Inductive logic in the context of Mathematical induction

Early Islamic philosophy or classical Islamic philosophy is a period of intense philosophical development beginning in the 2nd century AH of the Islamic calendar (early 9th century CE) and lasting until the 6th century AH (late 12th century CE). The period is known as the Islamic Golden Age, and the achievements of this period had a crucial influence in the development of modern philosophy and science. For Renaissance Europe, "Muslim maritime, agricultural, and technological innovations, as well as much East Asian technology via the Muslim world, made their way to western Europe in one of the largest technology transfers in world history." This period starts with al-Kindi in the 9th century and ends with Averroes (Ibn Rushd) at the end of 12th century. The death of Averroes effectively marks the end of a particular discipline of Islamic philosophy usually called the Peripatetic Arabic School, and philosophical activity declined significantly in Western Islamic countries, namely in Islamic Spain and North Africa, though it persisted for much longer in the Eastern countries, in particular Persia and India where several schools of philosophy continued to flourish: Avicennism, Illuminationist philosophy, Mystical philosophy, and Transcendent theosophy.

Intellectual innovations, achievements, and advancements of this period included, within jurisprudence, the development of ijtihad, a method or methodological approach to legal reasoning, interpretation, and argument based on independent inquiry and analogical deduction; within science and the philosophy of science, the development of empirical research methods emphasizing controlled experimentation, observational evidence, and reproducibility, as well as early formulations of empiricist epistemologies; commentaries and developments in Aristotelian logic, as well as innovations in non-Aristotelian temporal modal logic and inductive logic; and developments in research practice and methodology, including, within medicine, the first documented peer review process and within jurisprudence and theology, a strict science of citation, the isnad or "backing".

Ceteris paribus (also spelled caeteris paribus) (Classical Latin pronunciation: [ˈkeːtɛ.riːs ˈpa.rɪ.bʊs]) is a Latin phrase, meaning "other things equal"; some other English translations of the phrase are "all other things being equal", "other things held constant", "all else unchanged", and "all else being equal". A statement about a causal, empirical, moral, or logical relation between two states of affairs is ceteris paribus if it is acknowledged that the statement, although usually accurate in expected conditions, can fail because of, or the relation can be abolished by, intervening factors.

A ceteris paribus assumption is often key to scientific inquiry, because scientists seek to eliminate factors that perturb a relation of interest. Thus epidemiologists, for example, may seek to control independent variables as factors that may influence dependent variables—the outcomes of interest. Likewise, in scientific modeling, simplifying assumptions permit illustration of concepts considered relevant to the inquiry. An example in economics is "If the price of milk falls, ceteris paribus, the quantity of milk demanded will rise." This means that, if other factors, such as deflation, pricing objectives, utility, and marketing methods, do not change, the decrease in the price of milk will lead to an increase in demand for it.

Reflective equilibrium is a state of balance or coherence among a set of beliefs arrived at by a process of deliberative mutual adjustment among general principles and particular judgements. Although he did not use the term, philosopher Nelson Goodman introduced the method of reflective equilibrium as an approach to justifying the principles of inductive logic (this is now known as Goodman's method). The term reflective equilibrium was coined by John Rawls and popularized in his A Theory of Justice as a method for arriving at the content of the principles of justice.

Dietmar Hübner (de) has pointed out that there are many interpretations of reflective equilibrium that deviate from Rawls' method in ways that reduce the cogency of the idea. Among these misinterpretations, according to Hübner, are definitions of reflective equilibrium as "(a) balancing theoretical accounts against intuitive convictions; (b) balancing general principles against particular judgements; (c) balancing opposite ethical conceptions or divergent moral statements".

The raven paradox, also known as Hempel's paradox, Hempel's ravens or, rarely, the paradox of indoor ornithology, is a paradox arising from the question of what constitutes evidence for the truth of a statement. Observing objects that are neither black nor ravens may formally increase the likelihood that all ravens are black even though, intuitively, these observations are unrelated.

This problem was proposed by the logician Carl Gustav Hempel in the 1940s to illustrate a contradiction between inductive logic and intuition.

The Existence of God is a 1979 book by British philosopher of religion Richard Swinburne, claiming the existence of the Abrahamic God on rational grounds. The argument rests on an updated version of natural theology with biological evolution using scientific inference, mathematical probability theory, such as Bayes' theorem, and of inductive logic. In 2004, a second edition was released under the same title.

Swinburne discusses the intrinsic probability of theism, with an everlastingly omnipotent, omniscient and perfectly free God. He states various reasons for the existence of God, such as cosmological and teleological arguments, arguments from the consciousness of the higher vertebrates including humans, morality, providence, history, miracles and religious experience. Swinburne claims that the occurrence of evil does not diminish the probability of God, and that the hiddenness of God can be explained by his allowing free choice to humans. He concludes that on balance it is more probable than not that God exists, with a probability larger than 0.5, on a scale of 0.0 (impossible) to 1.0 (absolutely sure).