Question 1 Consider the following model of the quantified relational logic of a hypothetical love triangle. U = {Alice, Bob, Carol, Dominic} a=Alice,b=Bob,c=Carol,d=Dominic [L] = {< Alice, Alice >, < Bob, Bob >, < Dominic, Dominic >, < Alice, Dominic >, < Dominic, Carol >, < Carol, Alice >, < Alice, Carol >, < Bob, Alice >, < Bob, Dominic >} Determine which of the following sentences are true in the model. You may read the formula Lxy as “x loves y”. If an existential sentence is true, provide at least one witness. If a universal sentence is false, provide at least one counterexample. (a) Lac (b) Lca (c) ∃z∃y(Lzy ∧ Lyz) (d) ∀zLzz (e) ∀z∃y(Lzz ∨ Lyz) (f) ∀z∀y(Lzy → Lyz) (g) ∀z∃y(Lzz → Lyz) (h) ∃z∀y(Lzz ∧ ¬Lyz) (i) ∃z∃y(Lzy ∧ ¬Lyz) (j) ∃x∃y∃z((Lxz ∧ Lxy) ∧ Lxx)
Question 1
Consider the following model of the quantified relational logic of a hypothetical love triangle.
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U = {Alice, Bob, Carol, Dominic}
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a=Alice,b=Bob,c=Carol,d=Dominic
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[L] = {< Alice, Alice >, < Bob, Bob >, < Dominic, Dominic >, < Alice, Dominic >,
< Dominic, Carol >, < Carol, Alice >, < Alice, Carol >, < Bob, Alice >, < Bob, Dominic >}
Determine which of the following sentences are true in the model. You may read the formula Lxy as “x loves y”. If an existential sentence is true, provide at least one witness. If a universal sentence is false, provide at least one counterexample.
(a) Lac
(b) Lca
(c) ∃z∃y(Lzy ∧ Lyz)
(d) ∀zLzz
(e) ∀z∃y(Lzz ∨ Lyz)
(f) ∀z∀y(Lzy → Lyz)
(g) ∀z∃y(Lzz → Lyz)
(h) ∃z∀y(Lzz ∧ ¬Lyz)
(i) ∃z∃y(Lzy ∧ ¬Lyz)
(j) ∃x∃y∃z((Lxz ∧ Lxy) ∧ Lxx)
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