Charles Sanders Peirce, the father of both semeiotics and the logic of relations, argued that logic should be generalized to semeiotics, the philosophical theory of all varieties signs. He also averred that arguments are rational signs and, as such, are the most complex variety of signs. From early on in his inquiries Peirce recognized that there are precisely three summa genera of simple arguments, that is, arguments composed of three and only three propositions: two premisses and a conclusion. Such simple arguments are principally distinguished by virtue of their forms. In this talk only Aristotelean syllogisms will be considered. For each such deductive syllogism there is a counterpart abductive syllogism and a counterpart inductive syllogism. Following Kant, Peirce held that any deductive syllogism involved a rule (the major premiss), a case (the minor premiss) subsumed under that rule, and a result (the conclusion) drawn from the subsumption of that case under that rule. The corresponding abductive and inductive syllogisms are permutations of these three components of any deductive syllogism. These permutations are transformations; that is, they yield new forms. In abduction, a case follows from both a rule and a result, while in induction a rule follows from both a result and a case. Formal semeiotics utilizes concepts and tools from appropriate branches of mathematics to broaden and deepen our understanding of the nature and function of signs, in particular, group theory and category theory. Requisite algebraical formulas and geometrical diagrams shall be deployed throughout. The above contentions about arguments will be explicated and justified in two ways – first, group-theoretically and second, category-theoretically. The key group-theoretical insight is that abductive and inductive syllogisms are the results of the group action of either the permutation group S3 acting algebraically on the constituent categorical propositions of a deductive syllogism or the dihedral group D3 acting geometrically on the same. Since, S3, ≅ D3, the results of the group actions by these two groups agree, though their applications differ. This approach illumines the formal syntax of Peirce’s trichotomy of argument kinds. Abduction and induction can also be characterized category-theoretically. They are, respectively, cases of what pioneering category-theorist William Lawvere calls determination and choice problems or in abstract algebra, extension and lifting problems. These are two related varieties of generalized division problems. Wherever there is a binary operation over a set, that is, an operation of the form X ∘ Y = Z, there are two possible division problems – a left division (determination/extension) problem and a right division (choice/lifting) problem. Since according to Peirce, following de Morgan, a syllogism involves the binary operation of the colligation of its two premisses yielding a conclusion, it has two companion kinds of division problems. I will argue that abduction is a lifting and induction is an extension. Consequently, it again follows that there are exactly three major kinds of syllogisms. This account further illuminates the formal semantics of Peirce’s triad.