Deductive reasoning Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from, i.e., there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning. Simple example An example of a deductive argument: All men are mortal.Socrates is a man.Therefore, Socrates is mortal. Law of detachment P → Q.
Abductive reasoning Form of logical inference which seeks the simplest and most likely explanation Abductive reasoning (also called abduction, abductive inference, or retroduction) is a form of logical inference that seeks the simplest and most likely conclusion from a set of observations. It was formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century. Abductive reasoning, unlike deductive reasoning, yields a plausible conclusion but does not definitively verify it. Abductive conclusions do not eliminate uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". In the 1990s, as computing power grew, the fields of law, computer science, and artificial intelligence research spurred renewed interest in the subject of abduction. Diagnostic expert systems frequently employ abduction. Deduction, induction, and abduction Deduction Deductive reasoning allows deriving from only where
Inference Act or process of deriving logical conclusions from premises known or assumed to be true Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and cognitive psychology; artificial intelligence researchers develop automated inference systems to emulate human inference. Definition The process by which a conclusion is inferred from multiple observations is called inductive reasoning. This definition is disputable (due to its lack of clarity. Two possible definitions of "inference" are: A conclusion reached on the basis of evidence and reasoning.The process of reaching such a conclusion. Examples Example for definition #1 Ancient Greek philosophers defined a number of syllogisms, correct three part inferences, that can be used as building blocks for more complex reasoning. All humans are mortal.All Greeks are humans.All Greeks are mortal. ? (where ? ?
Abductive reasoning Abductive reasoning (also called abduction, abductive inference or retroduction) is a form of logical inference that goes from an observation to a hypothesis that accounts for the observation, ideally seeking to find the simplest and most likely explanation. In abductive reasoning, unlike in deductive reasoning, the premises do not guarantee the conclusion. One can understand abductive reasoning as "inference to the best explanation". The fields of law, computer science, and artificial intelligence research renewed interest in the subject of abduction. Diagnostic expert systems frequently employ abduction. History 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 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, but not necessary, for allows deriving
Proof assistant An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right. In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer. Comparison of systems User interface See also Notes Jump up ^ Hunt, Warren; Matt Kaufmann; Robert Bellarmine Krug; J Moore; Eric W. References Henk Barendregt and Herman Geuvers (2001). External links Catalogues
Incorporeality State or quality of being bodiless Incorporeality is "the state or quality of being incorporeal or bodiless; immateriality; incorporealism. In the problem of universals, universals are separable from any particular embodiment in one sense, while in another, they seem inherent nonetheless. The notion that a causally effective incorporeal body is even coherent requires the belief that something can affect what's material, without physically existing at the point of effect. Philosophy Plato depicted by Raphael. Pre-Socratic "The Love and Strife of Empedokles are no incorporeal forces. "Zeller holds, indeed, that Anaxagoras meant to speak of something incorporeal ; but he admits that he did not succeed in doing so, and that is historically the important point. On the whole of ancient philosophy and incorporeal, Zeller writes: Aristotelian Flannery in A Companion to Philosophy of Religion writes: Platonic Renehan (1980) writes: Theology See also References
Logically Speaking Graham Priest interviewed by Richard Marshall. Graham Priest is one of the giants of philosophical logic. He has written many books about this, including Doubt Truth to be a Liar, Towards Non-Being: the Logic and Metaphysics of Intentionality, Beyond the Limits of Thought, In Contradiction: A Study of the Transconsistent and Introduction to Non-Classical Logic. He can be found in Melbourne and New York, and sometimes in St. Andrews. 3:AM: You’re famous for denying that propositions have to be either true or false (and not both or neither) but before we get to that, can you start by saying how you became a philosopher? Graham Priest: Well, I was trained as a mathematician. 3:AM: Now, you’re interested in the very basis of how we think. GP: Well, first a clarification. 3:AM: So paraconsistent logic is a logic that tries to work out how we might formally understand treating some propositions as being both true and false at the same time. So for ‘logic’. But more should be said.
Logical consequence Relationship between statements that hold true when one logically follows from another Logicians make precise accounts of logical consequence regarding a given language , either by constructing a deductive system for or by formal intended semantics for language . Formal accounts The most widely prevailing view on how best to account for logical consequence is to appeal to formality. All X are Y All Y are Z Therefore, all X are Z. This is in contrast to an argument like "Fred is Mike's brother's son. A priori property of logical consequence If it is known that follows logically from , then no information about the possible interpretations of or will affect that knowledge. is a logical consequence of cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. A formula of . .
Infinity (philosophy) Philosophical concept ... It is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. This is often called potential infinity; however, there are two ideas mixed up with this. , which reads, "for any integer n, there exists an integer m > n such that P(m)". Sed omne continuum est actualiter existens. The parts are actually there, in some sense. Among the scholastics, Aquinas also argued against the idea that infinity could be in any sense complete or a totality. The Jain upanga āgama Surya Prajnapti (c. 400 BC) classifies all numbers into three sets: enumerable, innumerable, and infinite. Enumerable: lowest, intermediate and highestInnumerable: nearly innumerable, truly innumerable and innumerably innumerableInfinite: nearly infinite, truly infinite, infinitely infinite Jain theory of numbers (See IIIrd section for various infinities) The Jains were the first to discard the idea that all infinities were the same or equal. ...