Transcript CA 208 Logic - DCU School of Computing

CA 208 Logic Ex6
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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)).
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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}
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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:
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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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