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FOPL Natural Deduction Proofs: Inferences and Axiomatization in Logic

This document outlines the proofs in First-Order Predicate Logic (FOPL) using a provided set of constants, variables, and predicates. It covers inferences such as the existence of likes among students and relationships of being broke and happy, translated into English sentences. Additionally, it presents the axiomatization of relationships regarding age and height among three individuals, requiring translations into FOPL. The goal is to prove various logical statements using the Natural Deduction proof system based on the established axioms.

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FOPL Natural Deduction Proofs: Inferences and Axiomatization in Logic

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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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