Syllogisms are arguments about the properties of entities. They consist of 2 premises and a conclusion, which can each be in 1 of 4 "moods": All A are B, Some A are B, No A are B, and Some A are not B. Their logical analysis began with Aristotle, and their psychological investigation began over 100 years ago. This article outlines the logic of inferences about syllogisms, which includes the evaluation of the consistency of sets of assertions. It also describes the main phenomena of reasoning about properties. There are 12 extant theories of such inferences, and the article outlines each of them and describes their strengths and weaknesses. The theories are of 3 main sorts: heuristic theories that capture principles that could underlie intuitive responses, theories of deliberative reasoning based on formal rules of inference akin to those of logic, and theories of deliberative reasoning based on set-theoretic diagrams or models. The article presents a meta-analysis of these extant theories of syllogisms using data from 6 studies. None of the 12 theories provides an adequate account, and so the article concludes with a guide-based on its qualitative and quantitative analyses-of how best to make progress toward a satisfactory theory.

CHAPTER I ELEMENTS OF THE SYSTEM § I. The true form of the Aristotelian syllogism I § 2. Premisses and terms 3 § 3· Why singular terms were omitted by Aristotle . 5 § 4- Variables . . . . . 7 § 5· Syllogistic necessity. . . . 10 §6. What is formal logic? 12 § 7- What is formalism? 15 CHAPTER II THESES OF THE SYSTEM §8. Theses and rules of inference. 20 § 9· The syllogistic figures 23 § io. The major, middle, and minor terms . 28 § “ · The history of an error 30 § 12. The order of the premisses . 32 § 13· Errors of some modern commentators . 34 § Η· The four Galenian figures 38 CHAPTER III THE SYSTEM § 15· Perfect and imperfect syllogisms 43 § 16. The logic of terms and the logic of propositions 47 § 17· The proofs by conversion 5 1 § 18. The proofs by reductio ad impossibile 54 § 19· The proofs by ecthesis 59 § 20. The rejected forms . 67 § 21. Some unsolved problems 72 CHAPTER IV A r i s t o t l e ’s system in s y m b o l ic fo r m § 2 2 . Explanation of the symbolism 77 § 23. Theory of deduction 79 § 24. Quantifiers . .. . 83 § 2 5 .Fundamentals of the syllogistic 88 §26. Deduction of syllogistic theses 90 § 2 7 . Axioms and rules for rejected expressions 94 §28. Insufficiency of our axioms and rules . 98 CHAPTER V THE PROBLEM OF DECISION § 29. The number of undecidable expressions IOO § 30. Slupecki's rule of rejection . 103 § 31· Deductive equivalence 106 § 32. Reduction to elementary expressions . I I I § 3 3 - Elementary expressions of the syllogistic 120 § 3 4 · An ari

But as all truth, real as well as formal, is consistent, formal rules of consistency become real rules of truth, when the premises are true and the consistent conclusion is therefore true. The science of inference again rightly emphasizes the formal thinking of the syllogism in which the combination of premises involves the conclusion. But the combinations of premises in analogical and inductive inference, although the combination does not involve the conclusion, yet causes us to infer it, and in so similar a way that the science of inference is not complete without investigating all the combinations which characterize different kinds of inference. The question of logic is how we infer in fact, as well as perfectly; and we cannot understand inference unless we consider inferences of probability of all kinds. Moreover, the study of analogical and inductive inference is necessary to that of the syllogism itself, because they discover the premises of syllogism. The formal thinking of syllogism alone is merely necessary consequence; but when its premises are necessary principles, its conclusions are not only necessary consequents but also necessary truths.

the "subject", and which is capable of truth or falsity. The syllogism is an inference in which one proposition (the "conclusion") follows of necessity In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, the Peripatetics. It was revived after the third century CE by Porphyry's Isagoge. Term logic revived in medieval times, first in Islamic l providing it with mathematical foundations involving equations; extending the class of problems it could treat– from assessing validity to solving equations; and expanding the range of applications it could handle– e.g. from propositions having only two terms to those having arbitrarily many. More specifically, Boole agreed with what Aristotle said; Boole's ‘disagreements’, if they might be called that, concern…

Reasoning, conscious deliberate inference; the activity or process of reasoning.: Gregories Church in St. Pauls Church-yard|year=1658|page=196|pageurl=https://books.google.com/books?id=hEVPAQAAIAAJ&pg=PA196|oclc=11086058|passage=But it will be apparent from the refutation of the ſecond Falſification, wherewith you charge the Author of the Letters, that theſe miſchievous conſequences are rightly drawn from the wicked principle layd down by Vaſquez [ ] in the ſame place, and accordingly, that that Jeſuit hath not done violence to the rules of ratiocination, but to thoſe of the Goſpel.}} #* #* {{RQ:Mill System of Logic|chapter=Of Inference, or Reasoning, in General|volume=I|section=3|page=223|passage=Reasoning, in the extended sense in which I use the term, and in which it is synonymous with Inference, is popularly said to be of two kinds: reasoning from particulars to generals, and reasoning from generals to particulars; the former being called Induction, the latter Ratiocination or Syllogism. The meaning intended by these expressions is, that Induction is inferring a proposition from propositions less general than itself, and Ratiocination is inferring a proposition from propositions equally or more general.}} #* #*

Through out this paper, a broader class of conjunction functions and the class of their right-residual implication functions are treated. Various examples of such functions are enumerated, and a class of non-commutative conjunctions is proposed. As the main results of this work, the authors show some conditions for satisfaction of modus ponens, modus tollens and syllogism under "the compositional rule of inference" and "the translating rule" represented by such conjunctions and implications, respectively, and discuss the adequate combinations of conjunctions and implications for fuzzy reasoning.

‘P’ by using the inference rule of Modus (ponendo) ponens, i.e. referring to an inference schema. Alexander … assigned as an accident. Once the questioner has found out that the predicate has been assigned as an accident … that a topos is a universal proposition and functions as a hypothetical premiss in hypothetical syllogisms