Deductive reasoning Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! == Related concepts and theories == === Deductivism === Deductivism is a philosophical position that gives primacy to deductive reasoning or arguments over their non-deductive counterparts.<ref name="Bermejo-Luque"/><ref name="Howson"/> It is often understood as the evaluative claim that only deductive inferences are ''good'' or ''correct'' inferences. This theory would have wide-reaching consequences for various fields since it implies that the rules of deduction are "the only acceptable standard of [[evidence]]".<ref name="Bermejo-Luque">{{cite journal |last1=Bermejo-Luque |first1=Lilian |title=What is Wrong with Deductivism? |journal=Informal Logic |date=2020 |volume=40 |issue=3 |pages=295–316 |doi=10.22329/il.v40i30.6214 |s2cid=217418605 |url=https://philpapers.org/rec/BERWIW-3|doi-access=free }}</ref> This way, the rationality or correctness of the different forms of inductive reasoning is denied.<ref name="Howson">{{cite book |last1=Howson |first1=Colin |title=Hume's Problem |date=2000 |publisher=Oxford University Press |isbn=978-0-19-825037-1 |url=https://oxford.universitypressscholarship.com/view/10.1093/0198250371.001.0001/acprof-9780198250371-chapter-6 |chapter=5. Deductivism|doi=10.1093/0198250371.001.0001 }}</ref><ref>{{cite book |last1=Kotarbinska |first1=Janina |title=Twenty-Five Years of Logical Methodology in Poland |date=1977 |publisher=Springer Netherlands |isbn=978-94-010-1126-6 |pages=261–278 |chapter-url=https://link.springer.com/chapter/10.1007/978-94-010-1126-6_15 |language=en |chapter=The Controversy: Deductivism Versus Inductivism|doi=10.1007/978-94-010-1126-6_15 }}</ref> Some forms of deductivism express this in terms of degrees of reasonableness or probability. Inductive inferences are usually seen as providing a certain degree of support for their conclusion: they make it more likely that their conclusion is true. Deductivism states that such inferences are not rational: the premises either ensure their conclusion, as in deductive reasoning, or they do not provide any support at all.<ref name="Stove"/> One motivation for deductivism is the [[problem of induction]] introduced by [[David Hume]]. It consists in the challenge of explaining how or whether inductive inferences based on past experiences support conclusions about future events.<ref name="Howson"/><ref>{{cite web |last1=Henderson |first1=Leah |title=The Problem of Induction |url=https://plato.stanford.edu/entries/induction-problem/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 March 2022 |date=2020}}</ref><ref name="Stove">{{cite journal |last1=Stove |first1=D. |title=Deductivism |journal=Australasian Journal of Philosophy |date=1970 |volume=48 |issue=1 |pages=76–98 |doi=10.1080/00048407012341481 |url=https://philpapers.org/rec/STOD}}</ref> For example, a chicken comes to expect, based on all its past experiences, that the person entering its coop is going to feed it, until one day the person "at last wrings its neck instead".<ref>{{cite book |last1=Russell |first1=Bertrand |title=The Problems of Philosophy |date=2009 |orig-date=1959 |publisher=Project Gutenberg |url=https://www.gutenberg.org/files/5827/5827-h/5827-h.htm |chapter=VI. On Induction}}</ref> According to [[Karl Popper]]'s falsificationism, deductive reasoning alone is sufficient. This is due to its truth-preserving nature: a theory can be falsified if one of its deductive consequences is false.<ref>{{cite web |last1=Thornton |first1=Stephen |title=Karl Popper: 4. Basic Statements, Falsifiability and Convention |url=https://plato.stanford.edu/entries/popper/#BasiStatFalsConv |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=14 March 2022 |date=2021}}</ref><ref>{{cite web |last1=Shea |first1=Brendan |title=Popper, Karl: Philosophy of Science |url=https://iep.utm.edu/pop-sci/ |website=Internet Encyclopedia of Philosophy |access-date=14 March 2022}}</ref> So while inductive reasoning does not offer positive evidence for a theory, the theory still remains a viable competitor until falsified by [[Empirical evidence|empirical observation]]. In this sense, deduction alone is sufficient for discriminating between competing hypotheses about what is the case.<ref name="Howson"/> [[Hypothetico-deductive model|Hypothetico-deductivism]] is a closely related scientific method, according to which science progresses by formulating hypotheses and then aims to falsify them by trying to make observations that run counter to their deductive consequences.<ref>{{cite web |title=hypothetico-deductive method |url=https://www.britannica.com/science/hypothetico-deductive-method |website=Encyclopedia Britannica |access-date=14 March 2022 |language=en}}</ref><ref>{{cite web |title=hypothetico-deductive method |url=https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095954755 |website=Oxford Reference |access-date=14 March 2022 |language=en }}</ref> === Natural deduction === The term "[[natural deduction]]" refers to a class of proof systems based on self-evident rules of inference.<ref name="IEPNatural"/><ref name="StanfordNatural">{{cite web |last1=Pelletier |first1=Francis Jeffry |last2=Hazen |first2=Allen |title=Natural Deduction Systems in Logic |url=https://plato.stanford.edu/entries/natural-deduction/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=15 March 2022 |date=2021}}</ref> The first systems of natural deduction were developed by [[Gerhard Gentzen]] and [[Stanislaw Jaskowski]] in the 1930s. The core motivation was to give a simple presentation of deductive reasoning that closely mirrors how reasoning actually takes place.<ref>{{cite journal |last1=Gentzen |first1=Gerhard |title=Untersuchungen über das logische Schließen. I |journal=Mathematische Zeitschrift |date=1934 |volume=39 |issue=2 |pages=176–210 |doi=10.1007/BF01201353 |s2cid=121546341 |url=https://gdz.sub.uni-goettingen.de/id/PPN266833020_0039?tify={%22pages%22:[180],%22panX%22:0.559,%22panY%22:0.785,%22view%22:%22info%22,%22zoom%22:0.411} |quote=Ich wollte nun zunächst einmal einen Formalismus aufstellen, der dem wirklichen Schließen möglichst nahe kommt. So ergab sich ein "Kalkül des natürlichen Schließens. (First I wished to construct a formalism that comes as close as possible to actual reasoning. Thus arose a "calculus of natural deduction".)}}</ref> In this sense, natural deduction stands in contrast to other less intuitive proof systems, such as [[Hilbert system|Hilbert-style deductive systems]], which employ axiom schemes to express [[logical truth]]s.<ref name="IEPNatural">{{cite web |last1=Indrzejczak |first1=Andrzej |title=Natural Deduction |url=https://iep.utm.edu/natural-deduction/ |website=Internet Encyclopedia of Philosophy |access-date=15 March 2022}}</ref> Natural deduction, on the other hand, avoids axioms schemes by including many different rules of inference that can be used to formulate proofs. These rules of inference express how [[logical constant]]s behave. They are often divided into [[Natural deduction#Introduction and elimination|introduction rules and elimination rules]]. Introduction rules specify under which conditions a logical constant may be introduced into a new sentence of the [[Formal proof|proof]].<ref name="IEPNatural"/><ref name="StanfordNatural"/> For example, the introduction rule for the logical constant {{nowrap|"<math>\land</math>"}} (and) is {{nowrap|"<math>\frac{A, B}{(A \land B)}</math>"}}. It expresses that, given the premises {{nowrap|"<math>A</math>"}} and {{nowrap|"<math>B</math>"}} individually, one may draw the conclusion {{nowrap|"<math>A \land B</math>"}} and thereby include it in one's proof. This way, the symbol {{nowrap|"<math>\land</math>"}} is introduced into the proof. The removal of this symbol is governed by other rules of inference, such as the elimination rule {{nowrap|"<math>\frac{(A \land B)}{A}</math>"}}, which states that one may deduce the sentence {{nowrap|"<math>A</math>"}} from the premise {{nowrap|"<math>(A \land B)</math>"}}. Similar introduction and elimination rules are given for other logical constants, such as the propositional operator {{nowrap|"<math>\lnot</math>"}}, the [[Logical connective|propositional connectives]] {{nowrap|"<math>\lor</math>"}} and {{nowrap|"<math>\rightarrow</math>"}}, and the [[Quantifier (logic)|quantifiers]] {{nowrap|"<math>\exists</math>"}} and {{nowrap|"<math>\forall</math>"}}.<ref name="IEPNatural"/><ref name="StanfordNatural"/> The focus on rules of inferences instead of axiom schemes is an important feature of natural deduction.<ref name="IEPNatural"/><ref name="StanfordNatural"/> But there is no general agreement on how natural deduction is to be defined. Some theorists hold that all proof systems with this feature are forms of natural deduction. This would include various forms of [[Sequent calculus|sequent calculi]]{{efn|name=natDeduc |1= In natural deduction, a simplified [[sequent]] consists of an environment <math>\Gamma</math> that yields (<math>\vdash</math>) a single conclusion <math>C</math>; a single sequent would take the form :"''Assumptions'' A1, A2, A3 etc. yield ''Conclusion'' C1"; in the symbols of [[natural deduction]], <math>\Gamma A_1, A_2, A_3 ... \vdash C_1</math> *However if the premises were true but the conclusion were false, a hidden assumption could be intervening; alternatively, a hidden process might be coercing the form of presentation, and so forth; then the task would be to unearth the hidden factors in an ill-formed syllogism, in order to make the form valid. *''see [[Deduction theorem]]'' }} or [[Method of analytic tableaux#Tableau calculi and their properties|tableau calculi]]. But other theorists use the term in a more narrow sense, for example, to refer to the proof systems developed by Gentzen and Jaskowski. Because of its simplicity, natural deduction is often used for teaching logic to students.<ref name="IEPNatural"/> === Geometrical method === The geometrical method is a method of [[philosophy]] based on deductive reasoning. It starts from a small set of [[self-evident]] axioms and tries to build a comprehensive logical system based only on deductive inferences from these first [[axiom]]s.<ref name="DalyHandbook">{{cite book |last1=Daly |first1=Chris |title=The Palgrave Handbook of Philosophical Methods |date=2015 |publisher=Palgrave Macmillan UK |isbn=978-1-137-34455-7 |pages=1–30 |chapter-url=https://link.springer.com/chapter/10.1057/9781137344557_1 |language=en |chapter=Introduction and Historical Overview|doi=10.1057/9781137344557_1 }}</ref> It was initially formulated by [[Baruch Spinoza]] and came to prominence in various [[rationalist]] philosophical systems in the modern era.<ref>{{cite web |last1=Dutton |first1=Blake D. |title=Spinoza, Benedict De |url=https://iep.utm.edu/spinoza/#H2 |website=Internet Encyclopedia of Philosophy |access-date=16 March 2022}}</ref> It gets its name from the forms of [[Mathematical proof|mathematical demonstration]] found in traditional [[geometry]], which are usually based on axioms, [[definition]]s, and inferred [[theorem]]s.<ref>{{cite web |last1=Goldenbaum |first1=Ursula |title=Geometrical Method |url=https://iep.utm.edu/geo-meth/ |website=Internet Encyclopedia of Philosophy |access-date=17 February 2022}}</ref><ref>{{cite book |last1=Nadler |first1=Steven |title=Spinoza's 'Ethics': An Introduction |date=2006 |publisher=Cambridge University Press |isbn=978-0-521-83620-3 |pages=35–51 |url=https://www.cambridge.org/core/books/abs/spinozas-ethics/geometric-method/08550AF622C78ACC388069710D37036E |chapter=The geometric method}}</ref> An important motivation of the geometrical method is to repudiate [[philosophical skepticism]] by grounding one's philosophical system on absolutely certain axioms. Deductive reasoning is central to this endeavor because of its necessarily truth-preserving nature. This way, the certainty initially invested only in the axioms is transferred to all parts of the philosophical system.<ref name="DalyHandbook"/> One recurrent criticism of philosophical systems build using the geometrical method is that their initial axioms are not as self-evident or certain as their defenders proclaim.<ref name="DalyHandbook"/> This problem lies beyond the deductive reasoning itself, which only ensures that the conclusion is true if the premises are true, but not that the premises themselves are true. For example, Spinoza's philosophical system has been criticized this way based on objections raised against the [[Causality|causal]] axiom, i.e. that "the knowledge of an effect depends on and involves knowledge of its cause".<ref>{{cite book |last1=Doppelt |first1=Torin |title=Spinoza's Causal Axiom: A Defense |date=2010 |url=https://qspace.library.queensu.ca/bitstream/handle/1974/6052/Doppelt_Torin_201009_MA.pdf |chapter=3: The Truth About 1A4}}</ref> A different criticism targets not the premises but the reasoning itself, which may at times implicitly assume premises that are themselves not self-evident.<ref name="DalyHandbook"/> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. 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