Question

Explain in your own words the difference between the formulas (∃x)(A(x)→T(x)) and (∃x)(A(x)∧T(x)). Afterwards, justify that if the domain of interpretation is a fixed group of people, A(x) is the predicate that x is an ambassador, and T(x) is that x travels, then the sentence "some ambassador travels" should really be translated into (∃x)(A(x)∧T(x)), rather than (∃x)(A(x)→T(x)).

Answer 1

The formula (∃x)(A(x)→T(x)) represents the statement 'there exists an x such that if x is an ambassador, then x travels'. The formula (∃x)(A(x)∧T(x)) represents the statement 'there exists an x such that x is an ambassador and x travels'. If the domain of interpretation is a fixed group of people, the sentence 'some ambassador travels' should be translated into (∃x)(A(x)∧T(x))

Step-by-step explanation:

The formula (∃x)(A(x)→T(x)) represents the statement 'there exists an x such that if x is an ambassador, then x travels'. This formula means that there is at least one person who, if they are an ambassador, then they travel.

The formula (∃x)(A(x)∧T(x)) represents the statement 'there exists an x such that x is an ambassador and x travels'. This formula means that there is at least one person who is both an ambassador and travels.

If the domain of interpretation is a fixed group of people, where A(x) represents that x is an ambassador and T(x) represents that x travels, then the sentence 'some ambassador travels' should be translated into (∃x)(A(x)∧T(x)) because it implies that there is someone who is both an ambassador and travels, rather than implying a conditional relationship between being an ambassador and traveling.