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❋❙▲❚ ❙❡♠❛♥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 h1 ❂ ✏❤❛✈❡ ❛ ❞♦❣✑ ❛♥❞ h2 ❂ ✏❤❛✈❡ ❛ ❝❛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) ✷

♣ q ➡q ♣∨➡q ✭♣∨➡q✮∧q ✶ ✶ ✵ ✶ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ✵ ❚❤✐s st❛t❡♠❡♥t ✐s ❢❛❧s❡ ✇❤❡♥ q ✐s tr✉❡✱ ✐♥ ♦t❤❡r ❝❛s❡s ✐t ✐s tr✉❡✱ ✐✳❡✳ ✐t ✐s ❝♦♥t✐♥❣❡♥t✳ ❜✳ ((p ∧ q) → (p ∨ r)) ♣ q ♣∧q r ♣∨r ✭♣∧q✮✙✭♣∨r✮ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✵ ✶ ✶ ✶ ✵ ✵ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✶ ✶ ✵ ✵ ✵ ✵ ✵ ✶ ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧✇❛②s tr✉❡✱ ✐✳❡✳ ✐t ✐s ✈❛❧✐❞✳ ❝✳ (➡p ∧ ➡(p → q)) ♣ ➡♣ q ♣✙q ➡✭♣✙q✮ ➡♣∧➡✭♣✙q✮ ✶ ✵ ✶ ✶ ✵ ✵ ✶ ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✶ ✶ ✵ ✵ ✵ ✶ ✵ ✶ ✵ ✵ ❚❤✐s st❛t❡♠❡♥t ✐s ❛❧✇❛②s ❢❛❧s❡✱ ✐✳❡✳ ❝♦♥tr❛❞✐❝t♦r②✳ ✸✳ ❈❤❡❝❦ ✇✐t❤ t❤❡ tr✉t❤✲t❛❜❧❡ ♠❡t❤♦❞ ✇❤❡t❤❡r ❡♥t❛✐❧♠❡♥t ❤♦❧❞s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s✿ ✳ ❛✳ (p → ➡q), (r → q), (➡r → q) | = ➡p❄ ♣ q ➡q ♣ ✙ ➡q r r ✙ q ➡r ➡r ✙ q ➡♣ ✶ ✶ ✶ ✵ ✵ ✶ ✶ ✵ ✶ ✵ ✷ ✶ ✶ ✵ ✵ ✵ ✶ ✶ ✶ ✵ ✸ ✶ ✵ ✶ ✶ ✶ ✵ ✵ ✶ ✵ ✹ ✶ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✵ ✺ ✵ ✶ ✵ ✶ ✶ ✶ ✵ ✶ ✶ ✻ ✵ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✶ ✼ ✵ ✵ ✶ ✶ ✶ ✵ ✵ ✶ ✶ ✽ ✵ ✵ ✶ ✶ ✵ ✶ ✶ ✵ ✶ ❆❧❧ t❤❡ ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt ❛r❡ tr✉❡ ✐♥ ❝❛s❡s ✺ ❛♥❞ ✻✳ ➡p ✐s tr✉❡ ✐♥ t❤❡s❡ ❝❛s❡s ❛s ✇❡❧❧✳ ❚❤❡ ❡♥t❛✐❧♠❡♥t ❤♦❧❞s✳ ❜✳ (q ∨ r), ((q ∧ r) → s) | = (q → s)❄ ✸

q r q ∨ r q ∧ r s ✭✭q∧ r✮ ✙ s✮ ✭q ✙ s✮ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✶ ✷ ✶ ✶ ✶ ✶ ✵ ✵ ✵ ✸ ✶ ✵ ✶ ✵ ✶ ✶ ✶ ✹ ✶ ✵ ✶ ✵ ✵ ✶ ✵ ✺ ✵ ✶ ✶ ✵ ✶ ✶ ✶ ✻ ✵ ✶ ✶ ✵ ✵ ✶ ✶ ✼ ✵ ✵ ✵ ✵ ✶ ✶ ✶ ✽ ✵ ✵ ✵ ✵ ✵ ✶ ✶ ❆❧❧ t❤❡ ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt ❛r❡ tr✉❡ ✐♥ ❝❛s❡s ✶✱ ✸✱ ✹✱ ✺✱ ✻✳ ❍♦✇❡✈❡r✱ t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ r✐❣❤t ♣❛rt ✐s ❢❛❧s❡ ✐♥ t❤❡ ❢♦rt❤ ❝❛s❡✳ ❚❤❡ ❡♥t❛✐❧♠❡♥t ❞♦❡s♥✬t ❤♦❧❞✳ ✹✳ ❚r❛♥s❧❛t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡♥t❡♥❝❡s t♦ ❋❖▲✳ ✳ ❛✳ ❏♦❤♥ ❛❞♠✐r❡s s♦♠❡♦♥❡✳ ∃x(admire(John, x)) ❜✳ ❏♦❤♥ ❛❞♠✐r❡s ❤✐♠s❡❧❢✳ admire(John, John) ❝✳ ❇✐❧❧ ❛♥❞ ▼❛r② ❤❡❧♣ ❡❛❝❤ ♦t❤❡r✳ help(Bill, Mary)&help(Mary, Bill) ❞✳ ❆ st✉❞❡♥t r❡❛❞s ❛♥ ✐♥t❡r❡st✐♥❣ ❜♦♦❦ ∃x∃y(student(x)&interesting(y)&book(y)&read(x, y)) ❡✳ P❡t❡r r❡❛❞s ♦♥❧② ✐♥t❡r❡st✐♥❣ ❜♦♦❦s✳ ∀x(read(Peter, x) → (interesting(x)&book(x))) ❢✳ ◆♦ ♦♥❡ ✐s ❧♦✈❡❞ ❜② ❡✈❡r②♦♥❡✳ ∀y∃x(¬love(y, x)) ❣✳ ❆❧❧ ❜✉t ♦♥❡ st✉❞❡♥t ♣❛ss❡❞ ✭t❤❡ ❡①❛♠✮✳ ∃x(¬pass(x)&student(x)&∀y(x = y → pass(y))) ❤✳ ❖♥❧② P❡t❡r ✢✉♥❦❡❞✳ ∃x(flunk(x)&x = Peter&∀y(x = y → ¬flunk(y))) ✐✳ ❊①❛❝t❧② ♦♥❡ st✉❞❡♥t ✢✉♥❦❡❞✳ ∃!x(flunk(x)) ✺✳ ❆r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛❡ ❧♦❣✐❝❛❧❧② ✈❛❧✐❞✱ ❝♦♥tr❛❞✐❝t♦r② ✭❢❛❧s❡ ✐♥ ❛❧❧ ♠♦❞❡❧ str✉❝t✉r❡s✮✱ ♦r ❝♦♥t✐♥❣❡♥t ✭♥❡✐t❤❡r ✈❛❧✐❞ ♥♦r ❝♦♥tr❛❞✐❝t♦r②✮❄ ✳ ❛✳ ∃x(F(x) ∧ ➡F(x)) ❚❤✐s ❢♦r♠✉❧❛ ✐s ❝♦♥tr❛❞✐❝t♦r②✳ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛ ❢♦r♠✉❧❛ ❛♥❞ ✐ts ♥❡❣❛t✐♦♥ ❝❛♥ ♥❡✈❡r ❜❡ tr✉❡✱ ❜❡❝❛✉s❡ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ✐s ♦♥❧② ✈❛❧✐❞ ✇❤❡♥ ❜♦t❤ ♦❢ ✐ts ♠❡♠✲ ❜❡rs ❛r❡ ✈❛❧✐❞✱ ❛♥❞ ❛s ✐t ❢♦❧❧♦✇s ❢r♦♠ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ♥❡❣❛t✐♦♥ ♦♣❡r❛t♦r✱ t❤❡r❡ ✐s ♥♦ s✉❝❤ ❝❛s❡ ✇❤❡♥ ❛ ❢♦r♠✉❧❛ ❛♥❞ ✐ts ♥❡❣❛t✐♦♥ ❛r❡ ❜♦t❤ ✈❛❧✐❞✳ ❜✳ (∃xF(x) ∨ ∃x➡F(x)) ❚❤✐s ❢♦r♠✉❧❛ ✐s ❛❧✇❛②s tr✉❡✱ ✐✳❡✳ ✈❛❧✐❞✳ ❙✉♣♣♦s❡ ✐t ✐s ♥♦t t❤❡ ❝❛s❡✳ ❚❤❡♥✱ t❤❡r❡ s❤♦✉❧❞ ❜❡ ❛ ♠♦❞❡❧ ✇❤❡r❡ ✐t ✐s ♥♦t tr✉❡✳ ■ ✇✐❧❧ tr② t♦ ❝♦♥str✉❝t ✐t✳ ❋♦r t❤❡ ❢♦r♠✉❧❛ t♦ ❜❡ ❢❛❧s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t❤❛t ∃xF(x) = 0 ❛♥❞ ∃x➡F(x) = 0✳ ❇✉t ✐❢ ∃xF(x) = 0✱ t❤❡♥¬(∃xF(x)) = 1 ✱ t❤❡♥ ✹

∃x➡F(x) = 1 ❛♥❞ ✈✐❝❡ ✈❡rs❛✳ ❚❤❡r❡ ✐s ♥♦ s✉❝❤ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ t❤❡② ❜♦t❤ ❛r❡ ❢❛❧s❡✳ ■t ♠❡❛♥s t❤❛t t❤❡ ❢♦r♠✉❧❛ ✐s ❛❧✇❛②s tr✉❡✱ ✐✳❡✳ ✈❛❧✐❞✳ ❝✳ (∀xF(x) ∨ ∀x➡F(x)) ❚❤✐s ❢♦r♠✉❧❛ ✐s ❝♦♥t✐❣❡♥t✳ ❚♦ ♣r♦✈❡ t❤✐s✱ ■ ♥❡❡❞ t♦ ❝♦♥str✉❝t ❛ ♠♦❞❡❧ ✇❤❡r❡ ✐t ✐s tr✉❡ ❛♥❞ ❛ ♠♦❞❡❧ ✇❤❡r❡ ✐t ✐s ❢❛❧s❡✳ ❚♦ ❝♦♥str✉❝t t❤❡ ✜rst ♠♦❞❡❧✱ ■ ♦♥❧② ♥❡❡❞ t♦ ❛❝❝❡♣t t❤❡ ❝♦♥❞✐t✐♦♥ t❤❛t ∀xF(x) ✐s ♥♦t tr✉❡✳ ❚♦ ❝♦♥str✉❝t t❤❡ s❡❝♦♥❞ ♠♦❞❡❧✱ ■ ♥❡❡❞ t♦ ❛❝❝❡♣t t❤❛t ∀xF(x) ✐s ♥♦t t✉r❡ ❛♥❞∀x➡F(x) ✐s ♥♦t tr✉❡ ✐♥ t❤❡ s❛♠❡ ♠♦❞❡❧✳ ■t ♠❡❛♥s t❤❛t ❢♦r s♦♠❡ ① F(x) s❤♦✉❧❞ ❜❡ tr✉❡ ❛♥❞ ❢♦r s♦♠❡ ♦t❤❡r ① F(x) s❤♦✉❧❞ ❜❡ ❢❛❧s❡✱ ✐✳❡✳ ■ s❤♦✉❧❞ ✜♥❞ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❣✐✈❡s ❞✐✛❡r❡♥t r❡s✉❧ts ❢♦r ❞✐✛❡r❡♥t ① ✈❛❧✉❡s✳ ❙❛②✱ F = nasty✳ ■ ✇✐❧❧ t❛❦❡ t❤❡ ♠♦❞❡❧✱ ✇❤❡r❡ nasty(John) ✐s tr✉❡ ❛♥❞ nasty(Mary) ✐s ❢❛❧s❡✳ ❚❤❡♥✱ ❢♦r ♦♥❡ x = John F(x) ✇✐❧❧ ❜❡ tr✉❡ ❛♥❞ ❢♦r x = Mary F(x) ✇✐❧❧ ❜❡ ❢❛❧s❡✳ ❈❤❡❝❦ ✇❤❡t❤❡r t❤❡ ❡♥t❛✐❧♠❡♥t ❤♦❧❞s✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❛s❡s ✭t❤r♦✉❣❤ s❡♠❛♥t✐❝ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ t❤❡ ✐♥✈♦❧✈❡❞ ❢♦r♠✉❧❛s✮✿ ✳ ❛✳ ∀xF(x), G(a) | = ∃x(F(x) ∧ G(x)) ❙❛② ✇❡ ❤❛✈❡ ❛✮ t❤❡ ♠♦❞❡❧ ▼❂④❉✱■⑥✱ ✇❤❡r❡ ❉ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ ❝♦♥st❛♥ts ■ ✐s t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❢✉♥❝t✐♦♥ ❜✮ ✈❛r✐❛❜❧❡ ❛ss✐❣♥♠❡♥t ❢✉♥❝t✐♦♥ ❣✳ ❚r✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt✿ [∀xF(x)]M,g = 1 ✐✛ ❢♦r ❡❛❝❤ ❝♦♥st❛♥t d ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ❝♦♥st❛♥ts D [F(x)]M,g = 1 ✐✛ ❢♦r ❡❛❝❤ d ∈ D [x]M,g[d/x] ∈ [F]M,g[d/x] ✐✛ ❢♦r ❡❛❝❤ d ∈ D d ∈ I(F)✳ [G(a)]M,g = 1 ✐✛ I(a) ∈ I(G)✳ ❚r✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ r✐❣❤t ♣❛rt✿ ∃x(F(x) ∧ G(x)) = 1 ✐✛ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ d ∈ D s✉❝❤ t❤❛t [F(x) ∧ G(x)]M,g = 1 ✐✛ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ d ∈ D s✉❝❤ t❤❛t [F(x)]M,g = 1 ❛♥❞ [G(x)]M,g = 1 ✐✛ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ d ∈ D s✉❝❤ t❤❛t [x]M,g[d/x] ∈ [F]M,g[d/x] ❛♥❞ [x]M,g[d/x] ∈ [G]M,g[d/x] ✐✛ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ d ∈ D s✉❝❤ t❤❛t d ∈ I(F) ❛♥❞ d ∈ I(G)✳ ❇♦t❤ ❢♦r♠✉❧❛❡ ❢r♦♠ t❤❡ r✐❣❤t ♣❛rt ❛r❡ tr✉❡ ✇❤❡♥ t❤❡ ✜rst ✐s tr✉❡ ❛♥❞ t❤❡ s❡❝♦♥❞ ✐s tr✉❡✳ ■❢ ❢♦r ❡❛❝❤ d ∈ D d ∈ I(F)✱ t❤❡♥ a ∈ I(F)✳ ❚❤❡♥✱ ❝♦♥❥✉♥❝t✐♦♥ ♦❢ t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❜♦t❤ ❢♦r♠✉❧❛❡ ✐s a ∈ I(F) ❛♥❞a ∈ I(G)✳ ❙✐♥❝❡ a ✐s ❛ ❝♦♥st❛♥t ❢r♦♠ t❤❡ ❞♦♠❛✐♥ D✱ t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ r✐❣❤t ♣❛rt ❛r❡ s❛t✐s✜❡❞✳ ❚❤❛t ✐s✱ t❤❡ ❡♥t❛✐❧♠❡♥t ❜❡t✇❡❡❡♥ t❤❡ ❧❡❢t ❛♥❞ t❤❡ r✐❣❤t ♣❛rts ❤♦❧❞s✳ ❜✳ F(a), ∃x(F(x) ∧ G(x)) | = G(a) ❲❡ ❤❛✈❡ ❛✮ t❤❡ ♠♦❞❡❧ ▼❂④❉✱■⑥✱ ✇❤❡r❡ ✺

❉ ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ ❝♦♥st❛♥ts ■ ✐s t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❢✉♥❝t✐♦♥ ❜✮ ✈❛r✐❛❜❧❡ ❛ss✐❣♥♠❡♥t ❢✉♥❝t✐♦♥ ❣✳ ❚r✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt✿ ∃x(F(x) ∧ G(x)) = 1 ✐✛ t❤❡r❡ ❡①✐sts ❛t ❧❡❛st ♦♥❡ d ∈ D s✉❝❤ t❤❛t d ∈ I(F) ❛♥❞ d ∈ I(G)✳ [F(a)]M,g = 1 ✐✛ I(a) ∈ I(F)✳ ❚❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ r✐❣❤t ♣❛rt✿ [G(a)]M,g = 1 ✐✛ I(a) ∈ I(G)✳ ❙✉♣♣♦s❡ ✇❡ ❝❤♦♦s❡ t❤❡ ♠♦❞❡❧ s✉❝❤ t❤❛t t❤❡r❡ ❛r❡ ❝♦♥st❛♥ts ❛ ❛♥❞ ❜ s✉❝❤ t❤❛t I(b) = I(a) ❛♥❞ I(a) ∈ I(F) ❛♥❞ I(b) ∈ I(F) ❛♥❞ I(b) ∈ I(G)✳ ■♥ t❤✐s ♠♦❞❡❧✱ I(a) / ∈ I(G) ✱ ❜✉t t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❜♦t❤ ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt ❛r❡ tr✉❡✳ ❋♦r s✉❝❤ ♠♦❞❡❧✱ t❤❡ ❡♥t❛✐❧♠❡♥t ❞♦❡s♥✬t ❤♦❧❞✱ ✐✳❡✳ ✐t ❞♦❡s♥✬t ❤♦❧❞ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✳ ❝✳ ∀x(F(x) ↔ ➡G(x)), F(a), G(b) | = ➡a = b ❚r✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt✿ ∀x(F(x) ↔ ➡G(x)) = 1✐✛ ❢♦r ❡❛❝❤ d ∈ D [F(x) ↔ ➡G(x)]M,g = 1 ✐✛ ❢♦r ❡❛❝❤ d ∈ D [F(x)]M,g = [➡G(x)]M,g✐♥ ✷ ❝❛s❡s ♦♥❧②✿ ✶✳ ❢♦r ❡❛❝❤ d ∈ D [F(x)]M,g = 1 ❛♥❞ [➡G(x)]M,g = 1 ♦r ✷✳ ❢♦r ❡❛❝❤ d ∈ D [F(x)]M,g = 0 ❛♥❞ [➡G(x)]M,g = 0 ✐✛ ✶✳ ❢♦r ❡❛❝❤ d ∈ D d ∈ I(F) ❛♥❞ d / ∈ I(G) ♦r ✷✳ ❢♦r ❡❛❝❤ d ∈ D d / ∈ I(F) ❛♥❞ d ∈ I(G) F(a)✐✛ I(a) ∈ I(F) G(b)✐✛ I(b) ∈ I(G) ❚r✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❢♦r♠✉❧❛ ✐♥ t❤❡ r✐❣❤t ♣❛rt✿ ➡a = b✐✛ I(a) = I(b)✳ ▲❡t✬s ❝❤❡❝❦ ✇❤❛ ❛r❡ t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ❝♦♥❥✉❝♥t✐♦♥ ♦❢ t❤❡ t❤r❡❡ ❢♦r♠✉❧❛❡ ✐♥ t❤❡ ❧❡❢t ♣❛rt✳ ■♥ ❉ t❤❡r❡ s❤♦✉❧❞ ❜❡ ❛t ❧❡❛st ✷ ❝♦♥st❛♥ts a ❛♥❞ b✳ ■ ❛♠ ♥♦✇ ✐♥s❡rt✐♥❣ t❤❡♠ ✐♥t♦ t❤❡ ✜rst ❢♦r♠✉❧❛ ❛♥❞ ❝❤❡❝❦ ✇❤❛t ✇✐❧❧ ❢♦❧❧♦✇✳ ❋♦r ❛✿ [F(x)]M,g[a/x] = 1 ❛♥❞ [➡G(x)]M,g[a/x] = 1 ✐✛ a ∈ I(F)❛♥❞ a / ∈ I(G) ♦r [F(x)]M,g[a/x] = 0 ❛♥❞ [➡G(x)]M,g[a/x] = 0 ✐✛ a / ∈ I(F) ❛♥❞a ∈ I(G) ❋♦r ❜✿ [F(x)]M,g[b/x] = 1 ❛♥❞ [➡G(x)]M,g[b/x] = 1 ✐✛ b ∈ I(F)❛♥❞ b / ∈ I(G) ♦r [F(x)]M,g[b/x] = 0 ❛♥❞ [➡G(x)]M,g[b/x] = 0 ✐✛ b / ∈ I(F) ❛♥❞b ∈ I(G) ❚❤❡ ❝♦♥❥✉❝♥t✐♦♥ ✇✐t❤ t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ❢♦r t✇♦ ♦t❤❡r ❢♦r♠✉❧❛❡ ②✐❡❧❞ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡t ♦❢ ❝♦♥❞✐t✐♦♥s✿ a ∈ I(F)❛♥❞ a / ∈ I(G) ❛♥❞ b / ∈ I(F) ❛♥❞b ∈ I(G)✳ ✻

❙✐♥❝❡ I ✐s ❛ ❢✉♥❝t✐♦♥✱ ✐t s❤♦✉❧❞ ❛❧✇❛②s ❣✐✈❡ t❤❡ s❛♠❡ r❡s✉❧t ❢♦r t❤❡ s❛♠❡ ❛r❣✉♠❡♥t ❣✐✈❡♥ t❤❡ s❛♠❡ ♠♦❞❡❧ M ❛♥❞ t❤❡ s❛♠❡ ✈❛r✐❛❜❧❡ ❛ss✐❣♥♠❡♥t ❢✉♥❝t✐♦♥ g✱ ✐✳❡✳ ✐❢ (I(φ) = I(ϕ))✱ t❤❡♥ (φ = ϕ)✳ ❚❤❡♥✱ ✐❢ (I(φ) = I(ϕ))✱ t❤❡♥ (φ = ϕ)✳ ❋r♦♠ t❤✐s✱ ✐t ❢♦❧❧♦✇s t❤❛t ✐❢ a ∈ I(F) ❛♥❞ b / ∈ I(F) ✱ t❤❡♥ I(F(a)) = I(F(b))✳ ❙✐♥❝❡ I(F) ✐s t❤❡ s❛♠❡✱ I(a) = I(b)✱ t❤❛t ✐s t❤❡ tr✉t❤ ❝♦♥❞✐t✐♦♥s ♦❢ t❤❡ r✐❣❤t ♣❛rt✳ ❚❤❡ ❡♥t❛✐❧♠❡♥t ❤♦❧❞s✳ ✼