First-Order Logic
First-order logic is a bag of tools for studying the validity of arguments. At base it consists of a family of mathematically defined languages called first-order languages. Because these languages are constructed to be "logically perfect" (in Gottlob Frege's phrase), we can guarantee from their grammatical form that certain arguments written in these languages are valid. Separately from this we can study how arguments in English or any other natural language can be translated into an appropriate first-order language. It was Gottfried Wilhelm Leibniz who in the 1680s first proposed to divide the study of arguments into a mathematical part and a translational part, though his notion of mathematical languages was barely adequate for the purpose. First-order languages first came to light in the work of Charles S. Peirce in the 1880s; his name for them was "first-intentional logic of relatives." It took some time to develop a satisfactory mathematical description of these languages. David Hilbert achieved this in his lectures at Göttingen in 1917–1922, which appeared in an edited form in his book Grundzüge der Theoretischen Logik with Wilhelm Ackermann. Many logicians reckon that the appearance of this book in 1928 marked the true birth of first-order logic.
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