❜❡❧❛✲✈❡r❞❛❞❡✿
p q ¬p p→ q ¬p ∨ q (p→ q)↔ (¬p ∨ q)
❱ ❱ ❋ ❱ ❱ ❱
❋ ❱ ❱ ❱ ❱ ❱
❱ ❋ ❋ ❋ ❋ ❱
❋ ❋ ❱ ❱ ❱ ❱
❊s❝r❡✈❡♠♦s✱ ❡♥tã♦✱ (p→ q) ≡ (¬p ∨ q)
❉❡❝♦rr❡ ✐♠❡❞✐❛t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡✱ s❡ ❞✉❛s ♣r♦♣♦s✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱
❡♥tã♦ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✱ ❡✱ r❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ ❞✉❛s ♣r♦♣♦s✐çõ❡s
tê♠ ❛ ♠❡s♠❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❉❡ ❢❛t♦✱ ✉♠❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧ é ❱ s❡
❡ s♦♠❡♥t❡ s❡ s❡✉s ❝♦♠♣♦♥❡♥t❡s tê♠ ♦s ♠❡s♠♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s❀ ❝♦♠♦ ❛ ❡①♣r❡ssã♦
t❛♠❜é♠ é ✉♠❛ t❛✉t♦❧♦❣✐❛✱ é ❱ ❡♠ t♦❞♦s ♦s ❝❛s♦s❀ ✐st♦ é✱ s❡✉s ❝♦♠♣♦♥❡♥t❡s tê♠ ♦
♠❡s♠♦ ✈❛❧♦r ❧ó❣✐❝♦ ❡♠ t♦❞♦s ♦s ❝❛s♦s✱ ♦✉ s❡❥❛✱ tê♠ ❛ ♠❡s♠❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✳
❉❡❝♦rr❡ ❛✐♥❞❛ ❞❛ ❞❡✜♥✐çã♦ q✉❡ t♦❞❛s ❛s t❛✉t♦❧♦❣✐❛s✱ ❜❡♠ ❝♦♠♦ t♦❞❛s ❛s ❝♦♥✲
tr❛❞✐çõ❡s✱ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❡♥tr❡ s✐✳
P♦❞❡♠♦s ♠♦str❛r t❛♠❜é♠ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♣♦ss✉✐ ❛s ♣r♦♣r✐❡❞❛❞❡s✿
❼ ❘❡✢❡①✐✈❛✿ A ≡ A
❼ ❙✐♠étr✐❝❛✿ ❙❡ A ≡ B ❡♥tã♦ B ≡ A
❼ ❚r❛♥s✐t✐✈❛✿ ❙❡ A ≡ B ❡ B ≡ C ❡♥tã♦ A ≡ C
❊♠ t❡r♠♦s t❡①t✉❛✐s✱ ❞✉❛s ♣r♦♣♦s✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s q✉❛♥❞♦ tr❛❞✉③❡♠ ❛ ♠❡s♠❛
✐❞é✐❛✱ ❞✐❢❡r✐♥❞♦ ❛♣❡♥❛s ❛ ❢♦r♠❛ ❞❡ ❛♣r❡s❡♥t❛r ❡ss❛ ✐❞é✐❛✳
◆♦ ❆♣ê♥❞✐❝❡ ❆ sã♦ ❧✐st❛❞❛s ❛❧❣✉♠❛s ❞❛s ❡q✉✐✈❛❧ê♥❝✐❛ ♠❛✐s ✐♠♣♦rt❛♥t❡s ✭❡ út❡✐s✮
❞❛ ▲ó❣✐❝❛❀ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ♣♦❞❡ s❡r ♣r♦✈❛❞❛✱ s✐♠♣❧❡s♠❡♥t❡ ♠♦str❛♥❞♦ q✉❡ ❛ ❜✐✲
❝♦♥❞✐❝✐♦♥❛❧ ❝♦rr❡s♣♦♥❞❡♥t❡ é ✉♠❛ t❛✉t♦❧♦❣✐❛✱ ❜❛st❛♥❞♦✱ ♣❛r❛ ✐ss♦✱ ❝♦♥str✉✐r s✉❛
t❛❜❡❧❛✲✈❡r❞❛❞❡✳
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✷✾
❖ ❝♦♥❝❡✐t♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ♥♦s ♣❡r♠✐t❡ ♠♦str❛r ❛✐♥❞❛ q✉❡ sã♦ s✉✜❝✐❡♥t❡s ❛s
♦♣❡r❛çõ❡s ❞❡ ♥❡❣❛çã♦ ❡ ✉♠❛ ❞❛s ❞✉❛s✱ ❝♦♥❥✉♥çã♦ ♦✉ ❞✐s❥✉♥çã♦✱ ♣❛r❛ r❡♣r❡s❡♥t❛r
q✉❛❧q✉❡r ❡①♣r❡ssã♦ ♣r♦♣♦s✐❝✐♦♥❛❧✳ P❛r❛ ✐ss♦✱ ♥❡❝❡ss✐t❛♠♦s ❞❛s s❡❣✉✐♥t❡s ❡q✉✐✈❛❧ê♥✲
❝✐❛s✿
❼ ❊❧✐♠✐♥❛♥❞♦ ❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧✿ A↔ B ≡ (A ∧B) ∨ (¬A ∧ ¬B)
❼ ❊❧✐♠✐♥❛♥❞♦ ❛ ❝♦♥❞✐❝✐♦♥❛❧✿ A→ B ≡ ¬A ∨ B
❼ ❊s❝r❡✈❡♥❞♦ ❛ ❞✐s❥✉♥çã♦ ❡♠ t❡r♠♦s ❞❡ ❝♦♥❥✉♥çã♦✿ A ∨B ≡ ¬(¬A ∧ ¬B)
❼ ❊s❝r❡✈❡♥❞♦ ❛ ❝♦♥❥✉♥çã♦ ❡♠ t❡r♠♦s ❞❡ ❞✐s❥✉♥çã♦✿ A ∧B ≡ ¬(¬A ∨ ¬B)
❱❡❥❛ ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿ ❡s❝r❡✈❡r ❛ ♣r♦♣♦s✐çã♦ (p q) p ❡♠ t❡r♠♦s ❞❡↔ → ¬
♥❡❣❛çã♦ ❡ ❞✐s❥✉♥çã♦✿
❼ ❊❧✐♠✐♥❛♥❞♦ ❛ ❝♦♥❞✐❝✐♦♥❛❧✿ ¬(p↔ q) ∨ ¬p
❼ ❊❧✐♠✐♥❛♥❞♦ ❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧✿ ¬((p ∧ q) ∨ (¬p ∧ ¬q)) ∨ ¬p
❼ ❊s❝r❡✈❡♥❞♦ ❛ ❝♦♥❥✉♥çã♦ ❡♠ t❡r♠♦s ❞❡ ❞✐s❥✉♥çã♦✿ ¬(¬(¬p∨¬q)∨¬(p∨q))∨¬p
✷✳✹ ❋♦r♠❛s ♥♦r♠❛✐s
❯♠❛ ♣r♦♣♦s✐çã♦ ❡stá ♥❛ ❢♦r♠❛ ♥♦r♠❛❧ s❡ ❡ s♦♠❡♥t❡ s❡ ❝♦♥t❡♠✱ q✉❛♥❞♦ ♠✉✐t♦✱
♦s ❝♦♥❡❝t✐✈♦s ¬✱ ∧ ❡ ∨✳ ❚♦❞❛ ♣r♦♣♦s✐çã♦ ♣♦❞❡ s❡r ❧❡✈❛❞❛ à ❢♦r♠❛ ♥♦r♠❛❧ ♣❡❧❛
❡❧✐♠✐♥❛ç❛♦ ❞♦s ❝♦♥❡❝t✐✈♦s ❡ ✱ s❡ ❡①✐st✐r❡♠✳ ❊①✐st❡♠ ❞✉❛s ❢♦r♠❛s ♥♦r♠❛✐s✿ ❛→ ↔
❢♦r♠❛ ♥♦r♠❛❧ ❝♦♥❥✉♥t✐✈❛ ❡ ❛ ❢♦r♠❛ ♥♦r♠❛❧ ❞✐s❥✉♥t✐✈❛✳
❋♦r♠❛ ♥♦r♠❛❧ ❝♦♥❥✉♥t✐✈❛
❯♠❛ ♣r♦♣♦s✐çã♦ ❡stá ♥❛ ❢♦r♠❛ ♥♦r♠❛❧ ❝♦♥❥✉♥t✐✈❛ ✭❋◆❈✮ s❡ ❡ s♦♠❡♥t❡ s❡✿
❼ ❝♦♥té♠✱ q✉❛♥❞♦ ♠✉✐t♦✱ ♦s ❝♦♥❡❝t✐✈♦s ¬✱ ∧ ❡ ∨ ✭✐st♦ é✱ ❡stá ♥❛ ❢♦r♠❛ ♥♦r♠❛❧✮❀
❼ ❛ ♥❡❣❛çã♦ só t❡♠ ❛❧❝❛♥❝❡ s♦❜r❡ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ✭♦✉ s❡❥❛✱ s♦❜r❡ ❧❡tr❛s
♣r♦♣♦s✐❝✐♦♥❛✐s✮❀
{ ¬ ¬
✸✵❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❼ ❛ ♣r♦♣♦s✐çã♦ ❡stá ❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ ❝♦♥❥✉♥çã♦ ❞❡ ❞✐s❥✉♥çõ❡s✳
❙ã♦ ❡①❡♠♣❧♦s ❞❡ ♣r♦♣♦s✐çõ❡s ♥❛ ❢♦r♠❛ ♥♦r♠❛❧ ❝♦♥❥✉♥t✐✈❛✿
p
¬q
p ∧ ¬q
p ∨ ¬q
(p ∨ ¬q) ∧ ¬r
(p ∨ ¬q) ∧ (¬r ∨ s) ∧ (q ∨ r)
P❛r❛ ♦❜t❡r ❛ ❢♦r♠❛ ♥♦r♠❛❧ ❝♦♥❥✉♥t✐✈❛ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦✿
✶✳ ❊❧✐♠✐♥❡ ♦s ❝♦♥❡❝t✐✈♦s ❡ ✱ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉✐✈❛❧ê♥❝✐❛s✿→ ↔
❼ ❈♦♥❞✐❝✐♦♥❛❧✿ A→ B ≡ ¬A ∨ B
❼ ❇✐❝♦♥❞✐❝✐♦♥❛❧✿ A↔ B ≡ (¬A ∨B) ∧ (¬B ∨ A)
✷✳ ❘❡❞✉③❛ ♦ ❡s❝♦♣♦ ❞❛ ♥❡❣❛çã♦ ♥❛ ♣r♦♣♦s✐çã♦ s✐♠♣❧❡s✱ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉✐✈❛❧ê♥✲
❝✐❛s✿
❼ ❉✉♣❧❛ ◆❡❣❛çã♦✿ ( A) ≡ A
❼ ❉❡▼♦r❣❛♥✿
¬(A ∧ B) ≡ ¬A ∨ ¬B
¬(A ∨ B) ≡ ¬A ∧ ¬B
✸✳ ❊s❝r❡✈❛ ❛ ♣r♦♣♦s✐ã♦ s♦❜ ❢♦r♠❛ ❞❡ ❝♦♥❥✉♥çã♦ ❞❡ ❞✐s❥✉♥çõ❡s✱ ✉t✐❧✐③❛♥❞♦ ❛ ❡q✉✐✈✲
❛❧ê♥❝✐❛✿
❼ ❉✐str✐❜✉t✐✈❛✿ A ∨ (B ∧ C) ≡ (A ∨B) ∧ (A ∨ C)
❊①❡♠♣❧♦s✿
✶✳ (q ∨ r)→ p ≡ ¬(q ∨ r) ∨ p ≡ (¬q ∧ ¬r) ∨ p ≡ (¬q ∨ p) ∧ (¬r ∨ p)
✷✳ ¬p↔ (q∧r) ≡ (¬¬p∨(q∧r))∧(¬(q∧r)∨¬p) ≡ (p∨(q∧r))∧((¬q∨¬r)∨¬p) ≡
((p ∨ q) ∧ (p ∨ r)) ∧ (¬q ∨ ¬r ∨ ¬p)
✸✶❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❋♦r♠❛ ♥♦r♠❛❧ ❞✐s❥✉♥t✐✈❛
❯♠❛ ♣r♦♣♦s✐çã♦ ❡stá ♥❛ ❢♦r♠❛ ♥♦r♠❛❧ ❞✐s❥✉♥t✐✈❛ ✭❋◆❉✮ s❡ ❡ s♦♠❡♥t❡ s❡✿
❼ ❝♦♥té♠✱ q✉❛♥❞♦ ♠✉✐t♦✱ ♦s ❝♦♥❡❝t✐✈♦s ¬✱ ∧ ❡ ∨ ✭✐st♦ é✱ ❡stá ♥❛ ❢♦r♠❛ ♥♦r♠❛❧✮❀
❼ ❛ ♥❡❣❛ç❛♦ só t❡♠ ❛❧❝❛♥❝❡ s♦❜r❡ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ✭♦✉ s❡❥❛✱ s♦❜r❡ ❧❡tr❛s
♣r♦♣♦s✐❝✐♦♥❛✐s✮❀
❼ ❛ ♣r♦♣♦s✐çã♦ ❡stá ❡s❝r✐t❛ ❝♦♠♦ ✉♠❛ ❞✐s❥✉♥çã♦ ❞❡ ❝♦♥❥✉♥çõ❡s✳
❙ã♦ ❡①❡♠♣❧♦s ❞❛ ❢♦r♠❛ ♥♦r♠❛❧ ❞✐s❥✉♥t✐✈❛✿
p ∨ q
¬p ∧ q
(¬p ∧ q) ∨ ¬r
(¬p ∧ q) ∨ (r ∧ s) ∨ (p ∧ ¬s)
P❛r❛ ♦❜t❡r ❛ ❢♦r♠❛ ♥♦r♠❛❧ ❞✐s❥✉♥t✐✈❛ ❞❡ ✉♠❛ ♣r♦♣♦s✐çã♦✿
✶✳ ❊❧✐♠✐♥❡ ♦s ❝♦♥❡❝t✐✈♦s ❡ ✱ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉✐✈❛❧ê♥❝✐❛s✿→ ↔
❼ ❈♦♥❞✐❝✐♦♥❛❧✿ A→ B ≡ ¬A ∨ B
❼ ❇✐❝♦♥❞✐❝✐♦♥❛❧✿ A↔ B ≡ (¬A ∨B) ∧ (¬B ∨ A)
✷✳ ❘❡❞✉③❛ ♦ ❡s❝♦♣♦ ❞❛ ♥❡❣❛çã♦ ❛ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s✱ ✉t✐❧✐③❛♥❞♦ ❛s ❡q✉✐✈❛❧ê♥✲
❝✐❛s✿
❼ ❉✉♣❧❛ ◆❡❣❛çã♦✿ ¬(¬A) ≡ A
❼ ❉❡▼♦r❣❛♥✿
¬(A ∨ B) ≡ ¬A ∧ ¬B
¬(A ∧ B) ≡ ¬A ∨ ¬B
✸✳ ❊s❝r❡✈❛ ❛ ♣r♦♣♦s✐ç❛♦ s♦❜ ❢♦r♠❛ ❞❡ ❞✐s❥✉♥çã♦ ❞❡ ❝♦♥❥✉♥çõ❡s✱ ✉t✐❧✐③❛♥❞♦ ❛
❡q✉✐✈❛❧ê♥❝✐❛✿
❼ ❉✐str✐❜✉t✐✈❛✿ A ∧ (B ∨ C) ≡ (A ∧B) ∨ (A ∧ C)
❊①❡♠♣❧♦✿ ❛❝❤❛r ❛ ❋◆❉✱ ❛ ♣❛rt✐r ❞❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✱ ❞♦ ♦✉✲❡①❝❧✉s✐✈♦✳
❈❆P❮❚❯▲❖ ✷✳ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✸✷
✷✳✺ ■♥❢❡rê♥❝✐❛ ▲ó❣✐❝❛ ✭❈♦♥s❡q✉ê♥❝✐❛ ▲ó❣✐❝❛✮
❯♠❛ ✐♥❢❡rê♥❝✐❛ ❧ó❣✐❝❛✱ ♦✉✱ s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ ✐♥❢❡rê♥❝✐❛✱ é ✉♠❛ t❛✉t♦❧♦❣✐❛ ❞❛ ❢♦r♠❛
A B✳ ❆s ✐♥❢❡rê♥❝✐❛s ❧ó❣✐❝❛s✱ ♦✉ r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛✱ sã♦ r❡♣r❡s❡♥t❛❞❛s ♣♦r→
A |= B✳
❆s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛ sã♦✱ ♥❛ ✈❡r❞❛❞❡✱ ❢♦r♠❛s ✈á❧✐❞❛s ❞❡ r❛❝✐♦❝í♥✐♦✱ ✐st♦ é✱ sã♦
❢♦r♠❛s q✉❡ ♥♦s ♣❡r♠✐t❡♠ ❝♦♥❝❧✉✐r ♦ ❝♦♥s❡q✉❡♥t❡✱ ✉♠❛ ✈❡③ q✉❡ ❝♦♥s✐❞❡r❡♠♦s ♦ ❛♥✲
t❡❝❡❞❡♥t❡ ✈❡r❞❛❞❡✐r♦❀ ❡♠ t❡r♠♦s t❡①t✉❛✐s✱ ❝♦st✉♠❛♠♦s ✉t✐❧✐③❛r ♦ t❡r♠♦ ✏❧♦❣♦✑ ✭♦✉
s❡✉s s✐♥ô♥✐♠♦s✿ ✏♣♦rt❛♥t♦✑✱ ✏❡♠ ❝♦♥s❡q✉ê♥❝✐❛✑ ❡t❝✳✮ ♣❛r❛ ❝❛r❛❝t❡r✐③❛r ❛s ❘❡❣r❛s
❞❡ ■♥❢❡rê♥❝✐❛❀ ❛ ❡①♣r❡ssã♦ A |= B ♣♦❞❡ ❡♥tã♦ s❡r ❧✐❞❛✿ A❀ ❧♦❣♦✱ B✳
➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ ❛s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛ tê♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
❼ ❘❡✢❡①✐✈❛✿ A = A|
❼ ❚r❛♥s✐t✐✈❛✿ ❙❡ A = B ❡ B = C ❡♥tã♦ A = C| | |
◆♦ ❆♣ê♥❞✐❝❡ ❆ sã♦ ❧✐st❛❞❛s ❛❧❣✉♠❛s ❞❛s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛ ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❛
▲ó❣✐❝❛❀ ❞❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦ ❝❛s♦ ❞❛s ❡q✉✐✈❛❧ê♥❝✐❛s✱ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ♣♦❞❡ s❡r
♣r♦✈❛❞❛✱ ❜❛st❛♥❞♦ ♣❛r❛ ✐ss♦ ❝♦♥str✉✐r ❛ t❛❜❡❧❛✲✈❡r❞❛❞❡ ❞❛ ❝♦♥❞✐❝✐♦♥❛❧ ❝♦rr❡s♣♦♥✲
❞❡♥t❡❀ s❡ ❛ ❝♦♥❞✐❝✐♦♥❛❧ ❢♦r t❛✉t♦❧ó❣✐❝❛✱ s❡rá ✉♠❛ ✐♥❢❡rê♥❝✐❛✳
❱❛♠♦s ❡①❡♠♣❧✐✜❝❛r ❝♦♠ ❛ r❡❣r❛ ❞❡ ✐♥❢❡rê♥❝✐❛ ❝♦♥❤❡❝✐❞❛ ♣♦r ▼♦❞✉s P♦♥❡♥s✿
(p→ q), p |= q
p q p→ q (p→ q) ∧ p (p→ q)
∧ p→ q
❱ ❱ ❱ ❱ ❱
❋ ❱ ❱ ❋ ❱
❱ ❋ ❋ ❋ ❱
❋ ❋ ❱ ❋ ❱
❈❛♣ít✉❧♦ ✸
❉❡❞✉çã♦ ♥♦ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧
✸✳✶ ❆r❣✉♠❡♥t♦s
❈❤❛♠❛✲s❡ ❛r❣✉♠❡♥t♦ ❛ ❛✜r♠❛çã♦ ❞❡ q✉❡ ❞❡ ✉♠ ❞❛❞♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r♦♣♦s✐çõ❡s
P1, P2, ..., Pn ❝❤❛♠❛❞❛s ♣r❡♠✐ss❛s✱ ❞❡❝♦rr❡ ✉♠❛ ♣r♦♣♦s✐çã♦ Q✱ ❝❤❛♠❛❞❛ ❝♦♥❝❧✉sã♦✳
P♦❞❡ s❡r ❡①♣r❡ss♦ s✐♠❜♦❧✐❝❛♠❡♥t❡ ❝♦♠♦✿
P1, P2, ..., Pn ⊢ Q
❖ sí♠❜♦❧♦ ✏⊢✑✱ ❝❤❛♠❛❞♦ tr❛ç♦ ❞❡ ❛ss❡rçã♦✱ ❛✜r♠❛ q✉❡ ❛ ❢ór♠✉❧❛ à s✉❛ ❞✐r❡✐t❛
♣♦❞❡ s❡r ❞❡❞✉③✐❞❛ ✉t✐❧✐③❛♥❞♦ ❝♦♠♦ ♣r❡♠✐ss❛s s♦♠❡♥t❡ ❛s ❢ór♠✉❧❛s q✉❡ ❡stã♦ à s✉❛
❡sq✉❡r❞❛✳
❊✐s ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛r❣✉♠❡♥t♦✿
P1✿ ❙❡ ❏♦sé ♣❡❣♦✉ ❛s ❥ó✐❛s ♦✉ ❛ ❙r❛✳ ❑r❛s♦✈ ♠❡♥t✐✉✱ ❡♥tã♦ ♦❝♦rr❡✉
✉♠ ❝r✐♠❡✳
P2✿ ❙❡ ♦❝♦rr❡✉ ✉♠ ❝r✐♠❡ ❡♥tã♦ ♦ ❙r✳ ❑r❛s♦✈ ❡st❛✈❛ ♥❛ ❝✐❞❛❞❡✳
P3✿ ❖ ❙r✳ ❑r❛s♦✈ ♥ã♦ ❡st❛✈❛ ♥❛ ❝✐❞❛❞❡✳
Q✿ P♦rt❛♥t♦✱ ♦✉ ❏♦sé ♥ã♦ ♣❡❣♦✉ ❛s ❥ó✐❛s ♦✉ ❛ ❙r❛✳ ❑r❛s♦✈ ♥ã♦
♠❡♥t✐✉✳
❖ q✉❡ q✉❡r❡♠♦s ❞✐③❡r ❝♦♠ ❛ ❡①♣r❡ssã♦ ✏❛ ❝♦♥❝❧✉sã♦ ❞❡❝♦rr❡ ❞❛s ♣r❡♠✐ss❛s✑❄ ❇❛✲
s✐❝❛♠❡♥t❡✱ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❛ ❝♦♥❝❧✉sã♦ ❡stá ✐♥❝❧✉í❞❛ ♥❛ ✈❡r❛❝✐✲
❞❛❞❡ ❞❛s ♣r❡♠✐ss❛s❀ ✐st♦ é✱ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ s❡ ❛s ♣r❡♠✐ss❛s ❢♦r❡♠ ✈❡r❞❛❞❡✐r❛s✱
✸✸
✸✹❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❛ ❝♦♥❝❧✉sã♦ t❛♠❜é♠ ♦ s❡rá✳ ◗✉❛♥❞♦ ❛ ❝♦♥❝❧✉sã♦ r❡❛❧♠❡♥t❡ ❞❡❝♦rr❡ ❞❛s ♣r❡♠✐s✲
s❛s✱ ❞✐③❡♠♦s q✉❡ ♦ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦❀ q✉❛♥❞♦ ♥ã♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❛r❣✉♠❡♥t♦ é
✐♥✈á❧✐❞♦✳
❆❜❛✐①♦✱ ❞♦✐s ❡①❡♠♣❧♦s ❞❡ ❛r❣✉♠❡♥t♦s✿
❙❡ ❡✉ t✐✈❡r ❞✐♥❤❡✐r♦✱ ✈♦✉ ❛♦ ❝✐♥❡♠❛ ♦✉ ❛♦ t❡❛tr♦
▼❛s ❡✉ ♥ã♦ t❡♥❤♦ ❞✐♥❤❡✐r♦✳
▲♦❣♦✱ ♦✉ ♥ã♦ ✈♦✉ ❛♦ ❝✐♥❡♠❛ ♦✉ ♥ã♦ ✈♦✉ ❛♦ t❡❛tr♦✳
❆ ❝♦♥❝❧✉sã♦ ❛♣r❡s❡♥t❛❞❛ ♥ã♦ ❞❡❝♦rr❡ ❞❛s ♣r❡♠✐ss❛s✳ P♦rt❛♥t♦ ♦ ❛r❣✉♠❡♥t♦ é
✐♥✈á❧✐❞♦✳
❙❡ ❡✉ ❡st✉❞❛r✱ ✜❝♦ ❝❛♥s❛❞♦✳
❙❡ ❡✉ ✜❝❛r ❝❛♥s❛❞♦✱ ❞✉r♠♦✳
▲♦❣♦✱ s❡ ❡✉ ❡st✉❞❛r✱ ❞✉r♠♦✳
❈❧❛r❛♠❡♥t❡✱ ❛ ❝♦♥❝❧✉sã♦ ❞❡❝♦rr❡ ❞❛s ♣r❡♠✐ss❛s✱ ❡ ♦ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦✳
◆❛ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✱ ❛ ❡str✉t✉r❛ q✉❡ ♠❡❧❤♦r r❡♣r❡s❡♥t❛ ✉♠ ❛r❣✉♠❡♥t♦✱ é ❛
♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✿ ✉♠ ❛r❣✉♠❡♥t♦ é✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❝♦♥❞✐❝✐♦♥❛❧ ❞❛
❢♦r♠❛✿
P1 ∧ P2 ∧ ... ∧ Pn Q→
❆ ✈❛❧✐❞❛❞❡ ❞♦ ❛r❣✉♠❡♥t♦ ❞❡♣❡♥❞❡ ❡①❝❧✉s✐✈❛♠❡♥t❡ ❞♦ r❡❧❛❝✐♦♥❛♠❡♥t♦ ❧ó❣✐❝♦
❡♥tr❡ ❛s ♣r❡♠✐ss❛s ❡ ❛ ❝♦♥❝❧✉sã♦❀ ✐st♦ é✱ ♥ã♦ é ♦❝✉♣❛çã♦ ❞❛ ▲ó❣✐❝❛ ✈❡r✐✜❝❛r s❡
❛s ♣r❡♠✐ss❛s sã♦ ✈❡r❞❛❞❡✐r❛s✳ ❖ ♦❜❥❡t✐✈♦ ❞❛ ▲ó❣✐❝❛ ❋♦r♠❛❧ é ✈❡r✐✜❝❛r s❡ ♦ ❛r❣✉✲
♠❡♥t♦ é ❡str✉t✉r❛❞♦ ❞❡ ❢♦r♠❛ t❛❧ q✉❡✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦s ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞❛s
♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❡♥✈♦❧✈✐❞❛s✱ ❛ ✈❡r❛❝✐❞❛❞❡ ❞❛s ♣r❡♠✐ss❛s ✐♠♣❧✐❝❛ ♥❛ ✈❡r❛❝✐❞❛❞❡
❞❛ ❝♦♥❝❧✉sã♦✳
❊♠ t❡r♠♦s ❧ó❣✐❝♦s✱ ✐ss♦ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ s❡ ✉♠ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦✱ ❡♥tã♦ ❛
❝♦♥❞✐❝✐♦♥❛❧ q✉❡ ♦ r❡♣r❡s❡♥t❛ é s❡♠♣r❡ ✈❡r❞❛❞❡✐r❛✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞♦s ✈❛❧♦r❡s
❧ó❣✐❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s ❝♦♠♣♦♥❡♥t❡s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ ✉♠ ❛r❣✉♠❡♥t♦ é
✈á❧✐❞♦✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ q✉❡ ♦ r❡♣r❡s❡♥t❛ é ✉♠❛ t❛✉t♦❧♦❣✐❛✳
❆ t❛❜❡❧❛✲✈❡r❞❛❞❡ é ♦ ♠ét♦❞♦ ♠❛✐s s✐♠♣❧❡s ♣❛r❛ ✈❡r✐✜❝❛r s❡ ✉♠❛ ❢♦r♠❛ s✐♠✲
❜ó❧✐❝❛ é ♦✉ ♥ã♦ ✉♠❛ t❛✉t♦❧♦❣✐❛✱ ❡ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ❛ ✈❛❧✐❞❛❞❡
♦✉ ✐♥✈❛❧✐❞❛❞❡ ❞❡ ✉♠ ❛r❣✉♠❡♥t♦❀ ✈❡❥❛♠♦s ♦s ❞♦✐s ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛❞♦s✿
❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✸✺
❖ ♣r✐♠❡✐r♦ ❛r❣✉♠❡♥t♦✱ ✏s❡ ❡✉ t✐✈❡r ❞✐♥❤❡✐r♦✱ ✈♦✉ ❛♦ ❝✐♥❡♠❛ ♦✉ ❛♦ t❡❛tr♦❀ ♠❛s
❡✉ ♥ã♦ t❡♥❤♦ ❞✐♥❤❡✐r♦✳ ▲♦❣♦✱ ♦✉ ♥ã♦ ✈♦✉ ❛♦ ❝✐♥❡♠❛ ♦✉ ♥ã♦ ✈♦✉ ❛♦ t❡❛tr♦✑ ♣♦❞❡
s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r✿
(p→ q ∨ r) ∧ (¬p)→ (¬q ∨ ¬r)
♦♥❞❡✿
p✿ ❡✉ t❡♥❤♦ ❞✐♥❤❡✐r♦
q✿ ❡✉ ✈♦✉ ❛♦ ❝✐♥❡♠❛
r✿ ❡✉ ✈♦✉ ❛♦ t❡❛tr♦
❈♦♥str✉✐♥❞♦ ❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✿
p q r q ∨ r p→ q ∨ r ¬p ¬q ¬r ¬q ∨ ¬r (p→ q ∨ r) ∧ (¬p) (p→ q ∨ r) ∧ (¬p)→ (¬q ∨ ¬r)
❱ ❱ ❱ ❱ ❱ ❋ ❋ ❋ ❋ ❋ ❱
❋ ❱ ❱ ❱ ❱ ❱ ❋ ❋ ❋ ❱ ❋
❱ ❋ ❱ ❱ ❱ ❋ ❱ ❋ ❱ ❋ ❱
❋ ❋ ❱ ❱ ❱ ❱ ❱ ❋ ❱ ❱ ❱
❱ ❱ ❋ ❱ ❱ ❋ ❋ ❱ ❱ ❋ ❱
❋ ❱ ❋ ❱ ❱ ❱ ❋ ❱ ❱ ❱ ❱
❱ ❋ ❋ ❋ ❋ ❋ ❱ ❱ ❱ ❋ ❱
❋ ❋ ❋ ❋ ❱ ❱ ❱ ❱ ❱ ❱ ❱
❆ ❡①♣r❡ssã♦ ♥ã♦ é ✉♠❛ t❛✉t♦❧♦❣✐❛✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♦ ❛r❣✉♠❡♥t♦ ♥ã♦ é
✈á❧✐❞♦✳
❖ ❛r❣✉♠❡♥t♦ ✏s❡ ❡✉ ❡st✉❞❛r✱ ✜❝♦ ❝❛♥s❛❞♦❀ s❡ ❡✉ ✜❝❛r ❝❛♥s❛❞♦✱ ❞✉r♠♦✳ ▲♦❣♦✱
s❡ ❡✉ ❡st✉❞❛r✱ ❞✉r♠♦✑✱ ♣♦r s✉❛ ✈❡③ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞♦ ♣♦r✿
(p→ q) ∧ (q → r)→ (p→ r)
♦♥❞❡✿
p✿ ❡✉ ❡st✉❞♦
q✿ ❡✉ ✜❝♦ ❝❛♥s❛❞♦
✸✻❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
r✿ ❡✉ ❞✉r♠♦
❆ t❛❜❡❧❛✲✈❡r❞❛❞❡✿
p q r p→ q q → r p→ r (p→ q) ∧ (q → r) (p→ q) ∧ (q → r)→ (p→ r)
❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱
❋ ❱ ❱ ❱ ❱ ❱ ❱ ❱
❱ ❋ ❱ ❋ ❱ ❱ ❋ ❱
❋ ❋ ❱ ❱ ❱ ❱ ❱ ❱
❱ ❱ ❋ ❱ ❋ ❋ ❋ ❱
❋ ❱ ❋ ❱ ❋ ❱ ❋ ❱
❱ ❋ ❋ ❋ ❱ ❋ ❋ ❱
❋ ❋ ❋ ❱ ❱ ❱ ❱ ❱
❈♦♠♦ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ✉♠❛ t❛✉t♦❧♦❣✐❛✱ ♦ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦✳
◆ã♦ ❞❡✈❡ t❡r ♣❛ss❛❞♦ ❞❡s♣❡r❝❡❜✐❞♦ ❛♦ ❧❡✐t♦r ♠❛✐s ❛t❡♥t♦ q✉❡ ❡ss❡ ❛r❣✉♠❡♥t♦ é✱
♥❛ ✈❡r❞❛❞❡✱ ❛ r❡❣r❛ ❞❡ ✐♥❢❡rê♥❝✐❛ ❝❤❛♠❛❞❛ ❙✐❧♦❣✐s♠♦ ❍✐♣♦tét✐❝♦✳ ◗✉❛❧ ❛ ❞✐❢❡r❡♥ç❛✱
❡♥tã♦✱ ❡♥tr❡ ♦s ❛r❣✉♠❡♥t♦s ✈á❧✐❞♦s ❡ ❛s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛❄ ❋♦r♠❛❧♠❡♥t❡✱ ♥❡♥✲
❤✉♠❛✳ ◆❛ ♣rát✐❝❛✱ ❛s ❝♦♥❞✐❝✐♦♥❛✐s t❛✉t♦❧ó❣✐❝❛s ❝♦♠ ♣♦✉❝♦s ❛♥t❡❝❡❞❡♥t❡s ❡ ❝✉❥❛
❢♦r♠❛ ❥á é ❝♦♥❤❡❝✐❞❛✱ sã♦ ❝❤❛♠❛❞❛s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛✱ ❡♥q✉❛♥t♦ ❛s ❝♦♥❞✐❝✐♦♥❛✐s
♠❛✐♦r❡s✱ q✉❡ ❛✐♥❞❛ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ sã♦ t❛✉t♦❧♦❣✐❛s sã♦ ❝❤❛♠❛❞❛s ❛r❣✉♠❡♥t♦s✳
➚ ♣r♦♣ós✐t♦✱ ♦s ❛r❣✉♠❡♥t♦s ❝♦♠ ❞✉❛s ♣r❡♠✐ss❛s sã♦ ❝❤❛♠❛❞♦s s✐❧♦❣✐s♠♦s ✳
❊♥tr❡ ❛s ✈ár✐❛s ♥♦t❛çõ❡s ✉t✐❧✐③❛❞❛s ♣❛r❛ r❡♣r❡s❡♥t❛r ♦s ❛r❣✉♠❡♥t♦s✱ ✉♠❛ ❞❛s
♠❛✐s s✐♠♣❧❡s é r❡♣r❡s❡♥t❛r ❛s ♣r❡♠✐ss❛s ✉♠❛ ❡♠ ❝❛❞❛ ❧✐♥❤❛ ❡ s❡♣❛r❛♥❞♦✲❛s ❞❛
❝♦♥❝❧✉sã♦ ❛tr❛✈és ❞❡ ✉♠ tr❛ç♦ ❤♦r✐③♦♥t❛❧✳ ◆❡ss❛ ♥♦t❛çã♦✱ ♦ ❛r❣✉♠❡♥t♦
✏❙❡ ❏♦sé ♣❡❣♦✉ ❛s ❥ó✐❛s ♦✉ ❛ ❙r❛✳ ❑r❛s♦✈ ♠❡♥t✐✉✱ ❡♥tã♦ ♦❝♦rr❡✉ ✉♠
❝r✐♠❡✳ ❙❡ ♦❝♦rr❡✉ ✉♠ ❝r✐♠❡ ❡♥tã♦ ♦ ❙r✳ ❑r❛s♦✈ ❡st❛✈❛ ♥❛ ❝✐❞❛❞❡✳ ▼❛s
♦ ❙r✳ ❑r❛s♦✈ ♥ã♦ ❡st❛✈❛ ♥❛ ❝✐❞❛❞❡✳ P♦rt❛♥t♦✱ ♦✉ ❏♦sé ♥ã♦ ♣❡❣♦✉ ❛s
❥ó✐❛s ♦✉ ❛ ❙r❛✳ ❑r❛s♦✈ ♥ã♦ ♠❡♥t✐✉✳✑
❛♣r❡s❡♥t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ❛ss✉♠❡♠ ❛ ❢♦r♠❛✿
✸✼❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
p ∨ q r→
r s→
s¬
¬p ∨ ¬q
♦♥❞❡✿
p✿ ❏♦sé ♣❡❣♦✉ ❛s ❥ó✐❛s
q✿ ❛ ❙r❛✳ ❑r❛s♦✈ ♠❡♥t✐✉
r✿ ♦❝♦rr❡✉ ✉♠ ❝r✐♠❡
s✿ ♦ ❙r✳ ❑r❛s♦✈ ❡st❛✈❛ ♥❛ ❝✐❞❛❞❡
✸✳✷ ❉❡❞✉çã♦
❈♦♠♦ ✈✐♠♦s✱ ❡♥tã♦✱ ♦ ♠ét♦❞♦ ❞❛ t❛❜❡❧❛✲✈❡r❞❛❞❡ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ♠♦str❛r
q✉❡ ✉♠ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ✐♥✈á❧✐❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ❡ss❡ ♠ét♦❞♦ ❛♣r❡s❡♥t❛ ❞♦✐s
sér✐♦s ✐♥❝♦♥✈❡♥✐❡♥t❡s✿
✶✳ ❖ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❝r❡s❝❡ ♠✉✐t♦ r❛♣✐❞❛♠❡♥t❡ à ♠❡❞✐❞❛ q✉❡ ❛✉♠❡♥t❛ ♦
♥ú♠❡r♦ ❞❡ ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❡♥✈♦❧✈✐❞❛s ♥♦ ❛r❣✉♠❡♥t♦ ✕ ❝♦♠ ✶✵ ♣r♦♣♦s✐çõ❡s
❛ t❛❜❡❧❛ ♥❡❝❡ss✐t❛ ❞❡ ✶✵✷✹ ❧✐♥❤❛s✱ ❡ ❝♦♠ ✶✶✱ ♦ ♥✉♠❡r♦ ❞❡ ❧✐♥❤❛s ✈❛✐ ❛ ✷✵✹✽✳
❈♦♠ ♠❛✐s ✉♠❛s ♣♦✉❝❛s ♣r♦♣♦s✐çõ❡s✱ s✉❛ ❝♦♥str✉çã♦ s❡ t♦r♥❛ ✐♠♣r❛t✐❝á✈❡❧✳
✷✳ ◆♦ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s✱ q✉❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r✱ ♠✉✐t❛s ✈❡③❡s ♥ã♦ ❡①✐st❡ ✉♠
♣r♦❝❡❞✐♠❡♥t♦ q✉❡ ♣❡r♠✐t❛ ❡st❛❜❡❧❡❝❡r ♦ ✈❛❧♦r ❧ó❣✐❝♦ ❞❡ ✉♠❛ ❞❛❞❛ ❛✜r♠❛çã♦✱
♦ q✉❡ t♦r♥❛ ✐♠♣♦ssí✈❡❧ ❛ ❝♦♥str✉çã♦ ❞❛ t❛❜❡❧❛✲✈❡r❞❛❞❡✳
P♦r ✐ss♦✱ ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ♦✉tr♦s ♠ét♦❞♦s ♣❛r❛ q✉❡ s❡ ♣♦ss❛ ♠♦str❛r ❛ ✈❛❧✲
✐❞❛❞❡ ❞❡ ✉♠ ❛r❣✉♠❡♥t♦✳ ❚❛✐s ♠ét♦❞♦s sã♦ ❝❤❛♠❛❞♦s ♠ét♦❞♦s ❞❡❞✉t✐✈♦s✱ ❡ s✉❛
❛♣❧✐❝❛çã♦ ❝❤❛♠❛✲s❡ ❞❡❞✉çã♦✳ ❊♠ t❡r♠♦s ♠❛✐s ❢♦r♠❛✐s✱ ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡❞✉çã♦ ♣♦❞❡
s❡r ❛♣r❡s❡♥t❛❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
❉❛❞♦ ✉♠ ❛r❣✉♠❡♥t♦✿
✸✽❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
P1
P2
✳
✳
✳
Pn
Q
❝❤❛♠❛✲s❡ ❞❡♠♦♥str❛çã♦ ♦✉ ❞❡❞✉çã♦ ❞❡ Q ❛ ♣❛rt✐r ❞❛s ♣r❡♠✐ss❛s P1, ..., Pn✱ ❛
s❡q✉ê♥❝✐❛ ✜♥✐t❛ ❞❡ ♣r♦♣♦s✐çõ❡s X1, X2, ..., Xk✱ t❛❧ q✉❡ ❝❛❞❛ Xi ♦✉ é ✉♠❛ ♣r❡♠✐ss❛
♦✉ ❞❡❝♦rr❡ ❧♦❣✐❝❛♠❡♥t❡ ❞❡ ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s ❞❛ s❡qüê♥❝✐❛✱ ❡ ❞❡ t❛❧ ♠♦❞♦ q✉❡
❛ ú❧t✐♠❛ ♣r♦♣♦s✐çã♦ Xk s❡❥❛ ❛ ❝♦♥❝❧✉sã♦ Q ❞♦ ❛r❣✉♠❡♥t♦ ❞❛❞♦✳
❈❛❞❛ ♣r♦♣♦s✐çã♦ Xi q✉❡ ✐♥❝❧✉í♠♦s ♥❛ s❡q✉ê♥❝✐❛ ❞❡✈❡ ❞❡❝♦rr❡r ❧♦❣✐❝❛♠❡♥t❡ ❞❛s
❛♥t❡r✐♦r❡s❀ ✐ss♦ s✐❣♥✐✜❝❛ q✉❡ ❞❡✈❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❛ ❛t✉❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s
♦✉ ✐♥❢❡rê♥❝✐❛s s♦❜r❡ ✉♠❛ ♣r♦♣♦s✐çã♦ ♦✉ ✉♠❛ ❝♦♥❥✉♥çã♦ ❞❡ ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✳
❙❡ ❢♦r ♣♦ssí✈❡❧ ♦❜t❡r ❛ ❝♦♥❝❧✉sã♦ Q ❛tr❛✈és ❞♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ ❞❡❞✉çã♦✱ ♦
❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦❀ ❝❛s♦ ❝♦♥trár✐♦✱ ♥ã♦ é ✈á❧✐❞♦✳
❖ ♣r♦❝❡ss♦ ❞❡ ❞❡❞✉çã♦ ❝♦♥s✐st❡ ❜❛s✐❝❛♠❡♥t❡ ❞♦s s❡❣✉✐♥t❡s ♣❛ss♦s✿
✶✳ ❞❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦ P ❝♦♥st✐t✉í❞♦ ♣❡❧❛s ♣r❡♠✐ss❛s {P1, P2, ..., Pn}❀
✷✳ s♦❜r❡ ✉♠ ♦✉ ♠❛✐s ❡❧❡♠❡♥t♦s ❞♦ ❝♦♥❥✉♥t♦ ❢❛③❡♠♦s ❛t✉❛r ❡q✉✐✈❛❧ê♥❝✐❛s ❡ ✐♥✲
❢❡rê♥❝✐❛s ❝♦♥❤❡❝✐❞❛s✱ ♦❜t❡♥❞♦ ♥♦✈❛s ♣r♦♣♦s✐çõ❡s✱ ❡ ✐♥❝❧✉✐♥❞♦✲❛s ♥♦ ❝♦♥❥✉♥t♦
P ❀
✸✳ r❡♣❡t✐♠♦s ♦ ♣❛ss♦ ❛❝✐♠❛ ❛té q✉❡ ❛ ♣r♦♣♦s✐çã♦ ✐♥❝❧✉í❞❛ s❡❥❛ ♦ ❝♦♥s❡q✉❡♥t❡ Q✳
❱❛♠♦s ❡①❡♠♣❧✐✜❝❛r ♦ ♣r♦❝❡ss♦ ♣r♦✈❛♥❞♦ ♦ ❛r❣✉♠❡♥t♦ ❛♣r❡s❡♥t❛❞♦ ♥♦ ú❧t✐♠♦ ❡①✲
❡♠♣❧♦ ❞❛ s❡çã♦ ❛♥t❡r✐♦r✿
p ∨ q r→
r s→
s¬
¬p ∨ ¬q
❆ ❞❡❞✉çã♦ é ❛♣r❡s❡♥t❛❞❛ s♦❜ ❛ ❢♦r♠❛ ❞❡ ✉♠❛ t❛❜❡❧❛✱ ❝♦♠ três ❝♦❧✉♥❛s✱ ❞❛
s❡❣✉✐♥t❡ ❢♦r♠❛✿
✸✾❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❝♦❧✉♥❛ ✶ ♦s ♣❛ss♦s ❞❛❞♦s
♥❛ ❞❡❞✉çã♦ ✕ ❛ ❝❛❞❛ ♣❛ss♦ ♦❜t❡♠♦s ✉♠❛ ♣r♦♣♦s✐çã♦ q✉❡
é r❡❢❡r❡♥❝✐❛❞❛ ♥♦ r❡st❛♥t❡ ❞❛ ❞❡❞✉çã♦ ♣♦r ❡st❡ ♥ú♠❡r♦✳ ❖s ♣r✐♠❡✐r♦s
♣❛ss♦s sã♦✱ ♥❛t✉r❛❧♠❡♥t❡✱ ♦ ❡st❛❜❡❧❡❝✐♠❡♥t♦ ❞❛s ♣r❡♠✐ss❛s✳
❝♦❧✉♥❛ ✷ ❛ ♣r♦♣♦s✐çã♦ ♦❜t✐❞❛ ♥❛q✉❡❧❡ ♣❛ss♦ ✕ ❛ ú❧t✐♠❛ ❞❡✈❡ s❡r ❛ ❝♦♥❝❧✉sã♦ ❞♦
❛r❣✉♠❡♥t♦✳
❝♦❧✉♥❛ ✸ ✐♥❞✐❝❛çã♦ ❞❡ ❝♦♠♦ ❢♦✐ ♦❜t✐❞❛ ❛ ♣r♦♣♦s✐çã♦ ♥❛q✉❡❧❡ ♣❛ss♦ ✕ ♥♦r♠❛❧♠❡♥t❡
❛s ♣r♦♣♦s✐çõ❡s sã♦ ♦❜t✐❞❛s ❛♣❧✐❝❛♥❞♦✲s❡ ❡q✉✐✈❛❧ê♥❝✐❛s✱ r❡❣r❛s ❞❡ ✐♥❢❡r✲
ê♥❝✐❛ ♦✉ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s s♦❜r❡ ♣r♦♣♦s✐çõ❡s ❥á ♦❜t✐❞❛s ❡♠ ♣❛ss♦s
❛♥t❡r✐♦r❡s✳
✶✳ p ∨ q → r Pr❡♠✐ss❛ ✶
✷✳ r → s Pr❡♠✐ss❛ ✷
✸✳ ¬s Pr❡♠✐ss❛ ✸
✹✳ ¬r ✷✱✸✱▼❚
✺✳ ¬(p ∨ q) ✶✱✹✱▼❚
✻✳ ¬p ∧ ¬q ✺✱❉▼
✼✳ ¬p ✻✱❙■▼P
✽✳ ¬p ∨ ¬q ✼✱❆❉
➱ ♣♦ssí✈❡❧ ♠♦str❛r q✉❡ t♦❞♦ ♦ ❝♦r♣♦ ❞❛ ▲ó❣✐❝❛ ▼❛t❡♠át✐❝❛ ♣♦❞❡ s❡r ❡r❣✉✐❞♦
❛ ♣❛rt✐r ❞❡ ✉♠❛s ♣♦✉❝❛s r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛ ❛①✐♦♠át✐❝❛s✳ ◆♦ss♦ ♦❜❥❡t✐✈♦✱ ♥♦
❡♥t❛♥t♦✱ ♥ã♦ é ❛❧❝❛♥ç❛r ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ ▲ó❣✐❝❛✱ ♠❛s tã♦ s♦♠❡♥t❡ ❞❡s❝r❡✈❡r
♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦s ♠ét♦❞♦s ❞❡ ❞❡❞✉çã♦✳ ◆♦ ❆♣ê♥❞✐❝❡ ❆ ❡st❛❜❡❧❡❝❡♠♦s ✉♠
❝♦♥❥✉♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ ❧❡✐s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ r❡❣r❛s ❞❡ ✐♥❢❡rê♥❝✐❛✱ ♣❛r❛✱ ❛ ♣❛rt✐r
❞❡st❛s✱ ♠♦str❛r ❛ ✈❛❧✐❞❛❞❡ ❞❡ ❛r❣✉♠❡♥t♦s ♠❛✐s ❝♦♠♣❧❡①♦s✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♥♦
❞❡❝♦rr❡r ❞♦s ♣r♦❝❡ss♦s ❞❡❞✉t✐✈♦s ♣♦❞❡r❡♠♦s ❧❛♥ç❛r ♠ã♦ ❞❡st❛s✱ ❡ ❛♣❡♥❛s ❞❡st❛s✱
❡q✉✐✈❛❧ê♥❝✐❛s ❡ ✐♥❢❡rê♥❝✐❛s✱ ❝♦♥s✐❞❡r❛♥❞♦✲❛s ❥á ❝♦♥❤❡❝✐❞❛s✳
✸✳✸ ❙✐♠♣❧✐✜❝❛çã♦ ❞❛ ❈♦♥❝❧✉sã♦
❋♦r♠❛❧♠❡♥t❡✱ ♦ ♠ét♦❞♦ ❞❡ ❞❡❞✉çã♦ q✉❡ ❛♣r❡s❡♥t❛♠♦s é s✉✜❝✐❡♥t❡ ♣❛r❛ ♦❜t❡r ❛
❝♦♥❝❧✉sã♦ ❞❡ q✉❛❧q✉❡r ❛r❣✉♠❡♥t♦ ✈á❧✐❞♦❀ ♥♦ ❡♥t❛♥t♦✱ ♠✉✐t❛s ✈❡③❡s✱ q✉❛♥❞♦ ❛ ❝♦♥✲
❝❧✉sã♦ é ✉♠❛ ♣r♦♣♦s✐çã♦ ❝♦♠♣♦st❛✱ ❡♥✈♦❧✈❡♥❞♦ ✉♠❛ ♦✉ ♠❛✐s ♦♣❡r❛çõ❡s ❧ó❣✐❝❛s✱ ❛
✹✵❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❞❡❞✉çã♦ t♦r♥❛✲s❡ ♠❛✐s ❞✐❢í❝✐❧✳ ◆❡ss❡s ❝❛s♦s✱ ❝♦st✉♠❛✲s❡ s✐♠♣❧✐✜❝❛r ❛ ❝♦♥❝❧✉sã♦✱ ❞❡
❢♦r♠❛ ❛ ❢❛❝✐❧✐t❛r ❛ ❞❡❞✉çã♦✳
❈♦♥❝❧✉sã♦ ❞❛ ❢♦r♠❛ A ∧B
◗✉❛♥❞♦ ❛ ❝♦♥❝❧✉sã♦ é ✉♠❛ ❝♦♥❥✉♥çã♦✱ ❞❡✈❡♠♦s ♦❜t❡r✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ ❛s
♣❛r❝❡❧❛s A ❡ B✱ ❡ ❛ s❡❣✉✐r✱ ♦❜t❡r A ∧ B✱ ♣♦r ❈❖◆❏✳
❊①❡♠♣❧♦✿
❙❡ ❛ ♣r♦❝✉r❛ ❞♦ ♣r♦❞✉t♦ ❛✉♠❡♥t❛r✱ s❡✉ ♣r❡ç♦ s✉❜✐rá❀ s❡ ♦ ♣r❡ç♦ s✉❜✐r✱ ♦
♣r♦❞✉t♦ ♥ã♦ s❡rá ❡①♣♦rt❛❞♦❀ s❡ ♥ã♦ ❤♦✉✈❡r ✐♠♣♦rt❛çã♦ ♦✉ s❡ ❛ ♣r♦❞✉t♦
❢♦r ❡①♣♦rt❛❞♦✱ ♦ ♣r♦❞✉t♦ ❡s❝❛ss❡❛rá✳ ❆ ♣r♦❝✉r❛ ❞♦ ♣r♦❞✉t♦ ❛✉♠❡♥t♦✉
❡ ♥ã♦ ❤❛✈❡rá ✐♠♣♦rt❛çã♦✳ ▲♦❣♦✱ ♦ ♣r♦❞✉t♦ ♥ã♦ s❡rá ❡①♣♦rt❛❞♦ ❡ ❡s✲
❝❛ss❡❛rá✳
❋❛③❡♥❞♦✿
p✿ ❛ ♣r♦❝✉r❛ ❛✉♠❡♥t❛r
q✿ ♦ ♣r❡ç♦ s✉❜✐r
r✿ ♦ ♣r♦❞✉t♦ s❡r ❡①♣♦rt❛❞♦
s✿ ❤❛✈❡r ✐♠♣♦rt❛çã♦
t✿ ♦ ♣r♦❞✉t♦ ❡s❝❛ss❡❛r
t❡♠♦s ♦ ❛r❣✉♠❡♥t♦ ♥❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛ ❡ s✉❛ ❞❡❞✉çã♦✿
p q→
q r→ ¬
¬s ∨ r → t
p ∧ ¬s
¬r ∧ t
✹✶❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
✶✳ p→ q Pr❡♠✐ss❛ ✶
✷✳ q → ¬r Pr❡♠✐ss❛ ✷
✸✳ ¬s ∨ r → t Pr❡♠✐ss❛ ✸
✹✳ p ∧ ¬s Pr❡♠✐ss❛ ✹
✺✳ p ✹✱ ❙■▼P
✻✳ p→ ¬r ✶✱ ✷✱ ❙❍
✼✳ ¬r ✺✱ ✻✱ ▼P
✽✳ ¬s ✹✱ ❙■▼P
✾✳ ¬s ∨ r ✽✱ ❆❉
✶✵✳ t ✸✱✾✱ ▼P
✶✶✳ ¬r ∧ t ✼✱ ✶✵✱ ❈❖◆❏
❈♦♥❝❧✉sã♦ ❞❡ ❢♦r♠❛ A B→
❙✉♣♦♥❤❛ q✉❡ ♦ ❛r❣✉♠❡♥t♦ t❡♥❤❛ ❛ ❢♦r♠❛
P (B C)→ →
♦♥❞❡ P é ❛ ❝♦♥❥✉♥çã♦ ❞❡ ♣r❡♠✐ss❛s✳ ❖r❛✱ s❛❜❡♠♦s q✉❡✱ ♣♦r ❊■✱
A ∧B C ≡ A (B C)→ → →
♦ q✉❡ ♣❡r♠✐t❡ ❡s❝r❡✈❡r ♦ ❛r❣✉♠❡♥t♦ ❞❛❞♦ ♥❛ ❢♦r♠❛
P ∧B C→
P♦rt❛♥t♦✱ ♣❛r❛ ❞❡❞✉③✐r♠♦s ✉♠ ❛r❣✉♠❡♥t♦ ❝✉❥❛ ❝♦♥❝❧✉sã♦ é ❞❛ ❢♦r♠❛ B C✱→
✐♥❝❧✉í♠♦s B ♥♦ ❝♦♥❥✉♥t♦ ❞❡ ♣r❡♠✐ss❛s✱ ❡ ♣r♦❝✉r❛♠♦s ❞❡❞✉③✐r C✳ ❊st❡ ❛rt✐❢í❝✐♦ é
❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❉❡❞✉çã♦ ❞❛ ❈♦♥❞✐❝✐♦♥❛❧ ✳
❊①❡♠♣❧♦✿
❙❡ ❛ ❝❛s❛ ✜❝❛r ✈❛③✐❛ ♦✉ ❡✉ ❝♦♥s❡❣✉✐r ♦ ❡♠♣rést✐♠♦ ❡♥tã♦ ♣❛❣♦ ❛ ❞í✈✐❞❛
❡ ♠❡ ♠✉❞♦✳ ❙❡ ❡✉ ♠❡ ♠✉❞❛r ♦✉ P❡❞r♦ ✜❝❛r ❡♠ ❙ã♦ P❛✉❧♦ ❡♥tã♦ ✈♦❧t♦
❛ ❡st✉❞❛r✳ ▲♦❣♦✱ s❡ ❛ ❝❛s❛ ✜❝❛r ✈❛③✐❛✱ ✈♦❧t♦ ❛ ❡st✉❞❛r✳
❋❛③❡♥❞♦✿
✹✷❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
p✿ ❛ ❝❛s❛ ✜❝❛r ✈❛③✐❛
q✿ ❡✉ ❝♦♥s❡❣✉✐r ♦ ❡♠♣rést✐♠♦
r✿ ❡✉ ♣❛❣❛r ❛ ❞í✈✐❞❛
s✿ ♠❡ ♠✉❞❛r
t✿ P❡❞r♦ ✜❝❛r ❡♠ ❙ã♦ P❛✉❧♦
u✿ ✈♦❧t❛r ❛ ❡st✉❞❛r
t❡♠♦s ♦ ❛r❣✉♠❡♥t♦
p ∨ q → r ∧ s
s ∨ t→ u
p u→
❯t✐❧✐③❛♥❞♦ ❉❡❞✉çã♦ ❞❛ ❈♦♥❞✐❝✐♦♥❛❧✱ ✐♥❝❧✉♦ p ♥❛s ♣r❡♠✐ss❛s ❡ ❛ ❝♦♥❝❧✉sã♦ s❡
r❡❞✉③ ❛ u✿
p ∨ q → r ∧ s
s ∨ t→ u
p
u
✶✳ p ∨ q → r ∧ s Pr❡♠✐ss❛ ✶
✷✳ s ∨ t→ u Pr❡♠✐ss❛ ✷
✸✳ p Pr❡♠✐ss❛ ✸
✹✳ p ∨ q ✸✱ ❆❉
✺✳ r ∧ s ✹✱ ✶✱ ▼P
✻✳ s ✺✱ ❙■▼P
✼✳ s ∨ t ✻✱ ❆❉
✽✳ u ✼✱✷✱ ▼P
❈♦♥❝❧✉s❛♦ ❞❡ ❢♦r♠❛ A ∨B
❙❛❜❡♠♦s q✉❡✱ ♣♦r ❈❖◆❉✱ q✉❡
A ∨B ≡ ¬A→ B
✹✸❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
P♦rt❛♥t♦✱ s❡ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ❛r❣✉♠❡♥t♦ t❡♠ ❛ ❢♦r♠❛ A∨B✱ ♣♦❞❡♠♦s s✉❜st✐t✉í✲
❧❛ ♣♦r A B✱ ❡✱ ✉t✐❧✐③❛♥❞♦ ❉❡❞✉çã♦ ❞❛ ❈♦♥❞✐❝✐♦♥❛❧✱ ✐♥❝❧✉✐r A ♥❛s ♣r❡♠✐ss❛s¬ → ¬
❡ ❞❡❞✉③✐r B✳
❊①❡♠♣❧♦✿
❖✉ ♣❛❣❛♠♦s ❛ ❞í✈✐❞❛ ♦✉ ♦ ❞é✜❝✐t ❛✉♠❡♥t❛❀ s❡ ❛s ❡①♣♦rt❛çõ❡s ❝r❡s❝❡r❡♠✱
♦ ❞é✜❝✐t ♥ã♦ ❛✉♠❡♥t❛✳ ▲♦❣♦✱ ♦✉ ♣❛❣❛♠♦s ❛ ❞í✈✐❞❛ ♦✉ ❛s ❡①♣♦rt❛çõ❡s
♥ã♦ ❝r❡s❝❡♠✳
❋❛③❡♥❞♦✿
p✿ ♣❛❣❛r ❛ ❞í✈✐❞❛
q✿ ♦ ❞é✜❝✐t ❛✉♠❡♥t❛r
r✿ ❛s ❡①♣♦rt❛çõ❡s ❝r❡s❝❡r❡♠
t❡♠♦s ♦ ❛r❣✉♠❡♥t♦
p ∨ q
r q→ ¬
p ∨ ¬r
❈♦♠♦ ❛ ❝♦♥❝❧✉sã♦ p ∨ ¬r é ❡q✉✐✈❛❧❡♥t❡ ❛ ¬p → ¬r❀ ❡♥tã♦✱ ♣❡❧❛ ❉❡❞✉çã♦ ❞❛
❈♦♥❞✐❝✐♦♥❛❧✱ ♦ ❛r❣✉♠❡♥t♦ ❛ss✉♠❡ ❛ ❢♦r♠❛ ❛❜❛✐①♦✱ ❝♦♠ ❛ ❞❡❞✉çã♦ ❛ s❡❣✉✐r✿
p ∨ q
r q→ ¬
¬p
r¬
✶✳ p ∨ q Pr❡♠✐ss❛ ✶
✷✳ r → ¬q Pr❡♠✐ss❛ ✷
✸✳ ¬p Pr❡♠✐ss❛ ✸
✹✳ q ✶✱ ✸✱ ❙❉
✺✳ ¬r ✷✱ ✹✱ ▼❚
❯♠❛ ♦✉tr❛ ❢♦r♠❛ ❞❡ ❞❡❞✉③✐r ✉♠❛ ❞✐s❥✉♥çã♦✱ é ♦❜t❡r ✉♠ ❞♦s ❞✐s❥✉♥t♦s✱ ❡✱ ♣♦r
❛❞✐çã♦✱ ✐♥❝❧✉✐r ♦ ♦✉tr♦✳
✹✹❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ♦ ❛r❣✉♠❡♥t♦ ❛❜❛✐①♦✱ ❥á ♥❛ s✉❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛✿
( p ∨ q) ∧ (r s)¬ →
¬q
¬p ∨ s
❈♦♠♦ ❛ ❝♦♥❝❧✉sã♦ é p ∨ s✱ ♣♦❞❡♠♦s ❞❡❞✉③✐r p✱ ❡✱ ♣♦r ❛❞✐çã♦✱ p ∨ s✱ ♦✉¬ ¬ ¬
❞❡❞✉③✐r s✱ ❡✱ ♣♦r ❛❞✐çã♦✱ p ∨ s✳¬
✶✳ (¬p ∨ q) ∧ (r → s) Pr❡♠✐ss❛ ✶
✷✳ ¬q Pr❡♠✐ss❛ ✷
✸✳ ¬p ∨ q ✶✱ ❙■▼P
✹✳ ¬p ✷✱ ✸✱ ❙❉
✺✳ ¬p ∨ s ✹✱ ❆❉
❈♦♥❝❧✉sã♦ ❞❡ ❢♦r♠❛ A B↔
❙❛❜❡♠♦s✱ ♣♦r ❇■❈❖◆❉✱ q✉❡
A B ≡ (A B) ∧ (B A)↔ → →
❊♥tã♦✱ s❡ ❛ ❝♦♥❝❧✉sã♦ t❡♠ ❛ ❢♦r♠❛ A B✱ ♣♦❞❡♠♦s s✉❜st✐t✉í❂❧❛ ♣♦r (A↔ →
B) ∧ (B → A)✱ ♦ q✉❡ ✐♥❞✐❝❛ q✉❡ t❡♠♦s q✉❡ ❞❡❞✉③✐r✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡✱ A → B
❡ B A✳ ❯t✐❧✐③❛♥❞♦ ❉❡❞✉çã♦ ❞❛ ❈♦♥❞✐❝✐♦♥❛❧✱ ♣♦❞❡♠♦s✱ ♥❛ ❞❡❞✉çã♦ ❞❡ A B✱→ →
✐♥❝❧✉✐r A ♥❛s ♣r❡♠✐ss❛s ❡ ❞❡❞✉③✐r B✱ ❡✱ ♥❛ ❞❡❞✉çã♦ ❞❡ B A✱ ✐♥❝❧✉✐r B ♥❛s→
♣r❡♠✐ss❛s ❡ ❞❡❞✉③✐r A✳
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ♦ ❛r❣✉♠❡♥t♦ ❛❜❛✐①♦✱ ❥á ♥❛ s✉❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛✿
p ∧ q → r
r ∨ q → ¬p ∨ s
s q→
p
r s↔
❚❡♠♦s ❡♥tã♦ q✉❡ ❢❛③❡r ❞✉❛s ❞❡❞✉çõ❡s✱ ✉♠❛ ♣❛r❛ r s✱ ♣❛r❛ ❛ q✉❛❧ ✐♥❝❧✉í♠♦s→
r ♥❛s ♣r❡♠✐ss❛s ❡ ❞❡❞✉③✐♠♦s s✱ ❡ ♦✉tr❛ ♣❛r❛ s r✱ ♥❛ q✉❛❧ ✐♥❝❧✉í♠♦s s ♥❛s→
♣r❡♠✐ss❛s ❡ ❞❡❞✉③✐♠♦s r✳ ❆s ❞✉❛s ❞❡❞✉çõ❡s ❞❡✈❡♠ s❡r r❡❛❧✐③❛❞❛s ❡♠ s❡♣❛r❛❞♦✱
♣♦✐s ♦s r❡s✉❧t❛❞♦s ✐♥t❡r♠❡❞✐ár✐♦s ❞❡ ✉♠❛ ♥ã♦ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♥❛ ♦✉tr❛✳
✹✺❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
✶✳ p ∧ q → r Pr❡♠✐ss❛ ✶
✷✳ r ∨ q → ¬p ∨ s Pr❡♠✐ss❛ ✷
✸✳ s→ q Pr❡♠✐ss❛ ✸
✹✳ p Pr❡♠✐ss❛ ✹
✺✳ r Pr❡♠✐ss❛ ✺
✻✳ r ∨ q ✺✱ ❆❉
✼✳ ¬p ∨ s ✷✱ ✺✱ ▼P
✽✳ s ✹✱ ✼✱ ❙❉
✶✳ p ∧ q → r Pr❡♠✐ss❛ ✶
✷✳ r ∨ q → ¬p ∨ s Pr❡♠✐ss❛ ✷
✸✳ s→ q Pr❡♠✐ss❛ ✸
✹✳ p Pr❡♠✐ss❛ ✹
✺✳ s Pr❡♠✐ss❛ ✺
✻✳ q ✸✱ ✺✱ ▼P
✼✳ p ∧ q ✹✱ ✻✱ ❈❖◆❏
✽✳ r ✶✱ ✼✱ ▼P
❆❧t❡r♥❛t✐✈❛♠❡♥t❡✱ ♣♦❞❡rí❛♠♦s✱ ♥❛ ❞❡❞✉çã♦ ❞❛ ❜✐❝♦♥❞✐❝✐♦♥❛❧✱ ✉t✐❧✐③❛r ❛ ♦✉tr❛
❡q✉✐✈❛❧ê♥❝✐❛ ❇■❈❖◆❉✱
A↔ B ≡ (A ∧ B) ∨ (¬A ∧ ¬B)
◆❡ss❡ ❝❛s♦✱ ❜❛st❛r✐❛ ❞❡❞✉③✐r A ∧ B✱ ♦✉ ❞❡❞✉③✐r ¬A ∧ ¬B✱ ❡ ✐♥❝❧✉✐r ♦ ♦✉tr♦
❞✐s❥✉♥t♦ ♣♦r ❆❞✐çã♦✳
❊①❡♠♣❧♦✿
p ∧ q
p (q (s ∧ t))→ →
s t↔
✶✳ p ∧ q Pr❡♠✐ss❛ ✶
✷✳ p→ (q → (s ∧ t)) Pr❡♠✐ss❛ ✷
✸✳ p ∧ q → s ∧ t ✷✱ ❊■
✹✳ s ∧ t ✶✱ ✸✱ ▼P
✺✳ (s ∧ t) ∨ (¬s ∧ ¬t) ✹✱ ❆❉
✻✳ s↔ t ✺✱ ❇■❈❖◆❉
❉❡❞✉çã♦ ♣♦r ❆❜s✉r❞♦
❈♦♥s✐❞❡r❡ ♦ ❛r❣✉♠❡♥t♦ P Q✱ ♦♥❞❡ P é ❛ ❝♦♥❥✉♥çã♦ ❞❡ ♣r❡♠✐ss❛s✱ ❡ Q é ❛→
❝♦♥❝❧✉sã♦✳ ❊♥tã♦✱ s❡ ♦ ❛r❣✉♠❡♥t♦ ❢♦r ✈á❧✐❞♦✱ ✐st♦ é✱ s❡ P Q ❢♦r ✉♠❛ t❛✉t♦❧♦❣✐❛✱→
¬P ∨Q t❛♠❜é♠ ♦ s❡rá✱ ♣❡❧❛ ❡q✉✐✈❛❧ê♥❝✐❛ ❈❖◆❉❀ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s✉❛ ♥❡❣❛çã♦✱
¬(¬P ∨Q)✱ q✉❡✱ ♣♦r ❉❡ ▼♦r❣❛♥✱ é ❡q✉✐✈❛❧❡♥t❡ ❛ P ∧ ¬Q✱ s❡rá ✉♠❛ ❝♦♥tr❛❞✐çã♦✳
✹✻❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
❊♥tã♦✱ ♣❛r❛ ♠♦str❛r♠♦s q✉❡ ♦ ❛r❣✉♠❡♥t♦ P Q é ✈á❧✐❞♦✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r→
q✉❡ P ∧ ¬Q é ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❖✉ s❡❥❛✱ ♣❛r❛ ♠♦str❛r♠♦s q✉❡ ✉♠ ❛r❣✉♠❡♥t♦
é ✈á❧✐❞♦✱ ♣♦❞❡♠♦s ♥❡❣❛r ❛ ❝♦♥❝❧✉sã♦✱ ✐♥❝❧✉í✲❧❛ ♥❛s ♣r❡♠✐ss❛s ❡ ❞❡❞✉③✐r F ✱ q✉❡
r❡♣r❡s❡♥t❛ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❊ss❛ ❢♦r♠❛ ❞❡ ❞❡❞✉③✐r ✉♠ ❛r❣✉♠❡♥t♦ é ❝♦♥❤❡❝✐❞❛
♣♦r ❉❡❞✉çã♦ ♣♦r ❆❜s✉r❞♦✳
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ♦ ❛r❣✉♠❡♥t♦ ❛❜❛✐①♦✱ ❥á ♥❛ s✉❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛✿
p q ∨ r→
q p→ ¬
s r→ ¬
¬(p ∧ s)
❯t✐❧✐③❛♥❞♦ ❉❡♠♦♥str❛çã♦ ♣♦r ❛❜s✉r❞♦✱ ✐♥❝❧✉í♠♦s p ∧ s ♥❛s ♣r❡♠✐ss❛s ❡ ❞❡❞✉③✲
✐♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✿
p q ∨ r→
q p→ ¬
s r→ ¬
p ∧ s
F
✶✳ p→ q ∨ r Pr❡♠✐ss❛ ✶
✷✳ q → ¬p Pr❡♠✐ss❛ ✷
✸✳ s→ ¬r Pr❡♠✐ss❛ ✸
✹✳ p ∧ s Pr❡♠✐ss❛ ✹
✺✳ p ✹✱ ❙■▼P
✻✳ s ✹✱ ❙■▼P
✼✳ q ∨ r ✺✱ ✶✱ ▼P
✽✳ ¬r ✻✱ ✸✱ ▼P
✾✳ q ✼✱ ✽✱ ❙❉
✶✵✳ ¬p ✾✱ ✷✱ ▼P
✶✶✳ ¬p ∧ p ✶✵✱ ✺✱ ❈❖◆❏
✶✷✳ F ✶✶✱ ❊❈
❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲ ✹✼
✸✳✹ ■♥✈❛❧✐❞❛❞❡
❖s ♠ét♦❞♦s ❞❡ ❞❡❞✉çã♦ q✉❡ ❡①❛♠✐♥❛♠♦s sã♦ ❝❛♣❛③❡s ❞❡ ♠♦str❛r ❛ ✈❛❧✐❞❛❞❡ ❞❡ ✉♠
❛r❣✉♠❡♥t♦✱ ❛tr❛✈és ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛s ❡ ✐♥❢❡rê♥❝✐❛s✱ q✉❡✱ ❛ ♣❛rt✐r
❞❛s ♣r❡♠✐ss❛s✱ ♣r♦❞✉③ ✉♠❛ sér✐❡ ❞❡ ❝♦♥❝❧✉sõ❡s ♣❛r❝✐❛✐s✱ ❛té ❝❤❡❣❛r à ❝♦♥❝❧✉sã♦
✜♥❛❧ ❞♦ ❛r❣✉♠❡♥t♦✳
❊ss❡ ♣r♦❝❡ss♦✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ s❡r✈❡ ♣❛r❛ ♣r♦✈❛r ❛ ✐♥✈❛❧✐❞❛❞❡ ❞❡ ✉♠ ❛r❣✉♠❡♥t♦❀
❞❡ ❢❛t♦✱ s❡✱ ❞✉r❛♥t❡ ♦ ♣r♦❝❡ss♦ ❞❡ ❞❡❞✉çã♦✱ ♥ã♦ ❝♦♥s❡❣✉✐r♠♦s ❝❤❡❣❛r à ❝♦♥❝❧✉sã♦✱
♥ã♦ ♣♦❞❡♠♦s ✐♥❢❡r✐r q✉❡ ♥ã♦ é ♣♦ssí✈❡❧ ♦❜t❡r t❛❧ ❝♦♥❝❧✉sã♦❀ t❛❧✈❡③ ❛♣❡♥❛s ♥ã♦
s❛✐❜❛♠♦s ❝♦♠♦ ❝❤❡❣❛r ❛ ❡❧❛✳
P♦rt❛♥t♦✱ ♣❛r❛ ♠♦str❛r ❛ ✐♥✈❛❧✐❞❛❞❡ ❞❡ ✉♠ ❛r❣✉♠❡♥t♦✱ ♥❡❝❡ss✐t❛♠♦s ❞❡ ✉♠
♦✉tr♦ ♠ét♦❞♦✳ ❯♠ ❛r❣✉♠❡♥t♦ é✱ ♥❛ ✈❡r❞❛❞❡✱ ✉♠❛ ♦♣❡r❛çã♦ ❞❡ ❝♦♥❞✐❝✐♦♥❛♠❡♥t♦✳
❙❡ ♦ ❛r❣✉♠❡♥t♦ ❢♦r ✈á❧✐❞♦✱ ❡ss❛ ❝♦♥❞✐❝✐♦♥❛❧ é t❛✉t♦❧ó❣✐❝❛✱ ✐st♦ é✱ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛
q✉❛❧q✉❡r ❝♦♠❜✐♥❛çã♦ ♣♦ssí✈❡❧ ❞❡ ✈❛❧♦r❡s ❧ó❣✐❝♦s ❞❛s ♣r♦♣♦s✐çõ❡s q✉❡ ❝♦♥st✐t✉❡♠ ♦
❛r❣✉♠❡♥t♦❀ s❡✱ ♥♦ ❡♥t❛♥t♦✱ ❡①✐st✐r ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ✈❛❧♦r❡s ❧ó❣✐❝♦s
❞❛s ♣r♦♣♦s✐çõ❡s q✉❡ t♦r♥❡ ❛ ❝♦♥❞✐❝✐♦♥❛❧ ❢❛❧s❛✱ ♦ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞♦✳
❖r❛✱ ❡♠ q✉❡ ❝♦♥❞✐çõ❡s ✉♠❛ ❝♦♥❞✐❝✐♦♥❛❧ é ❢❛❧s❛❄ ❙ó ❡①✐st❡ ✉♠❛ ♣♦ss✐❜✐❧✐❞❛❞❡✿
q✉❛♥❞♦ ♦ ❛♥t❡❝❡❞❡♥t❡ é ✈❡r❞❛❞❡✐r♦ ❡ ♦ ❝♦♥s❡q✉❡♥t❡ é ❢❛❧s♦✳ ▼❛s✱ ❡♠ ✉♠ ❛r❣✉♠❡♥t♦✱
♦ ❛♥t❡❝❡❞❡♥t❡ é ✉♠❛ ❝♦♥❥✉♥çã♦ ❞❡ ♣r❡♠✐ss❛s✱ ❡ ♦ ❝♦♥s❡q✉❡♥t❡ é ❛ ❝♦♥❝❧✉sã♦❀ ❡♥tã♦✱
♣❛r❛ q✉❡ ♦ ❛♥t❡❝❡❞❡♥t❡ s❡❥❛ ✈❡r❞❛❞❡✐r♦✱ é ♥❡❝❡ssár✐♦ q✉❡ t♦❞❛s ❛s ♣r❡♠✐ss❛s s❡❥❛♠
✈❡r❞❛❞❡✐r❛s✱ ❡ ♣❛r❛ q✉❡ ♦ ❝♦♥s❡q✉❡♥t❡ s❡❥❛ ❢❛❧s♦✱ é ♥❡❝❡ssár✐♦ q✉❡ ❛ ❝♦♥❝❧✉sã♦ s❡❥❛
❢❛❧s❛✳
❊♥tã♦✱ ♣❛r❛ ♠♦str❛r q✉❡ ✉♠ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞♦✱ é s✉✜❝✐❡♥t❡ ❡♥❝♦♥tr❛r ✉♠❛
❝♦♠❜✐♥❛çã♦ ❞❡ ✈❛❧♦r❡s ❧ó❣✐❝♦s ♣❛r❛ ❛s ♣r♦♣♦s✐çõ❡s s✐♠♣❧❡s ❡♥✈♦❧✈✐❞❛s✱ ❞❡ ❢♦r♠❛
q✉❡ t♦r♥❡ ❝❛❞❛ ♣r❡♠✐ss❛ ✈❡r❞❛❞❡✐r❛✱ ❡ ❛ ❝♦♥❝❧✉sã♦ ❢❛❧s❛✳
❈♦♥s✐❞❡r❡ ♦ s❡❣✉✐♥t❡ ❛r❣✉♠❡♥t♦✿
❙❡ ❏♦sé ❝♦♠♣r❛r ❛çõ❡s ❡ ♦ ♠❡r❝❛❞♦ ❜❛✐①❛r✱ ❡❧❡ ♣❡r❞❡rá s❡✉ ❞✐♥❤❡✐r♦✳ ❖
♠❡r❝❛❞♦ ♥ã♦ ✈❛✐ ❜❛✐①❛r✳ ▲♦❣♦✱ ♦✉ ❏♦sé ❝♦♠♣r❛ ❛çõ❡s ♦✉ ♣❡r❞❡rá s❡✉
❞✐♥❤❡✐r♦✳
❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ ♦ ❛r❣✉♠❡♥t♦ ✜❝❛ r❡♣r❡s❡♥t❛❞♦ ♣♦r
✹✽❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
p ∧ q r→
¬q
p ∨ r
♦♥❞❡✿
p✿ ❏♦sé ❝♦♠♣r❛ ❛çõ❡s
q✿ ♦ ♠❡r❝❛❞♦ ❜❛✐①❛
r✿ ❏♦sé ♣❡r❞❡ ❞✐♥❤❡✐r♦
P❛r❛ ♠♦str❛r s✉❛ ✐♥✈❛❧✐❞❛❞❡ ❞❡✈❡♠♦s ❢❛③❡r ❛s ♣r❡♠✐ss❛s ✈❡r❞❛❞❡✐r❛s ❡ ❛ ❝♦♥❝❧✉sã♦
❢❛❧s❛❀ ✐st♦ é✿
❛✮ p ∧ q → r : V
❜✮ q : V¬
❝✮ p ∨ r : F
❉❡ ✭❜✮ ✈❡♠ q : F ❀ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭❛✮✱ t❡♠♦s✿
❞✮ p ∧ F → r : V
❈♦♠♦ p ∧ F : F ✱ q✉❛❧q✉❡r q✉❡ s❡❥❛ p✱ ❞❡ ✭❞✮✱ ✈❡♠ r : F ❀ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭❝✮✱ ✈❡♠
p ∨ F : F
❖ q✉❡ ✐♠♣❧✐❝❛ q✉❡ p : F
❚❡♠♦s ❡♥tã♦ q✉❡✱ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s ❧ó❣✐❝♦s✿ p : F, q : F, r : F ✱ ❛
❝♦♥❞✐❝✐♦♥❛❧ é ❢❛❧s❛✱ ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ ♦ ❛r❣✉♠❡♥t♦ é ✐♥✈á❧✐❞♦✳
❯♠ ♦✉tr♦ ❡①❡♠♣❧♦❀ ❝♦♥s✐❞❡r❡ ♦ ❛r❣✉♠❡♥t♦
❖✉ ❡st✉❞♦ ♦✉ tr❛❜❛❧❤♦ ♦✉ ✈♦✉ à ♣r❛✐❛❀ s❡ ❡st✉❞♦ s♦✉ ❛♣r♦✈❛❞♦❀ ♥ã♦
tr❛❜❛❧❤♦✳ ▲♦❣♦✱ s♦✉ ❛♣r♦✈❛❞♦✳
❙❡ ✜③❡r♠♦s✿
p✿ ❡✉ ❡st✉❞♦
q✿ ❡✉ tr❛❜❛❧❤♦
r✿ ❡✉ ✈♦✉ à ♣r❛✐❛
s✿ ❡✉ s♦✉ ❛♣r♦✈❛❞♦
✹✾❈❆P❮❚❯▲❖ ✸✳ ❉❊❉❯➬➹❖ ◆❖ ❈➪▲❈❯▲❖ P❘❖P❖❙■❈■❖◆❆▲
♦ ❛r❣✉♠❡♥t♦ ✜❝❛ r❡♣r❡s❡♥t❛❞♦ s✐♠❜♦❧✐❝❛♠❡♥t❡ ♣♦r✿
p ∨ q ∨ r
p s→
¬q
s
P❛r❛ q✉❡ ♦ ❛r❣✉♠❡♥t♦ s❡❥❛ ✐♥✈á❧✐❞♦ é ♥❡❝❡ssár✐♦✿
❛✮ p ∨ q ∨ r : V
❜✮ p s : V→
❝✮ q : V¬
❞✮ s : F
❉❡ ✭❝✮ ❡ ✭❞✮ ✈❡♠✱ ✐♠❡❞✐❛t❛♠❡♥t❡✱ q : F ❡ s : F ❀ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭❛✮ ❡ ✭❜✮✱ ✈❡♠
❡✮ p ∨ F ∨ r : V
❢✮ p F : V→
❉❡ ✭❢✮✱ ✈❡♠ p : F ❀ s✉❜st✐t✉✐♥❞♦ ❡♠ ✭❡✮✱ ✈❡♠ r : V ✳
❊♥❝♦♥tr❛♠♦s ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞❡ ✈❛❧♦r❡s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ❞❡ ✐♥✈❛❧✐✲
❞❛❞❡✳ ❖ ❛r❣✉♠❡♥t♦ é✱ ♣♦rt❛♥t♦✱ ✐♥✈á❧✐❞♦✳
❈❛♣ít✉❧♦ ✹
❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s
✹✳✶ Pr❡❞✐❝❛❞♦s ❡ ❚❡r♠♦s
◆♦s ❝❛♣ít✉❧♦s ❛♥t❡r✐♦r❡s ❡①❛♠✐♥❛♠♦s ✉♠❛ ♣❛rt❡ ❞❛ ▲ó❣✐❝❛ ❝❤❛♠❛❞❛ ▲ó❣✐❝❛ ❞❛s
Pr♦♣♦s✐çõ❡s✱ ♦✉ ❈á❧❝✉❧♦ Pr♦♣♦s✐❝✐♦♥❛❧✱ ♥❛ q✉❛❧ ❛♣r❡♥❞❡♠♦s té❝♥✐❝❛s q✉❡ ♥♦s ♣❡r✲
♠✐t✐r❛♠ ✈❡r✐✜❝❛r s❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ t✐♣♦ ❞❡ ❛r❣✉♠❡♥t♦ é ✈á❧✐❞♦ ♦✉ ✐♥✈á❧✐❞♦✳ ◆♦s
❛r❣✉♠❡♥t♦s ❡st✉❞❛❞♦s✱ ♦s ❡♥✉♥❝✐❛❞♦s s✐♠♣❧❡s ❡r❛♠ ❝♦♠❜✐♥❛❞♦s ❛tr❛✈és ❞♦s ❝♦♥❡❝✲
t✐✈♦s✱ ❢♦r♠❛♥❞♦ ❡♥✉♥❝✐❛❞♦s ❝♦♠♣♦st♦s✱ ❡ ❛ ✈❛❧✐❞❛❞❡ ❞❡ss❡s ❛r❣✉♠❡♥t♦s ❞❡♣❡♥❞✐❛✱
❡ss❡♥❝✐❛❧♠❡♥t❡✱ ❞❛ ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ♦s ❡♥✉♥❝✐❛❞♦s ❝♦♠♣♦st♦s s❡ ❛♣r❡s❡♥t❛✈❛♠✳
◆ã♦ é ❞✐❢í❝✐❧✱ ♥♦ ❡♥t❛♥t♦✱ ❡♥❝♦♥tr❛r ❛r❣✉♠❡♥t♦s ❞❡ ✉♠ t✐♣♦ ❞✐st✐♥t♦❀ ♣♦r ❡①❡♠✲
♣❧♦✱ ♦ ❛r❣✉♠❡♥t♦
❚♦❞♦s ♦s ❤✉♠❛♥♦s sã♦ ♠♦rt❛✐s
❙ó❝r❛t❡s é ✉♠ ❤✉♠❛♥♦
▲♦❣♦✱ ❙ó❝r❛t❡s é ♠♦rt❛❧
é ❝❧❛r❛♠❡♥t❡ ✈á❧✐❞♦✱ ♠❛s s✉❛ ✈❛❧✐❞❛❞❡ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❢♦r♠❛ ♣❡❧❛ q✉❛❧ ♦s ❡♥✉♥✲
❝✐❛❞♦s s✐♠♣❧❡s s❡ ❝♦♠♣õ❡♠✱ ✉♠❛ ✈❡③ q✉❡✱ ♥❡st❡ ❛r❣✉♠❡♥t♦✱ ♥ã♦ ❤á ❡♥✉♥❝✐❛❞♦s
❝♦♠♣♦st♦s✳ P♦❞❡✲s❡ ♣❡r❝❡❜❡r q✉❡ s✉❛ ✈❛❧✐❞❛❞❡ ❞❡♣❡♥❞❡✱ ♥❛ ✈❡r❞❛❞❡✱ ❞❛ ❡str✉t✉r❛
✐♥t❡r♥❛ ❞♦s ❡♥✉♥❝✐❛❞♦s q✉❡ ❝♦♥st✐t✉❡♠ ♦ ❛r❣✉♠❡♥t♦✳ ❆ ❝♦♥str✉çã♦ ❞❡ ♠ét♦❞♦s
♣❛r❛ ❛♥❛❧✐s❛r ❛r❣✉♠❡♥t♦s ❝♦♠♦ ❡ss❡ ✈❛✐✱ ♣♦rt❛♥t♦✱ ❡①✐❣✐r ❛ ❝r✐❛çã♦ ❞❡ té❝♥✐❝❛s
♣❛r❛ ❞❡s❝r❡✈❡r ❡ s✐♠❜♦❧✐③❛r ❛ ❡str✉t✉r❛ ✐♥t❡r♥❛ ❞♦s ❡♥✉♥❝✐❛❞♦s✳
❈♦♥s✐❞❡r❡ ❛ ♣r❡♠✐ss❛ ✏❙ó❝r❛t❡s é ❤✉♠❛♥♦✑✳ ❊ss❡ ❡♥✉♥❝✐❛❞♦ é ✉♠❛ ❞❡❝❧❛r❛çã♦
❞❡ q✉❡ ❞❡t❡r♠✐♥❛❞♦ ✐♥❞✐✈í❞✉♦ ✭❙ó❝r❛t❡s✮ ♣♦ss✉✐ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❡s♣❡❝í✜❝❛ ✭é
✺✵
✺✶❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙
❤✉♠❛♥♦✮✳ ◆❛ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧✱ ♦ ✐♥❞✐✈í❞✉♦ q✉❡ ♣♦ss✉✐ ❛ ♣r♦♣r✐❡❞❛❞❡ é ❝❤❛♠❛❞♦
s✉❥❡✐t♦✱ ❡♥q✉❛♥t♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❝r✐t❛ é ❝❤❛♠❛❞❛ ♣r❡❞✐❝❛❞♦✳
❖ ♣r❡❞✐❝❛❞♦✱ ♥❛ ✈❡r❞❛❞❡✱ ❡①♣❧✐❝✐t❛ ❝❡rt❛s q✉❛❧✐❞❛❞❡s q✉❡ ♦ s✉❥❡✐t♦ ♣♦ss✉✐ ❡
q✉❡ ♣❡r♠✐t❡ ✐♥❝❧✉í✲❧♦ ❡♠ ✉♠❛ ❝❛t❡❣♦r✐❛❀ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ ❞✐③❡♠♦s ✏❙ó❝r❛t❡s
é ❤✉♠❛♥♦✑ q✉❡r❡♠♦s ❞✐③❡r q✉❡ ♦ ♦❜❥❡t♦ ❝❤❛♠❛❞♦ ✏❙ó❝r❛t❡s✑ ♣♦ss✉✐ ❝❡rt❛s ❝❛r❛❝✲
t❡ríst✐❝❛s q✉❡ ♣❡r♠✐t❡♠ ✐♥❝❧✉í✲❧♦ ♥♦ ❝♦♥❝❡✐t♦ q✉❡ ❢❛③❡♠♦s ❞❛q✉✐❧♦ q✉❡ ❝❤❛♠❛♠♦s
✏❤✉♠❛♥♦✑✳
❊♠ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ♣r❡❞✐❝❛❞♦ ♣♦r s✉❛ ✐♥✐❝✐❛❧ ♠❛✐ús✲
❝✉❧❛✱ ❡ ♦ s✉❥❡✐t♦ ❛ s❡❣✉✐r✱ ❡♥tr❡ ♣❛rê♥t❡s❡s✳ ❆ss✐♠✱ ✏❙ó❝r❛t❡s é ❤✉♠❛♥♦✑ é r❡♣r❡✲
s❡♥t❛❞♦ ♣♦r
H(❙ó❝r❛t❡s)
❆ ❧✐♥❣✉❛❣❡♠ ♥❛t✉r❛❧ ♣❡r♠✐t❡ ❛✐♥❞❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♦✉tr♦ t✐♣♦ ❞❡ s❡♥t❡♥ç❛✱
❝♦♠♦ ✏❡❧❡ ❢♦✐ ♣r❡s✐❞❡♥t❡ ❞♦ ❇r❛s✐❧✑ ♥❛ q✉❛❧ ♦ s✉❥❡✐t♦ ♥ã♦ é ✉♠ s✉❜st❛♥t✐✈♦✱ ♠❛s
✉♠ ♣r♦♥♦♠❡✱ ✐st♦ é✱ ✉♠ t❡r♠♦ q✉❡ ✜❝❛ ♥♦ ❧✉❣❛r ❞♦ ♥♦♠❡✳
❊♠ ▲ó❣✐❝❛ ❙✐♠❜ó❧✐❝❛✱ t❛♠❜é♠ ❡①✐st❡♠ t❡r♠♦s q✉❡ ♦❝✉♣❛♠ ♦ ❧✉❣❛r ❞♦s ♥♦♠❡s❀
sã♦ ❝❤❛♠❛❞♦s ✈❛r✐á✈❡✐s✱ ❡ ❝♦st✉♠❛♠ s❡r r❡♣r❡s❡♥t❛❞♦s✱ ❝♦♠♦ ♥❛ ▼❛t❡♠át✐❝❛✱ ♣❡❧❛s
ú❧t✐♠❛s ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ ❡♠ ♠✐♥ús❝✉❧❛s✱ x✱ y✱ z✱ w ❡t❝✳ ❯t✐❧✐③❛♥❞♦ ❛ ✈❛r✐á✈❡❧ x
♥♦ ❧✉❣❛r ❞❡ ✏❡❧❡✑✱ ❛ s❡♥t❡♥ç❛ ❛ss✉♠❡ ❛ ❢♦r♠❛
x ❢♦✐ ♣r❡s✐❞❡♥t❡ ❞♦ ❇r❛s✐❧
❘❡♣r❡s❡♥t❛♥❞♦ ♦ ♣r❡❞✐❝❛❞♦ ✏❢♦✐ ♣r❡s✐❞❡♥t❡ ❞♦ ❇r❛s✐❧✑ ♣♦r P ✱ ❡ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥t❛
q✉❡ x é ♦ s✉❥❡✐t♦✱ t❡♠♦s ❛ ❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛
P (x)
❊♠ ▲ó❣✐❝❛✱ ♣♦rt❛♥t♦✱ ✉♠ ❡♥✉♥❝✐❛❞♦ é s✐♠❜♦❧✐③❛❞♦ ♣❡❧♦ ♣r❡❞✐❝❛❞♦ s❡❣✉✐❞♦ ♣♦r
t❡r♠♦s ✭❝♦♥st❛♥t❡s ♦✉ ✈❛r✐á✈❡✐s✮ ❡♥tr❡ ♣❛rê♥t❡s❡s✳
❯♠❛ ❢r❛s❡ ♥❛ q✉❛❧ ♦ t❡r♠♦ é ✉♠❛ ❝♦♥st❛♥t❡✱ ❝♦♠♦ ✏❙ó❝r❛t❡s é ❤✉♠❛♥♦✑✱ ♣♦❞❡
s❡r ✈❡r❞❛❞❡✐r❛ ♦✉ ❢❛❧s❛ ❡✱ ♥❡st❡ ❝❛s♦✱ tr❛t❛✲s❡ ❞❡ ✉♠ ❡♥✉♥❝✐❛❞♦ s✐♥❣✉❧❛r ✳ ▼❛s s❡
♦ t❡r♠♦ ❢♦r ✉♠❛ ✈❛r✐á✈❡❧✱ ❝♦♠♦ ❡♠ ✏❡❧❡ ❢♦✐ ♣r❡s✐❞❡♥t❡ ❞♦ ❇r❛s✐❧✑ ✱ ❛ ❢r❛s❡ ♥ã♦ é
✈❡r❞❛❞❡✐r❛ ♥❡♠ ❢❛❧s❛ ❡ ❞❡♣❡♥❞❡ ❞♦ ♥♦♠❡ q✉❡ ❛ss✉♠❛ ♦ ❧✉❣❛r ❞♦ ♣r♦♥♦♠❡✱ ♥ã♦
s❡♥❞♦✱ ♣♦rt❛♥t♦✱ ✉♠ ❡♥✉♥❝✐❛❞♦ s✐♥❣✉❧❛r✳
✺✷❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙
❖s ❡♥✉♥❝✐❛❞♦s s✐♥❣✉❧❛r❡s sã♦ ❝❤❛♠❛❞♦s s❡♥t❡♥ç❛s ❢❡❝❤❛❞❛s✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡✱
❢❡❝❤❛❞♦s ✱ ❡♥q✉❛♥t♦ q✉❡ ❢r❛s❡s ❝♦♠♦ ✏x ❢♦✐ ♣r❡s✐❞❡♥t❡ ❞♦ ❇r❛s✐❧✑ ✱ ✏y ❡s❝r❡✈❡✉ ❖s
▲✉sí❛❞❛s✑ ❡ ✏z ✈✐❛❥♦✉ ♣❛r❛ ♦s ❊st❛❞♦s ❯♥✐❞♦s✑ sã♦ ❝❤❛♠❛❞❛s s❡♥t❡♥ç❛s ❛❜❡rt❛s✱
♦✉✱ s✐♠♣❧❡s♠❡♥t❡✱ ❛❜❡rt♦s ✳
❖s ❛❜❡rt♦s ♥ã♦ sã♦ ✈❡r❞❛❞❡✐r♦s ♥❡♠ ❢❛❧s♦s❀ ♣♦❞❡♠♦s ❞✐③❡r ❛♣❡♥❛s q✉❡ sã♦ s❛t✐s✲
❢❡✐t♦s ♣❛r❛ ❝❡rt♦s ✈❛❧♦r❡s ❞❛s ✈❛r✐á✈❡✐s✱ ❡ ♥ã♦ s❛t✐s❢❡✐t♦s ♣❛r❛ ♦✉tr♦s✳ ❆ s✉❜st✐t✉✐çã♦
❞❛s ✈❛r✐á✈❡✐s ❞❡ ✉♠ ❛❜❡rt♦ ♣♦r ❝♦♥st❛♥t❡s ❝❤❛♠❛✲s❡ ✐♥st❛♥❝✐❛çã♦ ♦✉ ❡s♣❡❝✐✜❝❛çã♦❀
❛ ✐♥st❛♥❝✐❛çã♦ tr❛♥s❢♦r♠❛ ✉♠ ❛❜❡rt♦ ❡♠ ✉♠ ❡♥✉♥❝✐❛❞♦ s✐♥❣✉❧❛r✱ q✉❡✱ ❡st❡ s✐♠✱
♣♦❞❡ s❡r ✈❡r❞❛❞❡✐r♦ ♦✉ ❢❛❧s♦✳
❈❤❛♠❛✲s❡ ❈♦♥❥✉♥t♦✲❯♥✐✈❡rs♦ ✭U✮ ❞❡ ✉♠❛ ✈❛r✐á✈❡❧ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈❛❧♦r❡s q✉❡
❡❧❛ ♣♦❞❡ ❛ss✉♠✐r✳ ◆❛ ❧✐♥❣✉❛❣❡♠ ❝♦rr❡♥t❡✱ ♦ ✉♥✐✈❡rs♦ ✭às ✈❡③❡s ❝❤❛♠❛❞♦ ✉♥✐✈❡rs♦
❞♦ ❞✐s❝✉rs♦✮ ♥ã♦ é✱ ♠✉✐t❛s ✈❡③❡s✱ ❡①♣❧✐❝✐t❛❞♦❀ ✐♥t✉✐t✐✈❛♠❡♥t❡✱ ✐♥❝❧✉í♠♦s ♦s ♦❜❥❡t♦s
q✉❡ ♣♦❞❡♠ s✉❜st✐t✉✐r ♦ ♣r♦♥♦♠❡ ❡ ❞❡s❝❛rt❛♠♦s ❛q✉❡❧❡s ♦❜❥❡t♦s q✉❡ s❛❜❡♠♦s q✉❡
♥ã♦ ♣♦❞❡♠❀ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❢r❛s❡
✏✐st♦ ❡stá ✈❡r❞❡✑
s❛❜❡♠♦s q✉❡ ✏✐st♦✑
♣♦❞❡ s❡r ✉♠❛ ❢r✉t❛✱ ♦✉ ✉♠❛ ♣❛r❡❞❡✱ ♦✉ ♦ ♠❛r✱ ♠❛s q✉❡
❞✐✜❝✐❧♠❡♥t❡ s❡rá ✉♠ s❡r ❤✉♠❛♥♦✳
❊♠ ▲ó❣✐❝❛✱ ♦ ✉♥✐✈❡rs♦ ❞♦ ❞✐s❝✉rs♦✱ q✉❛♥❞♦ ♥ã♦ ❢♦r ❡①♣❧✐❝✐t❛❞♦✱ é ❞❡✜♥✐❞♦ ♣❡❧♦
♣ró♣r✐♦ ❝♦♥t❡①t♦✳ ▼✉✐t❛s ✈❡③❡s✱ ❛ ❞❡✜♥✐çã♦ ❞♦ ✉♥✐✈❡rs♦ ♣♦❞❡ ❛❢❡t❛r ❛ s❛t✐s❢❛t♦✲
r✐❡❞❛❞❡ ❞♦ ❛❜❡rt♦❀ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❛❜❡rt♦
x é ❢❡r♦③
♣♦❞❡ s❡r s❛t✐s❢❛tí✈❡❧ s❡ ♦ ✉♥✐✈❡rs♦ ❢♦r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❛♥✐♠❛✐s✱ ❡ ♥ã♦ s❛t✐s❢❛tí✈❡❧
s❡ ♦ ✉♥✐✈❡rs♦ ❢♦r ♦ ❝♦♥❥✉♥t♦ ❞❡ ❞✐s❝✐♣❧✐♥❛s ❞❡ ✉♠ ❝✉rs♦✳
❈❤❛♠❛✲s❡ ❈♦♥❥✉♥t♦✲❱❡r❞❛❞❡ ✭VP ✮ ❞❡ ✉♠ ❛❜❡rt♦ P (x) ♦ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥✲
t♦s ❞♦ ❈♦♥❥✉♥t♦✲❯♥✐✈❡rs♦ q✉❡✱ q✉❛♥❞♦ ✐♥st❛♥❝✐❛♠ ❛ ✈❛r✐á✈❡❧✱ s❛t✐s❢❛③❡♠ ✭t♦r♥❛♠
✈❡r❞❛❞❡✐r♦✮ ♦ ❡♥✉♥❝✐❛❞♦❀ ♦✉ s❡❥❛
VP = {x ∈ U |vl(P (x)) = V }
P♦r ❡①❡♠♣❧♦✱ s❡❥❛ U = {1, 2, 3, 4, 5, 6, 7} ❡ ❛ ❡①♣r❡ssã♦ ✏x é ♣r✐♠♦✑ r❡♣r❡s❡♥t❛❞❛
♣♦r P (x)✳ ❚❡♠♦s ❡♥tã♦ VP = {2, 3, 5, 7}✳
❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙ ✺✸
❖s ♣r❡❞✐❝❛❞♦s ♣♦❞❡♠ s❡r ♠♦♥á❞✐❝♦s ✭❞❡ ✉♠ só t❡r♠♦✮✱ ❞✐á❞✐❝♦s ✭❞❡ ❞♦✐s t❡r✲
♠♦s✮✱ tr✐á❞✐❝♦s ✭❞❡ três t❡r♠♦s✮ ♦✉ ♣♦❧✐á❞✐❝♦s ✭❞❡ q✉❛tr♦ ♦✉ ♠❛✐s t❡r♠♦s✮✳ ▼✉✐t♦s
❛✉t♦r❡s ♥♦ ❡♥t❛♥t♦✱ ♣r❡❢❡r❡♠ ♦ ❝❤❛♠❛r ♦s ♣r❡❞✐❝❛❞♦s ❞❡ ❞♦✐s ♦✉ ♠❛✐s t❡r♠♦s ❞❡
r❡❧❛çã♦✱ r❡s❡r✈❛♥❞♦ ♦ ♥♦♠❡ ♣r❡❞✐❝❛❞♦ ♣❛r❛ ♦s ♣r❡❞✐❝❛❞♦s ♠♦♥á❞✐❝♦s✳
❊✐s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ r❡❧❛çõ❡s✱ ❡ ✉♠❛ s✉❣❡stã♦ ❞❡ ❢♦r♠❛ s✐♠❜ó❧✐❝❛✿
x ❣♦st❛ ❞❡ y✿ G(x, y)
❏♦ã♦ é ❝❛s❛❞♦ ❝♦♠ ▼❛r✐❛✿ C(❏♦ã♦✱ ▼❛r✐❛)
x ❡stá ❡♥tr❡ y ❡ z✿ E(x, y, z)
❈❛♠õ❡s é ♦ ❛✉t♦r ❞❡ ❖s ▲✉sí❛❞❛s✿ A(❈❛♠õ❡s✱ ❖s ▲✉sí❛❞❛s)
◆❛s r❡❧❛çõ❡s✱ ❛ ♦r❞❡♠ ❞❛s ✈❛r✐á✈❡✐s é ✐♠♣♦rt❛♥t❡❀ ♥♦ ❡①❡♠♣❧♦ ❞❛❞♦✱ G(x, y) s✐❣✲
♥✐✜❝❛ ✏x ❣♦st❛ ❞❡ y✑ ♠❛s ♥ã♦ s✐❣♥✐✜❝❛ ✏y ❣♦st❛ ❞❡ x✑✳ ❊ss❡ ❢❛t♦ ❞❡✈❡ s❡r ❧❡✈✲
❛❞♦ ❡♠ ❝♦♥t❛ ♠❡s♠♦ ❡♠ ♣r❡❞✐❝❛❞♦s q✉❡ s❛❜❡♠♦s s❡r ❝♦♠✉t❛t✐✈♦s❀ ♥♦ ❡①❡♠♣❧♦✱
C(❏♦ã♦✱ ▼❛r✐❛) s✐❣♥✐✜❝❛ ✏❏♦ã♦ é ❝❛s❛❞♦ ❝♦♠ ▼❛r✐❛✑ ♠❛s ♥ã♦ s✐❣♥✐✜❝❛ ✏▼❛r✐❛ é
❝❛s❛❞❛ ❝♦♠ ❏♦ã♦✑✳ ❖ ♠♦t✐✈♦ ♣❛r❛ ✐ss♦ é q✉❡ ❛ ▲ó❣✐❝❛ ❋♦r♠❛❧ ❧❡✈❛ ❡♠ ❝♦♥t❛ ❛♣❡♥❛s
❛ ❢♦r♠❛ ❞❛s ❡①♣r❡ssõ❡s✱ ❡ ♥ã♦ s❡✉ s✐❣♥✐✜❝❛❞♦✳
◆❛ ✐♥st❛♥❝✐❛çã♦✱ ✈❛r✐á✈❡✐s ✐❣✉❛✐s ❞❡✈❡♠ s❡r s✉❜st✐t✉í❞❛s ♣♦r ♥♦♠❡s ✐❣✉❛✐s❀ ✈❛r✲
✐á✈❡✐s ❞✐st✐♥t❛s✱ ♥♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠ s❡r s✉❜st✐t✉í❞❛s ♣♦r ♥♦♠❡s ✐❣✉❛✐s ♦✉ ❞✐st✐♥t♦s✳
P♦r ❡①❡♠♣❧♦✱ ♦ ❛❜❡rt♦
x é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ y
♣❡r♠✐t❡ t❛♥t♦ ❛ ✐♥st❛♥❝✐❛çã♦ ✏✼ é ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✸✑ ❝♦♠♦ ❛ ✐♥st❛♥❝✐❛çã♦ ✏✼ é
♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ ✼✑✳
❊♠ r❡❧❛çõ❡s ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s✱ ♦ ❝♦♥❥✉♥t♦✲✉♥✐✈❡rs♦ é ❝♦♥st✐t✉í❞♦ ♣❡❧♦ ♣r♦❞✉t♦
❝❛rt❡s✐❛♥♦ ❞♦s ✉♥✐✈❡rs♦s ❞❛s ✈❛r✐á✈❡✐s❀ ♦ ❝♦♥❥✉♥t♦✲✈❡r❞❛❞❡ é ❝♦♥st✐t✉í❞♦ ♣❡❧♦s ♣❛r❡s
♦r❞❡♥❛❞♦s ❞♦s ✈❛❧♦r❡s q✉❡ s❛t✐s❢❛③❡♠ ❛ r❡❧❛çã♦✳
P♦r ❡①❡♠♣❧♦✱ ❝♦♥s✐❞❡r❡ ♦ ❛❜❡rt♦M(x, y) r❡♣r❡s❡♥t❛♥❞♦ ✏x é ♠❡t❛❞❡ ❞❡ y✑✱ ♦♥❞❡
Ux = {1, 2, 3} ❡ Uy = {4, 5, 6}✳ ❊♥tã♦ VM = {(2, 4), (3, 6)}✳
✹✳✷ ❖♣❡r❛çõ❡s ▲ó❣✐❝❛s
❚❛♠❜é♠ ♥♦ ❈á❧❝✉❧♦ ❞❡ Pr❡❞✐❝❛❞♦s ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s ♦♣❡r❛çõ❡s ❞❡ ❝♦♥❥✉♥çã♦✱
❞✐s❥✉♥çã♦✱ ♥❡❣❛çã♦✱ ❝♦♥❞✐❝✐♦♥❛❧ ❡ ❜✐❝♦♥❞✐❝✐♦♥❛❧✱ s♦❜r❡ ❡♥✉♥❝✐❛❞♦s ❡✴♦✉ ❛❜❡rt♦s✳
✺✹❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙
❈♦♥s✐❞❡r❡✱ ♣♦r ❡①❡♠♣❧♦✱ ♦s ❛❜❡rt♦s ✏x é ♠é❞✐❝♦✑✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r M(x)✱ ❡
✏x é ♣r♦❢❡ss♦r✑ ✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r P (x)❀ ♣♦❞❡♠♦s ❡♥tã♦ r❡♣r❡s❡♥t❛r ✏x é ♠é❞✐❝♦ ❡
♣r♦❢❡ss♦r✑ M(x) ∧ P (x)✳
❙❡❥❛ U ♦ ❈♦♥❥✉♥t♦✲❯♥✐✈❡rs♦ ❞❡ x❀ ♦s ✈❛❧♦r❡s ❞❡ U q✉❡ s❛t✐s❢❛③❡♠ M(x)∧P (x)
❞❡✈❡♠ s❛t✐s❢❛③❡r s✐♠✉❧t❛♥❡❛♠❡♥t❡ Mx ❡ Px❀ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
VM∧P = VM ∩ VP
❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ✏x é ♠é❞✐❝♦ ♦✉ ♣r♦❢❡ss♦r✑ ♣♦r M(x) ∨
P (x)✳ ❊st❡ ❛❜❡rt♦ é s❛t✐s❢❡✐t♦ ♣♦r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s q✉❡ sã♦ ♠é❞✐❝♦s ❡ ♣♦r t♦❞♦s
q✉❡ sã♦ ♣r♦❢❡ss♦r❡s❀ ♣♦rt❛♥t♦✱
VM∨P = VM ∪ VP
◆❛ ♦♣❡r❛çã♦ ❞❡ ♥❡❣❛çã♦✱ ♣♦❞❡♠♦s r❡♣r❡s❡♥t❛r ✏x ♥ã♦ é ♠é❞✐❝♦✑ ♣♦r M(x)✱ ❡¬
s❡✉ ❝♦♥❥✉♥t♦✲✈❡r❞❛❞❡ s❡rá ❝♦♥st✐t✉í❞♦ ♣♦r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞♦ ✉♥✐✈❡rs♦ q✉❡ ♥ã♦
s❛t✐s❢❛③❡♠ M(x)✱ ✐st♦ é✱ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ VM ✿
V M = U − VM¬
❯♠❛ ♥♦t❛çã♦ ❞❡ ✉s♦ ❣❡♥❡r❛❧✐③❛❞♦ ♣❛r❛ ♦ ❝♦♠♣❧❡♠❡♥t♦ ❞❡ VM é VM ✳
❈♦♥s✐❞❡r❡ ❛ ❡①♣r❡ssã♦ ✏s❡ x tr❛❜❛❧❤❛✱ ❡♥tã♦ x ✜❝❛ ❝❛♥s❛❞♦✑❀ r❡♣r❡s❡♥t❛♥❞♦
✏x tr❛❜❛❧❤❛✑ ♣♦r T (x)✱ ❡ ✏x ✜❝❛ ❝❛♥s❛❞♦✑ ♣♦r C(x)✱ t❡♠♦s q✉❡ ❛ ❡①♣r❡ssã♦ ❞❛❞❛
✜❝❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r T (x) C(x)✳ ❙❡✉ ❝♦♥❥✉♥t♦✲✈❡r❞❛❞❡ é ❝♦♥st✐t✉í❞♦ ♣♦r ❞✉❛s→
❝❧❛ss❡s ❞❡ ❡❧❡♠❡♥t♦s✿ ♣❡❧♦s q✉❡ tr❛❜❛❧❤❛♠ ❡ ✜❝❛♠ ❝❛♥s❛❞♦s ❡ ♣❡❧♦s q✉❡ ♥ã♦ tr❛✲
❜❛❧❤❛♠ ✭✉♠❛ ✈❡③ q✉❡ q✉❛♥❞♦ ♦ ❛♥t❡❝❡❞❡♥t❡ é ❢❛❧s♦✱ ❛ ❝♦♥❞✐❝✐♦♥❛❧ é ✈❡r❞❛❞❡✐r❛✮✳
❚❡♠♦s ❡♥tã♦ q✉❡
VT C = (VT ∩ VC) ∪ V T ✱ ✉t✐❧✐③❛♥❞♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞✐str✐❜✉t✐✈❛→ ¬
VT C = (VT ∪ V T ) ∩ (VC ∪ V T )✱ ❝♦♠♦ (VT ∪ V T ) = U→ ¬ ¬ ¬
VT C = U ∩ (VC ∪ V T )✱ ♦✉ s❡❥❛→ ¬
VT C = (V T ∪ VC)✱ ♦✉ ❛✐♥❞❛→ ¬
✺✺❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙
VT→C = VT ∪ VM
P❛r❛ ❛ ♦♣❡r❛çã♦ ❜✐❝♦♥❞✐❝✐♦♥❛❧✱ ❝♦♥s✐❞❡r❡ ❛ ❡①♣r❡ssã♦ ✏x tr❛❜❛❧❤❛ s❡ ❡ s♦♠❡♥t❡
s❡ ❣❛♥❤❛ ❞✐♥❤❡✐r♦✑❀ r❡♣r❡s❡♥t❛♥❞♦ ✏x tr❛❜❛❧❤❛✑ ♣♦r T (x)✱ ❡ ✏x ❣❛♥❤❛ ❞✐♥❤❡✐r♦✑
♣♦r G(x)✱ t❡♠♦s T (x) G(x)✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ❡❧❡♠❡♥t♦s q✉❡ s❛t✐s❢❛③❡♠ ❛ ❡ss❛↔
❡①♣r❡ssã♦ é ❝♦♥st✐t✉í❞♦ ♣❡❧❛ ✉♥✐ã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s ❞❛q✉❡❧❡s q✉❡ tr❛❜❛❧❤❛♠ ❡
❣❛♥❤❛♠ ❞✐♥❤❡✐r♦ ❡ ❞❛q✉❡❧❡s q✉❡ ♥ã♦ tr❛❜❛❧❤❛♠ ❡ ♥ã♦ ❣❛♥❤❛♠ ❞✐♥❤❡✐r♦❀ ❛ss✐♠✱
VT G = (VT ∩ VG) ∪ (VT ∩ VG)↔
❋♦r♠❛ ❙✐♠❜ó❧✐❝❛
❖❜t❡r ❛ ❢♦r♠❛ s✐♠❜ó❧✐❝❛ ❞❡ ✉♠❛ ❡①♣r❡ssã♦ ❡♠ ❧✐♥❣✉❛❣❡♠ t❡①t✉❛❧ ♥ã♦ é ❞✐❢í❝✐❧✱ ♠❛s
❡♥q✉❛♥t♦ ♥ã♦ s❡ ❛❞q✉✐r❡ ✉♠❛ ❝❡rt❛ ❤❛❜✐❧✐❞❛❞❡✱ ❞á ❛❧❣✉♠ tr❛❜❛❧❤♦❀ ♠✉✐t❛s ✈❡③❡s✱
♣❛r❛ ❢❛❝✐❧✐t❛r✱ ❝♦♥str✉í♠♦s ✉♠❛ ❢♦r♠❛ ✐♥t❡r♠❡❞✐ár✐❛✱ ❝❤❛♠❛❞❛ ❢♦r♠❛ ❧ó❣✐❝❛✱ ♦❜t✐❞❛
❛♣❡♥❛s ♣♦r ✐♥tr♦❞✉çã♦ ❞❡ ✈❛r✐á✈❡✐s ♥❛ ❢♦r♠❛ t❡①t✉❛❧✳
❱❛♠♦s ✈❡r ❛❧❣✉♥s ❡①❡♠♣❧♦s✱ ♦❜t❡♥❞♦ ❛ ❢♦r♠❛ ❧ó❣✐❝❛ ❡ s✐♠❜ó❧✐❝❛ ❞❡ ❡①♣r❡ssõ❡s
t❡①t✉❛✐s✱ ✉t✐❧✐③❛♥❞♦ ♦s ♣r❡❞✐❝❛❞♦s ❞❡✜♥✐❞♦s✿
✶✳ ●❛t♦s ❝❛ç❛♠ r❛t♦s ✭G(x)✿ x é ✉♠ ❣❛t♦❀ R(x)✿ x ❝❛ç❛ r❛t♦s✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠ ❣❛t♦✱ x ❝❛ç❛ r❛t♦s
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ G(x) R(x)→
✷✳ ❈❤✐♥❡s❡s ✈❡❧❤♦s sã♦ sá❜✐♦s ✭C(x)✿ x é ❝❤✐♥ês❀ V (x)✿ x é ✈❡❧❤♦❀ S(x)✿ x é
sá❜✐♦✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ❝❤✐♥ês ❡ x é ✈❡❧❤♦✱ ❡♥tã♦ x é sá❜✐♦
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ (C(x) ∧ V (x))→ S(x)
✸✳ ❆❜❛❝❛t❡s sã♦ ❞❡❧✐❝✐♦s♦s ❡ ♥✉tr✐t✐✈♦s ✭A(x)✿ x é ✉♠ ❛❜❛❝❛t❡❀ D(x)✿ x é ❞❡❧✐✲
❝✐♦s♦❀ N(x)✿ x é ♥✉tr✐t✐✈♦✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠ ❛❜❛❝❛t❡✱ ❡♥tã♦ x é ❞❡❧✐❝✐♦s♦ ❡ x é ♥✉tr✐t✐✈♦
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ A(x)→ (D(x) ∧N(x))
❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙ ✺✻
✹✳ ❆❜❛❝❛t❡s ❡ ❧❛r❛♥❥❛s sã♦ ❞❡❧✐❝✐♦s♦s ❡ ♥✉tr✐t✐✈♦s ✭A(x)✿ ① é ✉♠ ❛❜❛❝❛t❡❀ L(x)✿
① é ✉♠❛ ❧❛r❛♥❥❛❀ D(x)✿ x é ❞❡❧✐❝✐♦s♦❀ N(x)✿ x é ♥✉tr✐t✐✈♦✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠ ❛❜❛❝❛t❡ ♦✉ x é ✉♠❛ ❧❛r❛♥❥❛✱ ❡♥tã♦ x é ❞❡❧✐❝✐♦s♦ ❡ x
é ♥✉tr✐t✐✈♦
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ (A(x) ∨ L(x))→ (D(x) ∧N(x))
✺✳ ❙ã♦ r❛r♦s ♦s ♣♦❧ít✐❝♦s q✉❡ ♥ã♦ ♠❡♥t❡♠ ✭ R(x)✿ x é r❛r♦❀ P (x)✿ x é ♣♦❧ít✐❝♦❀
M(x)✿ x ♠❡♥t❡✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ♣♦❧ít✐❝♦ ❡ x ♥ã♦ ♠❡♥t❡✱ ❡♥tã♦ x é r❛r♦
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ (P (x) ∧ ¬M(x))→ R(x)
✻✳ ❈❛rr♦s só s❡ ❧♦❝♦♠♦✈❡♠ ❝♦♠ ❣❛s♦❧✐♥❛ ✭C(x)✿ x é ✉♠ ❝❛rr♦❀ L(x)✿ x s❡
❧♦❝♦♠♦✈❡❀ G(x)✿ x t❡♠ ❣❛s♦❧✐♥❛✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠ ❝❛rr♦✱ ❡♥tã♦ x s❡ ❧♦❝♦♠♦✈❡ s❡ ❡ s♦♠❡♥t❡ s❡ x t❡♠
❣❛s♦❧✐♥❛
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ C(x) (L(x) G(x))→ ↔
✼✳ ❊str❛❞❛s ❞❡ t❡rr❛ sã♦ tr❛❢❡❣á✈❡✐s ✉♥✐❝❛♠❡♥t❡ q✉❛♥❞♦ s❡❝❛s ✭E(x)✿ x é ✉♠❛
❡str❛❞❛ ❞❡ t❡rr❛❀ T (x)✿ x é tr❛❢❡❣á✈❡❧❀ S(x)✿ x ❡stá s❡❝❛✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠❛ ❡str❛❞❛ ❞❡ t❡rr❛✱ ❡♥tã♦ x é tr❛❢❡❣á✈❡❧ s❡ ❡ s♦♠❡♥t❡
s❡ x ❡stá s❡❝❛
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ E(x) (T (x) S(x))→ ↔
✽✳ ❍♦♠❡♥s só s❡ ❝❛s❛♠ ❝♦♠ ♠✉❧❤❡r❡s ✭H(x)✿ x é ❤♦♠❡♠❀ C(x, y)✿ x é ❝❛s❛❞♦
❝♦♠ y❀ M(x)✿ x é ♠✉❧❤❡r✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ❤♦♠❡♠✱ ❡ x é ❝❛s❛❞♦ ❝♦♠ y✱ ❡♥tã♦ y é ♠✉❧❤❡r
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ (H(x) ∧ C(x, y))→M(y)
✾✳ ●❛t♦s ♣r❡t♦s sã♦ ♠❡❧❤♦r❡s ❝❛ç❛❞♦r❡s q✉❡ ♦✉tr♦s ❣❛t♦s ✭G(x)✿ x é ✉♠ ❣❛t♦❀
P (x)✿ x é ♣r❡t♦❀ C(x, y)✿ x é ♠❡❧❤♦r ❝❛ç❛❞♦r q✉❡ y✮
❋♦r♠❛ ❧ó❣✐❝❛✿ s❡ x é ✉♠ ❣❛t♦ ❡ x é ♣r❡t♦ ❡ y é ✉♠ ❣❛t♦ ❡ y ♥ã♦ é ♣r❡t♦✱ ❡♥tã♦
x é ♠❡❧❤♦r ❝❛ç❛❞♦r q✉❡ y
❋♦r♠❛ s✐♠❜ó❧✐❝❛✿ (G(x) ∧ P (x) ∧G(y) ∧ ¬P (y))→ C(x, y)
6
❈❆P❮❚❯▲❖ ✹✳ ❈➪▲❈❯▲❖ ❉❊ P❘❊❉■❈❆❉❖❙ ✺✼
✹✳✸ ◗✉❛♥t✐✜❝❛❞♦r❡s
❉❛❞♦ ✉♠ ❛❜❡rt♦ P (x) ❡♠ ✉♠ ✉♥✐✈❡rs♦ U ✱ ♣♦❞❡ ♦❝♦rr❡r✿
❼ t♦❞♦s ♦s x ❡♠ U s❛t✐s❢❛③❡♠ P ❀ ✐st♦ é✱ VP = U
❼ ❛❧❣✉♥s x ❡♠ U s❛t✐s❢❛③❡♠ P ✱ ✐st♦ é✱ VP = ∅
❼ ♥❡♥❤✉♠ x ❡♠ U s❛t✐s❢❛③ P ✱ ✐st♦ é✱ VP = ∅
❈♦♥s✐❞❡r❡✱ ♣♦r ❡①❡♠♣❧♦✱ U = {2, 4, 6, 8}✳ ❙❡ ✜③❡r♠♦s P (x) r❡♣r❡s❡♥t❛r ✏x é ♣❛r✑✱
t❡♠♦s ♦ ♣r✐♠❡✐r♦ ❝❛s♦✿ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s s❛t✐s❢❛③❡♠ P ✱ ❡ VP = U ✳ P❛r❛ P (x)
r❡♣r❡s❡♥t❛♥❞♦ ✏x é ♠ú❧t✐♣❧♦ ❞❡ ✸✑✱ t❡♠♦s ❛♣❡♥❛s ✉♠ ❡❧❡♠❡♥t♦ q✉❡ s❛t✐s❢❛③ P ✱ ❡
VP = {6}✳ ❋✐♥❛❧♠❡♥t❡✱ s❡ P (x) r❡♣r❡s❡♥t❛r ✏x é ♠❛✐♦r q✉❡ ✶✵✑✱ ♥❡♥❤✉♠ ❡❧❡♠❡♥t♦
❞❡ U s❛t✐s❢❛③ P ✱ ❡✱ ♣♦rt❛♥t♦✱ VP = ∅✳
◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ✏♣❛r❛ t♦❞♦ x ❡♠ U ✱ P (x) é ✈❡r❞❛❞❡✐r♦✑✱