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Revison: Translation from English to awFOL

\emph{Exercise} Translate the following sentences of English into awFOL using the interpretation below:
\hspace{5mm} L(x,y) : x is a logical consequence of y
\hspace{5mm} N(x,y) : x is the negation of y
\hspace{5mm} S(x) : x is a sentence
\hspace{5mm} a : ‘Fire melts ice’
i. ‘Fire melts ice’ is a sentence
ii. There is a sentence
iii. There is a sentence which is the negation of ‘Fire melts ice’
iv. Some sentences are contradictions and all contradictions are logically equivalent.

i. ‘Fire melts ice’ is a sentence

S(a)

ii. There is a sentence

∃y S(y)

iii. There is a sentence which is the negation of ‘Fire melts ice’

∃y ( S(y) ∧ N(y,a) )

iv. Some sentences are contradictions and all contradictions are logically equivalent.

This kind of thing is just here to make you cry like a baby; you’re not supposed to be able to get 100% without being exceptionally good at alogic.
But let’s look at how you could get something approximately this complex ...

i. Some sentences are logically equivalent.

∃x ∃y ( S(y) ∧ S(x) ∧ LgcllyEqvlnt(x,y) )

LgcllyEqvlnt(x,y) = L(x,y) ∧ L(y,x)

ii. Some sentences are contradictions.

∃y ( S(y) ∧ Contradiction(y) )

Contradiction(y) = ∀x L(y,x)

iii. All contradictions are logically equivalent.

∀x ∀y ( Contradiction(x) ∧ Contradiction(y) → LgcllyEqvlnt(y) )