❋❙▲❚ ❙❡♠❛♥t✐❝s ❊①❡r❝✐s❡ ❉✉❡ ✹ ❏❛♥✉❛r② ◆✐❦✐t✐♥❛ ❖❧❣❛ ✶✳ ❋♦r♠❛❧✐s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥t❡♥❝❡s ✐♥ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝✦ ✭❚r❛♥s✲ ❧❛t❡ ❜❛s✐❝ s❡♥t❡♥❝❡s ❧✐❦❡ ✏✐t r❛✐♥s✑ ♦r ✏❙t❡✈❡ ❝♦♠❡s ❤♦♠❡ ❧❛t❡✑ t♦ ♣r♦♣♦s✐t✐♦♥❛❧ ❝♦♥st❛♥ts ♣✱ q✱ r✮ ✳ ❛✳ ❲❤❡♥ ✐t r❛✐♥s✱ ✐t ♣♦✉rs✳ ❚❤❡ s❡♥t❡♥❝❡ ❤❛s ✷ ♠❡❛♥✐♥❣s✿ t❡♠♣♦r❛❧ ❛♥❞ ✐♥❢❡r❡♥t✐❛❧✳ p ❂ ✏✐t r❛✐♥s✑ q ❂ ✏✐t ♣♦✉rs✑ ■♥❢❡r❡♥t✐❛❧ ♠❡❛♥✐♥❣ ✭❡q✉✐✈❛❧❡♥t t♦ ✏■❢ ✐t r❛✐♥s✱ ✐t ♣♦✉rs✑✮✿ p → q ❚❡♠♣♦r❛❧ ♠❡❛♥✐♥❣ ✭❡q✉✐✈❛❧❡♥t t♦ ✏❉✉r✐♥❣ t❤❡ t✐♠❡ ✐t r❛✐♥s✱ ✐t ♣♦✉rs✑✮✿ ▲❡t✬s ✐♥tr♦❞✉❝❡ ❛ t✐♠❡ ✈❛r✐❛❜❧❡ t ✱ t❤❛t ✇✐❧❧ ❦❡❡♣ tr❛❝❦ ♦❢ t❤❡ t✐♠❡ st❛t❡ ♦❢ t❤❡ ✇♦r❧❞ ✭❤♦✇❡✈❡r✱ t✐♠❡ ✐s ♥♦t ❛ ❝♦♥✈❡♥t✐♦♥❛❧ ♠❡❛♥✐♥❣ ♦❢ t❤✐s ✈❛r✐❛❜❧❡✱ ✐✳❡✳ ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ✐♠♣❧② t❤✐s ♠❡❛♥✐♥❣ ❢r♦♠ t❤❡ ❢♦r♠✉❧❛ ♦♥❧②✱ ❛♥❞ ❛ ❝♦♠♠❡♥t ✐s ❛❧✇❛②s ♥❡❡❞❡❞✮✿ ∀ t ( p & t ) → ( q & t ) ❜✳ ❙❛♠ ✇❛♥ts ❛ ❞♦❣✱ ❜✉t ❆❧✐❝❡ ♣r❡❢❡rs ❝❛ts✳ Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ❞♦❡s♥✬t ♣r♦✈✐❞❡ t♦♦❧s ❢♦r ❡①♣r❡ss✐♥❣ ❝♦♥tr❛❞✐❝t✐♦♥✱ ❜✉t ■ ❝❛♥ ✐♥tr♦❞✉❝❡ ❛ ❝♦♥st❛♥t d ✇❤✐❝❤ ❛♣♣❡❛r❛♥❝❡ ✐♥ t❤❡ ❢♦r♠✉❧❛ ✇♦✉❧❞ s✐❣♥✐❢② t❤❛t t❤❡r❡ ✐s ♥♦ ✇♦r❧❞ ✐♥ ✇❤✐❝❤ ❜♦t❤ ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥s t❤❛t ❛r❡ ❛r❣✉♠❡♥ts ♦❢ ✈❡r❜s ✏✇❛♥ts✑ ❛♥❞ ✏♣r❡❢❡rs✑ ❝❛♥ ❜❡ tr✉❡✳ ❆❣❛✐♥✱ t❤❡ ♠❡❛♥✐♥❣ ♦❢ t❤✐s ❝♦♥st❛♥t ✐s ♥♦t ❝♦♥✈❡♥t✐♦♥❛❧✳ ▼♦r❡♦✈❡r✱ t❤❡ ✇❛② t❤❡ ✈❛❧✉❡ ♦❢ t❤✐s ❝♦♥st❛♥t ✐s ❢♦r♠✉❧❛t❡❞ s❤♦✇s t❤❛t ✐t ✐♠♣♦ss✐❜❧❡ t♦ ❡①♣r❡ss t❤❡ ♠❡❛♥✐♥❣ ♦❢ ✏❜✉t✑ ✐♥ t❡r♠s ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝✱ ❜❡❝❛✉s❡ ♦♥❡ ✇♦✉❧❞ ♥❡❡❞ t♦ ❞❡❝♦♠♣♦s❡ p ❛♥❞ q ✐♥t♦ ♠❛✐♥ ♣r❡❞✐❝❛t❡s want ❛♥❞ prefer ❛♥❞ ♣r♦♣♦s✐t✐♦♥s h 1 ❂ ✏❤❛✈❡ ❛ ❞♦❣✑ ❛♥❞ h 2 ❂ ✏❤❛✈❡ ❛ ❝❛t✑✳ p ❂ ✏❙❛♠ ✇❛♥ts ❛ ❞♦❣✑ q ❂ ✏❆❧✐❝❡ ♣r❡❢❡rs ❝❛ts✑ p & q & d ❝✳ ■ ✇✐❧❧ ♠❛❦❡ t❤❡ ❞✐s❤❡s ✐❢ ②♦✉ ❝♦♦❦✳ p ❂ ✏■ ✇✐❧❧ ♠❛❦❡ t❤❡ ❞✐s❤❡s✑ q ❂ ✏②♦✉ ❝♦♦❦✑ q → p ❞✳ ■ ✇✐❧❧ ♠❛❦❡ t❤❡ ❞✐s❤❡s ♦♥❧② ✐❢ ②♦✉ ❝♦♦❦ p ❂ ✏■ ✇✐❧❧ ♠❛❦❡ t❤❡ ❞✐s❤❡s✑ q ❂ ✏②♦✉ ❝♦♦❦✑ ( q → p )&( ¬ q → ¬ p ) ■♥ ♦t❤❡r t❡r♠s✱ p ↔ q ❡✳ ▼❛rs❤❛ ✇♦♥✬t ❣♦ ♦✉t ✇✐t❤ ❏♦❤♥ ✉♥❧❡ss ❤❡ s❤❛✈❡s ♦✛ ❤✐s ❜❡❛r❞ ❛♥❞ st♦♣s ❞r✐♥❦✐♥❣✳ p ❂ ✏▼❛rs❤❛ ✇✐❧❧ ❣♦ ♦✉t ✇✐t❤ ❏♦❤♥✑ q ❂ ✏❤❡ s❤❛✈❡s ♦✛ ❤✐s ❜❡❛r❞✑ ✶
r ❂ ✏❤❡ st♦♣s ❞r✐♥❦✐♥❣✑ ( ¬ q & ¬ r → ¬ p )&( q & r → p ) ❢✳ ❚❤❡ st♦❝❦ ♠❛r❦❡t ❛❞✈❛♥❝❡s ✇❤❡♥ ♣✉❜❧✐❝ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ ❡❝♦♥♦♠② ✐s r✐s✐♥❣✳ ❙❛♠❡ r❡❛s♦♥✐♥❣ ❛s ✐♥ ✶ ❛♣♣❧✐❡s ❤❡r❡✳ ❚❤❡ s❡♥t❡♥❝❡ ❛❣❛✐♥ ❤❛s ❛♥ ✐♥❢❡r❡♥t✐❛❧ ❛♥❞ ❛ t❡♠♣♦r❛❧ ♠❡❛♥✐♥❣s✳ ■ ✇✐❧❧ ✐♥tr♦❞✉❝❡ t❤❡ s❛♠❡ t✐♠❡ ✈❛r✐❛❜❧❡ t ✇✐t❤ t❤❡ s❛♠❡ ♣r♦♣❡rt✐❡s✿ p ❂ ✏t❤❡ st♦❝❦ ♠❛r❦❡t ❛❞✈❛♥❝❡s✑ q ❂ ✏♣✉❜❧✐❝ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ ❡❝♦♥♦♠② ✐s r✐s✐♥❣✑ ■♥❢❡r❡♥t✐❛❧ ♠❡❛♥✐♥❣ ✭❡q✉✐✈❛❧❡♥t t♦ ✏■❢ ♣✉❜❧✐❝ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ ❡❝♦♥♦♠② ✐s r✐s✐♥❣✱ t❤❡ st♦❝❦ ♠❛r❦❡t ❛❞✈❛♥❝❡s✑✮✿ q → p ❚❡♠♣♦r❛❧ ♠❡❛♥✐♥❣ ✭❡q✉✐✈❛❧❡♥t t♦ ✏❉✉r✐♥❣ t❤❡ t✐♠❡s ✇❤❡♥ ♣✉❜❧✐❝ ❝♦♥✜❞❡♥❝❡ ✐♥ t❤❡ ❡❝♦♥♦♠② ✐s r✐s✐♥❣✱ t❤❡ st♦❝❦ ♠❛r❦❡t ❛❞✈❛♥❝❡s✮✿ ∀ t ( q & t ) → ( p & t ) ❣✳ ❏♦❤♥ ❛♥❞ ❇✐❧❧ ❛r❡ ❣♦✐♥❣ t♦ t❤❡ ♠♦✈✐❡s✱ ❜✉t ♥♦t ❚♦♠✳ p ❂ ✏❏♦❤♥ ❛♥❞ ❇✐❧❧ ❛r❡ ❣♦✐♥❣ t♦ t❤❡ ♠♦✈✐❡s✑ q ❂ ✏❚♦♠ ✐s ❣♦✐♥❣ t♦ t❤❡ ♠♦✈✐❡s✑ Pr♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ❞♦❡s♥✬t ❤❛✈❡ ♠❡❛♥s ❢♦r r❡♣r❡s❡♥t✐♥❣ ❞✐s❝♦✉rs❡ ❝♦♥tr❛st ❡♥❝❧♦s❡❞ ✐♥ ✏❜✉t✑✳ ▲♦❣✐❝❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ♠❡❛♥✐♥❣ ♦❢ t❤❡ s❡♥t❡♥❝❡ ✐s✿ p & ¬ q ❈♦♥tr❛st ❞♦❡s♥✬t ❜❡❧♦♥❣ t♦ t❤❡ ♣r♦♣♦s✐t✐♦♥❛❧ ❧❡✈❡❧ ♦❢ t❤❡ ♠❡❛♥✐♥❣✳ ■s ✐t ✇❤❛t ✐s ❝❛❧❧❡❞ ♣r❛❣♠❛t✐❝s❄ ❤✳ ■❢ ▼❛r② ❤❛s♥✬t ❣♦t ❧♦st ♦r ❤❛❞ ❛♥ ❛❝❝✐❞❡♥t✱ s❤❡ ✇✐❧❧ ❜❡ ❤❡r❡ ✐♥ ✺ ♠✐♥✉t❡s✳ p ❂ ✏▼❛r② ❤❛s ❣♦t ❧♦st✑ q ❂ ✏▼❛r② ❤❛s ❤❛❞ ❛♥ ❛❝❝✐❞❡♥t✑ r ❂ ✏▼❛r② ✇✐❧❧ ❜❡ ❤❡r❡ ✐♥ ✺ ♠✐♥✉t❡s✑ ( ¬ p & ¬ q ) → r ■ ❞♦♥✬t ❦♥♦✇ ♦❢ s♣❡❝✐❛❧ ♠❡❛♥s ♦❢ ♣r♦♣♦s✐t✐♦♥❛❧ ❧♦❣✐❝ ❢♦r r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❡♠♣♦r❛❧ ❛♥❞ ❛s♣❡❝t✉❛❧ s❡♠❛t✐❝s✳ ❖♥❡ ❣❡♥❡r❛❧ ♥♦t❡✿ ❙tr✐❦t❧② s♣❡❛❦✐♥❣✱ q → p ✐s ♥♦t ❡q✉✐✈❛❧❡♥t t♦ ❛ ❝♦♥❞✐t✐♦♥❛❧ s❡♥t❡♥❝❡ ✐♥ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡✳ ❋♦r ❝❛s❡s q = 1 ✱ t❤❡ ❢♦r♠✉❧❛ r❡♣r❡s❡♥ts t❤❡ ♥❛t✉r❛❧ ❧❛♥❣✉❛❣❡ s❡♥t❡♥❝❡ ❝♦rr❡❝t❧②✳ ❇✉t ❢♦r t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ❝♦♥❞✐t✐♦♥ ✐s ❢❛❧s❡✱ t❤❡ ❝♦rr❡❝t ✈❛❧✉❡ ♦❢ t❤❡ s❡♥t❡♥❝❡ ✇♦✉❧❞ ❜❡ ✏■ ❞♦♥✬t ❦♥♦✇✑✱ ❜✉t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢♦r♠✉❧❛ ✐s ✏tr✉❡✑✳ ✷✳ ❈❤❡❝❦ ✇✐t❤ t❤❡ tr✉t❤✲t❛❜❧❡ ♠❡t❤♦❞✱ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r✲ ♠✉❧❛❡ ❛r❡ ❧♦❣✐❝❛❧❧② ✈❛❧✐❞✱ ❝♦♥tr❛❞✐❝t♦r②✱ ♦r ❝♦♥t✐♥❣❡♥t ✭✐✳❡✳ ♥❡✐t❤❡r ✈❛❧✐❞ ♥♦r ❝♦♥tr❛❞✐❝t♦r②✮✦ ✳ ❛✳ (( p ∨ ➡ q ) ∧ q ) ✷
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