Formal logic is a set of rules and assumptions that allow for precise and rigorous deductions to be made based on given premises. While it has been used successfully in fields such as mathematics, philosophy, and computer science, there are some limitations to its usefulness in the business world.

One limitation is that formal logic is based on the assumption that all statements are either true or false, and that all deductions are valid if the premises are true. However, in real-world business scenarios, there may be uncertainty and ambiguity that cannot be easily captured within the confines of formal logic. For example, a business decision may depend on a range of factors, including market trends, customer preferences, and regulatory requirements. These factors may not be easily reducible to a set of precise, statements that can be analysed using formal logic.

Formal logic can be challenging to apply in situations where there are multiple interpretations of the premises or conclusions. For instance, the basic logical connectives in formal logic are typically taken to have a standard interpretation, but there are alternative interpretations that have been proposed by logicians and philosophers. Here are some examples:

1. Negation: The standard interpretation of negation is that it flips the truth value of a statement. For example, "It is not raining" is the negation of "It is raining". An alternative interpretation of negation is as denial. This interpretation views negation as denying the truth of a statement, rather than simply reversing its truth value.

2. Conjunction: The standard interpretation of conjunction is that it combines two statements and is true only if both statements are true. For example, "It is raining and the ground is wet" is true only if it is both raining and the ground is wet. An alternative interpretation of conjunction is as aggregation. This interpretation views conjunction as simply putting two statements together, without making any claim about their individual truth values.

3. Disjunction: The standard interpretation of disjunction is that it combines two statements and is true if either statement is true. For example, "It is raining or it is sunny" is true if it is either raining or sunny (or both). An alternative interpretation of disjunction is as alternative denial. This interpretation views disjunction as denying the conjunction of two statements. In other words, it asserts that at least one of the statements is false.

4. Implication: The standard interpretation of implication is that it expresses a conditional relationship between two statements. For example, "If it is raining, then the ground is wet" asserts that if it is raining, then the ground will be wet. An alternative interpretation of implication is as material implication. This interpretation views implication as a statement about the truth values of two statements, rather than their relationship. It asserts that if the antecedent (the first statement) is false, then the whole implication is true, regardless of the truth value of the consequent (the second statement).

It's worth noting that these alternative interpretations of the logical connectives are not universally accepted, and there is ongoing debate among logicians and philosophers about their merits and limitations. However, they demonstrate the flexibility and nuance of formal logic, and how it can be adapted and refined to suit different contexts and purposes.