Understanding First-order Logic 123 Wilfrid Augustine Hodges (b 27.05.1941) is a British logician and mathematician, a prominent expert in classical logic and model theory, an enthusiastic explorer of the history of logic, particularly of Arabic logic, author of some of the most popular logic textbooks, and passionate popularizer of logic in general Hodges studied at Oxford during 1959–65, where he received degrees in both Literae Humaniores and Theology Then he did doctoral studies in logic (model theory) with John Crossley, and in 1970 was awarded a doctorate He then lectured in both Philosophy and Mathematics at Bedford College, University of London In 1984 he moved to Queen Mary College, University of London, where he was a Professor of Mathematics until his retirement in 2008; he is now an Emeritus Professor Hodges has written several very influential textbooks in logic in a lively, rather informal, yet stimulating and thought-provoking style His 1997 Logic: An Introduction to Elementary Logic is still one of the best popular expositions of the subject, while his 1993 monograph Model Theory and the 1997 version A Shorter Model Theory are top references in that major field of logic His 2007 Mathematical Logic written with I Chiswell has become one of the most appreciated textbooks on the subject Hodges also developed compositional semantics for the so called Independence-Friendly logic Since the early 2000s Hodges has also been deeply involved in exploring and interpreting the work in logic and algorithmics of the Persian encyclopedic thinker of the Islamic Golden Age: Ibn Sin¯a (Avicenna) Hodges was, inter alia, President of the British Logic Colloquium, of the European Association for Logic, Language, and Information, and of the Division of Logic, Methodology, and Philosophy of Science of the International Union of History and Philosophy of Science In 2009 he was elected a Fellow of the British Academy 3.3 Basic grammar and use of first-order languages The best (and only) way to learn a new language is by using it, in particular, by practicing translation between that language and one that you know well In this section I discuss how to use first-order languages For that you will have to understand their basic grammar, but first let us look at some examples of translating statements from natural language to first-order languages 3.3.1 Translation from natural language to first-order languages: warm-up Let us start with translating “There is an integer greater than and less than 3.”