  The next logician of rank, if of lower rank, of whom we have sufficient evidence to speak is Galen (129–199 or 216 CE), who achieved greater fame as a physician. He studied logic with both Peripatetic and Stoic teachers, and recommended availing oneself of parts of either doctrine, as long as it could be used for scientific demonstration. He composed commentaries on logical works by Aristotle, Theophrastus, Eudemus and Chrysippus, as well as treatises on various logical problems and a major work entitled On Demonstration. All these are lost, except for some information in later texts, but his Introduction to Logic has come down to us almost in full. In On Demonstration, Galen developed, among other things, a theory of compound categorical syllogisms with four terms, which fall into four figures, but we do not know the details. He also introduced the so-called relational syllogisms, examples of which are ‘A is equal to B, B is equal to C; therefore A is equal to C’ and ‘Dio owns half as much as Theo; Theo owns half as much as Philo. Therefore Dio owns a quarter of what Philo owns’ (Galen Inst. Log, 17–18). All the relational syllogisms Galen mentions have in common that they are not reducible in either Aristotle’s or the Stoic syllogistic, but it is difficult to find further formal characteristics that unite them. In general, in his Introduction to Logic Galen merges Aristotelian Syllogistic with a strongly Peripatetic reinterpretation of Stoic propositional logic. This becomes apparent in particular in Galen’s emphatic denial that truth-preservation is sufficient for the validity or syllogismhood of an argument, and his insistence that, instead, knowledge-introduction or knowledge-extension is a necessary condition for something to count as a  syllogism.[5] 