Logic translation

Logic translation is the process of representing a text in the formal language of a logical system. If the original text is formulated in ordinary language then the term natural language formalization is often used. An example is the translation of the English sentence "some men are bald" into first-order logic as ∃ x ( M ( x ) ∧ B ( x ) ) {\displaystyle \exists x(M(x)\land B(x))} . The purpose is to reveal the logical structure of arguments. This makes it possible to use the precise rules of formal logic to assess whether these arguments are correct. It can also guide reasoning by arriving at new conclusions. Many of the difficulties of the process are caused by vague or ambiguous expressions in natural language. For example, the English word "is" can mean that something exists, that it is identical to something else, or that it has a certain property. This contrasts with the precise nature of formal logic, which avoids such ambiguities. Natural language formalization is relevant to various fields in the sciences and humanities. It may play a key role for logic in general since it is needed to establish a link between many forms of reasoning and abstract logical systems. The use of informal logic is an alternative to formalization since it analyzes the cogency of ordinary language arguments in their original form. Natural language formalization is distinguished from logic translations that convert formulas from one logical system into another, for example, from modal logic to first-order logic. This form of logic translation is specifically relevant for logic programming and metalogic. A major challenge in logic translation is determining the accuracy of translations and separating good from bad ones. The technical term for this is criteria of adequate translations. An often-cited criterion states that translations should preserve the inferential relations between sentences. This implies that if an argument is valid in the original text then the translated argument should also be valid. Another criterion is that the original sentence and the translation have the same truth conditions. Further suggested conditions are that a translation does not include additional or unnecessary symbols and that its grammatical structure is similar to the original sentence. Various procedures for translating texts have been suggested. Preparatory steps include understanding the meaning of the original text and paraphrasing it to remove ambiguities and make its logical structure more explicit. As an intermediary step, a translation may happen into a hybrid language. This hybrid language implements a logical formalism but retains the vocabulary of the original expression. In the last step, this vocabulary is replaced by logical symbols. Translation procedures are usually not exact algorithms and their application depends on intuitive understanding. Logic translations are often criticized on the grounds that they are unable to accurately represent all the aspects and nuances of the original text.

Definition A logic translation is a translation of a text into a logical system. For example, translating the sentence "all skyscrapers are tall" as ∀ x ( S ( x ) → T ( x ) ) {\displaystyle \forall x(S(x)\to T(x))} is a logic translation that expresses an English language sentence in the logical system known as first-order logic. The aim of logic translations is usually to make the logical structure of natural language arguments explicit. This way, the rules of formal logic can be used to assess whether the arguments are valid. Understood in a wide sense, a translation is a process that associates expressions belonging to a source language with expressions belonging to a target language. For example, in a sentence-by-sentence translation of an English text into French, English sentences are linked to their French counterparts. The hallmark of logic translations is that the target language belongs to a logical system. Logic translations differ from regular translations in that they are mainly concerned with expressing the logical structure of the original text and less with its concrete content. Regular translations, on the other hand, take various additional factors into account pertaining to the content, meaning, and style of the original expression. For this reason, some theorists, like Peregrin and Svoboda, have argued that it is not a form of translation. They tend to use other terms, such as "formalization", "symbolization", and "explication". This opinion is not shared by all logicians and some, like Mark Sainsbury, argue that successful logic translations preserve all the original meaning while making the logical structure explicit. Discussions on logic translations usually focus on the problem of expressing the logical structure of ordinary language sentences in a formal logical system. The term also covers cases where the translation happens from one logical system into another.

Basic concepts

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