Fallibilism Fallibilism (from medieval Latin fallibilis, "liable to err") is the philosophical principle that human beings could be wrong about their beliefs, expectations, or their understanding of the world, and yet still be justified in holding their incorrect beliefs. In the most commonly used sense of the term, this consists in being open to new evidence that would disprove some previously held position or belief, and in the recognition that "any claim justiﬁed today may need to be revised or withdrawn in light of new evidence, new arguments, and new experiences." This position is taken for granted in the natural sciences. In another sense, it refers to the consciousness of "the degree to which our interpretations, valuations, our practices, and traditions are temporally indexed" and subject to (possibly arbitrary) historical flux and change. Such "time-responsive" fallibilism consists in an openness to the confirmation of a possibility that one anticipates or expects in the future.
Certainty Certainty is perfect knowledge that has total security from error, or the mental state of being without doubt. Objectively defined, certainty is total continuity and validity of all foundational inquiry, to the highest degree of precision. Something is certain only if no skepticism can occur. Philosophy (at least historical Cartesian philosophy) seeks this state. History Pyrrho – ancient Greece Pyrrho is credited as being the first Skeptic philosopher. Al-Ghazali – Islamic theologian Al-Ghazali was a professor of philosophy in the 11th century. Ibn-Rushd - Averroes Latinized name Averroës Averroes was a defender of Aristotelian philosophy against Ash'ari theologians led by Al-Ghazali. Descartes – 17th century Descartes' Meditations on First Philosophy is a book in which Descartes first discards all belief in things which are not absolutely certain, and then tries to establish what can be known for sure. Ludwig Wittgenstein – 20th century Quotes
Second-order logic In logic and mathematics second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence Syntax and fragments A sort of variables that range over sets of individuals. Each of the variables just defined may be universally and/or existentially quantified over, to build up formulas. It's possible to forgo the introduction of function variables in the definition given above (and some authors do this) because an n-ary function variable can be represented by a relation variable of arity n+1 and an appropriate formula for the uniqueness of the "result" in the n+1 argument of the relation. or , where is a first-order formula. , or even as ∃SO. has the form
Solipsism Solipsism ( i/ˈsɒlɨpsɪzəm/; from Latin solus, meaning "alone", and ipse, meaning "self") is the philosophical idea that only one's own mind is sure to exist. As an epistemological position, solipsism holds that knowledge of anything outside one's own mind is unsure; the external world and other minds cannot be known and might not exist outside the mind. Varieties There are varying degrees of solipsism that parallel the varying degrees of serious skepticism.  Epistemological solipsism Epistemological solipsism is the variety of idealism according to which only the directly accessible mental contents of the solipsistic philosopher can be known. Epistemological solipsists claim that realism requires the question: assuming that there is a universe independent of an agent's mind and knowable only through the agent's senses, how is the existence of this independent universe to be scientifically studied? Methodological solipsism Main points History
Probabilism In theology and philosophy, probabilism (from Latin probare, to test, approve) refers to an ancient Greek doctrine of academic skepticism. It holds that in the absence of certainty, probability is the best criterion. It can also refer to a 17th-century religious thesis about ethics, or a modern physical-philosophical thesis. Theology This view was advanced by the Spanish theologian Bartolomé de Medina (1527–1581) and defended by many Jesuits such as Luis Molina (1528–1581). It was heavily criticised by Blaise Pascal in his Provincial Letters as leading to moral laxity. The doctrine became particularly popular at the start of the 17th century, as it could be used to support almost any position or council any advice. Philosophy Academic skeptics accept probabilism, while Pyrrhonian skeptics do not. In modern usage, a probabilist is someone who believes that central epistemological issues are best approached using probabilities. See also Sources and references
Higher-order logic In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics. Higher-order logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic. Higher-order simple predicate logic The term "higher-order logic", abbreviated as HOL, is commonly used to mean higher order simple predicate logic. There are two possible semantics for HOL. HOL with standard semantics is more expressive than first-order logic. The model-theoretic properties of HOL with standard semantics are also more complex than those of first-order logic. In Henkin semantics, a separate domain is included in each interpretation for each higher-order type. Examples See also References External links Miller, Dale, 1991, "Logic: Higher-order," Encyclopedia of Artificial Intelligence, 2nd ed.Herbert B.
Agnosticism Agnosticism is the view that the truth values of certain claims—especially claims about the existence or non-existence of any deity, as well as other religious and metaphysical claims—are unknown or unknowable. According to the philosopher William L. Rowe, in the popular sense, an agnostic is someone who neither believes nor disbelieves in the existence of a deity or deities, whereas a theist and an atheist believe and disbelieve, respectively. Thomas Henry Huxley, an English biologist, coined the word agnostic in 1869. Since the time that Huxley coined the term, many other thinkers have extensively written about agnosticism. Defining agnosticism Thomas Henry Huxley said: Agnosticism, in fact, is not a creed, but a method, the essence of which lies in the rigorous application of a single principle ... According to philosopher William L. Etymology Early Christian church leaders used the Greek word gnosis (knowledge) to describe "spiritual knowledge".
Non-monotonic logic Abductive reasoning Abductive reasoning is the process of deriving the most likely explanations of the known facts. An abductive logic should not be monotonic because the most likely explanations are not necessarily correct. For example, the most likely explanation for seeing wet grass is that it rained; however, this explanation has to be retracted when learning that the real cause of the grass being wet was a sprinkler. Reasoning about knowledge If a logic includes formulae that mean that something is not known, this logic should not be monotonic. Belief revision Belief revision is the process of changing beliefs to accommodate a new belief that might be inconsistent with the old ones. Proof-theoretic versus model-theoretic formalizations of non-monotonic logics See also References N. External links
Mathematical logic For Quine's theory sometimes called "Mathematical Logic", see New Foundations. Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. Subfields and scope The Handbook of Mathematical Logic makes a rough division of contemporary mathematical logic into four areas: History Mathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditional study of logic (Ferreirós 2001, p. 443).
Skepticism Skepticism or scepticism (see American and British English spelling differences) is generally any questioning attitude towards knowledge, facts, or opinions/beliefs stated as facts, or doubt regarding claims that are taken for granted elsewhere. Philosophical skepticism is an overall approach that requires all information to be well supported by evidence. Classical philosophical skepticism derives from the 'Skeptikoi', a school who "asserted nothing". Adherents of Pyrrhonism, for instance, suspend judgment in investigations. Skeptics may even doubt the reliability of their own senses. Religious skepticism, on the other hand, is "doubt concerning basic religious principles (such as immortality, providence, and revelation)". Definition In ordinary usage, skepticism (US) or scepticism (UK) (Greek: 'σκέπτομαι' skeptomai, to think, to look about, to consider; see also spelling differences) refers to: Philosophical skepticism Scientific skepticism Media
Monotonicity of entailment Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called monotonic logics in order to differentiate them from non-monotonic logics. Weakening rule To illustrate, starting from the natural deduction sequent: weakening allows one to conclude: Non-monotonic logics In most logics, weakening is either an inference rule or a metatheorem if the logic doesn't have an explicit rule. See also