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Logic HW#4

PHIL 1068

Fall 2023

This assignment is due Nov 10. You must turn the assignment in online via Moodle; .doc or .pdf is strongly preferred. If you prefer to handwrite, please scan your assignment into .doc or .pdf.

Here are the connectives for your copy/pasting convenience: & v ~ → ↔ ∀ ∃

1. Translate the following sentences into QL, assuming that UD = people, and using a=Adam, e=Eve, G=is boring, D=is bored. Since the UD contains only people, ‘everyone’ may be translated with ∀x and ‘someone’ may be translated with ∃x. [10 marks]

a. If everyone is boring, then Adam and Eve are not bored.

b. Someone is neither boring nor bored.

c. Bored people are boring.

d. Adam is bored if and only if someone is bored but not boring.

2. Translate the following sentences into QL, assuming that UD = people, and assuming that a=Antonius, c=Cleopatra, j=Caesar, J= is jealous, H=hates, L=loves. Since the UD contains only people, ‘everyone’ may be translated with ∀x and ‘someone’ may be translated with ∃x. [10 marks]

a. Cleopatra loves herself and hates nobody.

b. Only jealous people love themselves and hate everyone.  

c. Anyone who hates Antonius also loves Caesar.

d. Unless Caesar hates those whom nobody hates, Antonius loves those whom everyone loves.

3. Consider the following interpretation. UD = {Jules, Jim, Catherine}, o=Jules, h=Jim, j=Catherine, Extension(K) = {Catherine, Jim}, Extension(W) = {}, Extension(S) = {Catherine, Jules}. Are the following sentences true or false on this interpretation? [10 marks]

a. (~∃xSx→~∃xKx)

b. (∃x~Wx→∀x~Wx)

c. (∃xSx↔~∀xKx)

d. ∃x~(KxvSx)

4. Consider the following interpretation. UD = {Anne, Catherine, Henry}, a=Anne, c=Catherine, h=Henry, Extension(L) = {<Henry, Anne>, <Henry, Catherine>, <Henry, Henry>, <Catherine, Henry>}, Extension(H) = {<Henry, Catherine>, <Catherine, Anne>, <Catherine, Catherine>, <Anne, Catherine>, Extension(S) = {}. Are the following sentences true or false on this interpretation? [10 marks]

a. ~∃x∀yLxy  

b. (∀x∃yLxy→∃x(Hxx→~Sx))

c. ~∃x(Sx→Lxx)

d. ∃x~∃yHxy

5. Construct an interpretation to show that the set of sentences ∀x(Fx→Cx), ~Cl, and ∃xFx is consistent. Provide the UD, names, and predicate extensions. Note: there are infinite possible answers! [10 marks]

6. Construct an interpretation to show that ~∃x∀yDxy does not entail ~∃xDxx. Provide the UD, names, and predicate extensions. Note: there are infinite possible answers! [10 marks]