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CA 208 Logic Ex6

CA 208 Logic Ex6. Given a (specific) language of FOPL with CONST = {j,k,m}, VARS = {x,y,z} and PRED = {student ¹ , broke ¹ , happy ¹, like ² }, prove the following in the Natural Deduction proof system: {like(j,m)} |- x like(x,m) {like(j,m)} |- y like(y,m) {like(j,m)} |- x like(j,x)

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CA 208 Logic Ex6

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  1. CA 208 Logic Ex6 • Given a (specific) language of FOPL with CONST = {j,k,m}, VARS = {x,y,z} and PRED = {student¹, broke¹, happy¹, like²}, prove the following in the Natural Deduction proof system: • {like(j,m)} |- x like(x,m) • {like(j,m)} |- y like(y,m) • {like(j,m)} |- x like(j,x) • {like(j,m)} |- x y like(x,y) • {x y (like(x,y)} |- y x like(y,x) • {x y (like(x,y)} |- like(m,m) • {x y (like(x,y)} |- x like(x,x) • {x (student(x)  broke(x)), student(k)} |- broke(k) • {x y (like(x,y)  happy(y)), like(k,j)} |- happy(j) • {x (student(x)  broke(x)), x (broke(x)  happy(x))} |- z (student(z)  happy(z)) • Translate the FOPL inferences above into corresponding sentences in English (assume that j translates to John, m to Mary and k to Kate while the predicate symbols translate into the corresponding English verbs (like), nouns (student) and adjectives (broke, happy)).

  2. CA 208 Logic Ex6 • Axiomatise (i.e. describe) the following situation in FOPL with CONST = {j,k,m}, VARS = {x,y,z} and PRED = {as_old_as², taller²} • Kate is as old as John. John is as old Mary. • Everybody is as old as themselves. • If x is as old as y, and y is as old as z, then x is as old as z. • Kate is taller than John. John is taller than Mary. • Nobody is taller than themselves. • If x is taller than y, and y is taller than z, then x is taller than z. • Translate the following into FOPL and prove the resulting formulas from the axiomatisation above in the Natural Deduction proof system: • Kate is as old as Kate. • Kate is as old as Mary. • John is not taller than John. • Kate is taller than Mary.

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