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In propositional logic, modus ponens, also known as modus ponendo ponens or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore Q must also be true."
Abductive reasoning is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century. It starts with an observation or set of observations and then seeks the simplest and most likely conclusion from the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as inference to the best explanation, although not all usages of the terms abduction and inference to the best explanation are exactly equivalent.
Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.
In the philosophy of logic, A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion. For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic, in the sense that if the premises are true, then so is the conclusion.
In mathematical logic, a sequent is a very general kind of conditional assertion.
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word infer means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle. Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.
Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence, but not full assurance, of the truth of the conclusion. It is also described as a method where one's experiences and observations, including what is learned from others, are synthesized to come up with a general truth. Many dictionaries define inductive reasoning as the derivation of general principles from specific observations, although there are many inductive arguments that do not have that form.
The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.
Development of doctrine is a term used by John Henry Newman and other theologians influenced by him to describe the way Catholic teaching has become more detailed and explicit over the centuries, while later statements of doctrine remain consistent with earlier statements.
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that:
In logic, logical form of a statement is a precisely-specified semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly ambiguous statement into a statement with a precise, unambiguous logical interpretation with respect to a formal system. In an ideal formal language, the meaning of a logical form can be determined unambiguously from syntax alone. Logical forms are semantic, not syntactic constructs; therefore, there may be more than one string that represents the same logical form in a given language.
In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
An object of the mind is an object that exists in the imagination, but which, in the real world, can only be represented or modeled. Some such objects are abstractions, literary concepts, or fictional scenarios.
Probabilistic logic involves the use of probability and logic to deal with uncertain situations. The result is a richer and more expressive formalism with a broad range of possible application areas. Probabilistic logics attempt to find a natural extension of traditional logic truth tables: the results they define are derived through probabilistic expressions instead. A difficulty with probabilistic logics is that they tend to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as those of Dempster–Shafer theory in evidence-based subjective logic. The need to deal with a broad variety of contexts and issues has led to many different proposals.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and argument patterns symbolically, using formal systems such as first order logic. Within formal logic, mathematical logic studies the mathematical characteristics of logical systems, while philosophical logic applies them to philosophical problems such as the nature of meaning, knowledge, and existence. Systems of formal logic are also applied in other fields including linguistics, cognitive science, and computer science.
As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality. Arguments may be logical if they are "conducted or assessed according to strict principles of validity", while they are rational according to the broader requirement that they are based on reason and knowledge.
Illative sense is an epistemological concept coined by John Henry Newman (1801–1890) in his Grammar of Assent. For him it is the unconscious process of the mind, by which probabilities converge into certainty.
References
- Collins, James, ed. (1961). Philosophical Readings in Cardinal Newman. Chicago: Henry Regnery.CS1 maint: extra text: authors list (link)
- Cornwell, John (2010). Newman's Unquiet Grave: The Reluctant Saint. Continuum.
- Ker, Ian (1988). John Henry Newman: A Biography. New York, NY: Oxford. ISBN 0-19-282705-7.
- Newman, Jay (1986). The Mental Philosophy of John Henry Newman. Ontario, Canada: Wilfrid Laurier. ISBN 0-88920-186-2.
- Norris, Thomas J. (1977). Newman and His Theological Method: A Guide for the Theologian Today. Leiden: E. J. Brill. ISBN 90-04-04884-7.
- Walgrave, O.P., J. H., Newman the Theologian: The Nature of Belief and Doctrine as Exemplified in His Life and Works. Translated by A. V. Littledale (London: Geoffrey Chapman, 1960).
