M) and "is mortal" (here D): the sentence is given by the judgement A(M,D). In predicate logic, the sentence involves the same two nonlogical concepts, here analyzed as and , and the sentence is given by , involving the logical connectives for universal quantification and implication. • But equally, the modern view is more powerful. Medieval logicians recognized the problem of multiple generality, where Aristotelian logic is unable to satisfactorily render such sentences as "Some guys have all the luck", because both quantities "all" and "some" may be relevant in an inference, but the fixed scheme that Aristotle used allows only one to govern the inference. Just as linguists recognize recursive structure in natural languages, it appears that logic needs recursive structure. Deductive and inductive reasoning, and abductive inference Deductive reasoning concerns what follows necessarily from given premises (if a, then b). However, inductive reasoning—the process of deriving a reliable generalization from observations—has sometimes been included in the study of logic. Similarly, it is important to distinguish deductive validity and inductive validity (called "cogency"). An inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion false. An inductive argument can be neither valid nor invalid; its premises give only some degree of probability, but not certainty, to its conclusion. The notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics. Inductive validity on the other hand requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use mathematical models of probability. For the most part this discussion of logic deals only with deductive logic. Abduction is a form of logical inference that goes from observation to a hypothesis that accounts for the reliable data (observation) and seeks to explain relevant evidence. The American philosopher Charles Sanders Peirce (1839–1914) first introduced the term as "guessing". Peirce said that to abduce a hypothetical explanation from an observed surprising circumstance is to surmise that may be true because then would be a matter of course. Thus, to abduce from involves determining that is sufficient (or nearly sufficient), but not necessary, for . Consistency, validity, soundness, and completeness Among the important properties that logical systems can have: • Consistency, which means that no theorem of the system contradicts another. • Validity, which means that the system's rules of proof never allow a false inference from true premises. A logical system has the property of Page 25 of 45

soundness when the logical system has the property of validity and uses only premises that prove true (or, in the case of axioms, are true by definition). • Completeness, of a logical system, which means that if a formula is true, it can be proven (if it is true, it is a theorem of the system). • Soundness, the term soundness has multiple separate meanings, which creates a bit of confusion throughout the literature. Most commonly, soundness refers to logical systems, which means that if some formula can be proven in a system, then it is true in the relevant model/structure (if A is a theorem, it is true). This is the converse of completeness. A distinct, peripheral use of soundness refers to arguments, which means that the premises of a valid argument are true in the actual world. Some logical systems do not have all four properties. As an example, Kurt Gödel's incompleteness theorems show that sufficiently complex formal systems of arithmetic cannot be consistent and complete; however, first-order predicate logics not extended by specific axioms to be arithmetic formal systems with equality can be complete and consistent. Rival conceptions of logic Main article: Definitions of logic Logic arose (see below) from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality. Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations". By contrast, Immanuel Kant argued that logic should be conceived as the science of judgement, an idea taken up in Gottlob Frege's logical and philosophical work. But Frege's work is ambiguous in the sense that it is both concerned with the "laws of thought" as well as with the "laws of truth", i.e. it both treats logic in the context of a theory of the mind, and treats logic as the study of abstract formal structures. Page 26 of 45