Why is translating ordinary-language sentences into logical notation difficult?

Translating ordinary-language sentences into logical notation is difficult because natural languages and formal logical systems operate differently, and the process of mapping one to the other is not trivial.12

Key reasons for this difficulty include:

• Ambiguity of Natural Language: Early symbolic logicians, such as Gottlob Frege and Bertrand Russell, argued that natural-language verbs like "is" are multiply ambiguous, representing distinct concepts such as identity, predication, existence, or class inclusion. However, it is not clear that such words are truly ambiguous; they may instead possess a single sense that is interpreted differently based on context.132
• Complex Logical Forms: Theoretical linguists have struggled to formulate general rules to derive logical formulas from ordinary sentences. For example, "donkey sentences"—such as "If Peter owns a donkey, he beats it"—are notoriously difficult because the indefinite article "a" functions as an existential quantifier in English, yet the logical form requires a universal quantifier to capture the intended meaning.12
• Structural Differences: In symbolic logic, scopes (indicated by parentheses) define both variable binding and logical priority. Natural languages do not always follow these rigid structural rules, and it is possible to construct logical systems where these functions are separated or represented differently.2

Ultimately, while logical notation provides a precise framework for analyzing inference, it is often viewed as an instructive comparison to natural language rather than a direct replacement.2