Logic 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! ==Systems of logic== Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, [[Aristotelian logic]] was treated as the canon of logic in the Western world,{{sfnm|1a1=Jacquette|1y=2006|1loc=Introduction: Philosophy of logic today|1pp=1–12|2a1=Smith|2y=2022|3a1=Groarke}} but modern developments in this field have led to a vast proliferation of logical systems.{{sfn |Haack |1996 |loc=1. 'Alternative' in 'Alternative Logic'}} One prominent categorization divides modern formal logical systems into [[classical logic]], extended logics, and [[deviant logic]]s.{{sfnm|1a1=Haack|1y=1978|1loc=Philosophy of logics|1pp=1–10|2a1=Haack|2y=1996|2loc=1. 'Alternative' in 'Alternative Logic'|3a1=Wolf|3y=1978|3pp=327–340}} ===Aristotelian=== {{main|Aristotelian logic}} [[Aristotelian logic]] encompasses a great variety of topics. They include [[Metaphysics|metaphysical]] theses about [[Ontology|ontological]] categories and problems of scientific explanation. But in a more narrow sense, it is identical to [[term logic]] or syllogistics. A [[syllogism]] is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a [[Subject (grammar)|subject]], a predicate, and a [[Copula (linguistics)|copula]] connecting the subject to the predicate.{{sfnm|1a1=Smith|1y=2022|2a1=Groarke|3a1=Bobzien|3y=2020}} For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is".{{sfn |Groarke}} The subject and the predicate are the ''terms'' of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using a logical connective like "and" to form a new complex proposition.{{sfnm|1a1=Smith|1y=2022|2a1=Magnus|2y=2005|2loc=2.2 Connectives}} [[File:Square of opposition, set diagrams.svg|thumb|upright=1.4|alt=Diagram of the square of opposition|The [[square of opposition]] is often used to visualize the relations between the four basic [[categorical propositions]] in Aristotelian logic. It shows, for example, that the propositions "All S are P" and "Some S are not P" are contradictory, meaning that one of them has to be true while the other is false.]] In Aristotelian logic, the subject can be ''universal'', ''particular'', ''indefinite'', or ''singular''. For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates".{{sfnm|1a1=Smith|1y=2022|2a1=Bobzien|2y=2020|3a1=Hintikka|3a2=Spade|3loc=[https://www.britannica.com/topic/history-of-logic/Aristotle Aristotle]}} Aristotelian logic only includes predicates for simple [[Property (philosophy)|properties]] of entities. But it lacks predicates corresponding to [[Relations (philosophy)|relations]] between entities.{{sfn |Westerståhl |1989 |pp=577–585}} The predicate can be linked to the subject in two ways: either by affirming it or by denying it.{{sfnm|1a1=Smith|1y=2022|2a1=Groarke}} For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case.{{sfnm|1a1=Smith|1y=2022|2a1=Hurley|2y=2015|2loc=4. Categorical Syllogisms|3a1=Copi|3a2=Cohen|3a3=Rodych|3y=2019|3loc=[https://books.google.com/books?id=38bADwAAQBAJ&pg=PA187 6. Categorical Syllogisms]}} Thus, these three propositions contain three predicates, referred to as ''major term'', ''minor term'', and ''middle term''.{{sfnm|1a1=Groarke|2a1=Hurley|2y=2015|2loc=4. Categorical Syllogisms|3a1=Copi|3a2=Cohen|3a3=Rodych|3y=2019|3loc=[https://books.google.com/books?id=38bADwAAQBAJ&pg=PA187 6. Categorical Syllogisms]}} The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how the propositions are formed.{{sfnm|1a1=Smith|1y=2022|2a1=Groarke}}{{sfn |Hurley |2015 |loc=4. Categorical Syllogisms}} For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid.{{sfn |Spriggs |2012 |pp=20–2}} ===Classical=== {{main|Classical logic}} [[Classical logic]] is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in the sense that it is based on basic logical intuitions shared by most logicians.{{sfnm|1a1=Hintikka|1y=2019|1loc=§Nature and varieties of logic, §Alternative logics|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=27-8|3a1=Bäck|3y=2016|3p=317}} These intuitions include the [[law of excluded middle]], the [[double negation elimination]], the [[principle of explosion]], and the bivalence of truth.{{sfn |Shapiro |Kouri Kissel |2022}} It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are the contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future.{{sfn |Burgess |2009 |loc=1. Classical logic}} Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like [[ethics]] or epistemology that lie beyond the scope of mathematics.{{sfnm|1a1=Jacquette|1y=2006|1loc=Introduction: Philosophy of logic today|1pp=1–12|2a1=Borchert|2y=2006c|2loc=Logic, Non-Classical|3a1=Goble|3y=2001|3loc=Introduction}} ====Propositional logic==== {{main|Propositional calculus}} Propositional logic comprises formal systems in which formulae are built from [[atomic propositions]] using [[logical connectives]]. For instance, propositional logic represents the [[conjunction (logic)|conjunction]] of two atomic propositions <math>P</math> and <math>Q</math> as the complex formula <math>P \land Q</math>. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component.{{sfn |Brody |2006 |pp=535–536}} Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition.{{sfn |Klement|1995b}} ====First-order logic==== [[File:BS-12-Begriffsschrift Quantifier1-svg.svg|thumb|alt=Symbol introduced by Gottlob Frege for the universal quantifier|[[Gottlob Frege]]'s ''[[Begriffschrift]]'' introduced the notion of quantifier in a graphical notation, which here represents the judgment that <math>\forall x. F(x)</math> is true.]] {{main|First-order logic}} First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates the internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, [[Predicate (mathematical logic)|predicates]], which refer to properties and relations, and quantifiers, which treat notions like "some" and "all".{{sfnm|1a1=Shapiro|1a2=Kouri Kissel|1y=2022|2a1=Honderich|2y=2005|2loc=philosophical logic|3a1=Michaelson|3a2=Reimer|3y=2019}} For example, to express the proposition "this raven is black", one may use the predicate <math>B</math> for the property "black" and the singular term <math>r</math> referring to the raven to form the expression <math>B(r)</math>. To express that some objects are black, the existential quantifier <math>\exists</math> is combined with the variable <math>x</math> to form the proposition <math>\exists x B(x)</math>. First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer <math>\exists x B(x)</math> from <math>B(r)</math>.{{sfnm|1a1=Nolt|1y=2021|2a1=Magnus|2y=2005|2loc=4 Quantified logic}} ===Extended=== Extended logics are logical systems that accept the basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like [[metaphysics]], [[ethics]], and [[epistemology]].{{sfnm |1a1=Bunnin |1a2=Yu |1y=2009|1p=[https://books.google.com/books?id=M7ZFEAAAQBAJ&pg=PA179 179] |2a1=Garson |2y=2023 |2loc=[https://plato.stanford.edu/entries/logic-modal/ Introduction]}} ====Modal logic==== {{Main|Modal logic}} [[Modal logic]] is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: <math>\Diamond</math> expresses that something is possible while <math>\Box</math> expresses that something is necessary.{{sfnm|1a1=Garson|1y=2023|2a1=Sadegh-Zadeh|2y=2015|2p=983}} For example, if the formula <math>B(s)</math> stands for the sentence "Socrates is a banker" then the formula <math>\Diamond B(s)</math> articulates the sentence "It is possible that Socrates is a banker".{{sfn |Fitch |2014 |p=17}} To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something is necessary, then it is also possible. This means that <math>\Diamond A</math> follows from <math>\Box A</math>. Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that <math>\Box A</math> is equivalent to <math>\lnot \Diamond \lnot A</math>.{{sfnm|1a1=Garson|1y=2023|2a1=Carnielli|2a2=Pizzi|2y=2008|2p=3|3a1=Benthem}} Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, [[deontic logic]] concerns the field of ethics and introduces symbols to express the ideas of [[obligation]] and [[Permission (philosophy)|permission]], i.e. to describe whether an agent has to perform a certain action or is allowed to perform it.{{sfn |Garson |2023}} The modal operators in [[Temporal logic|temporal modal logic]] articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time.{{sfn |Garson |2023}} In epistemology, [[epistemic modal logic]] is used to represent the ideas of [[Knowledge|knowing]] something in contrast to merely [[Belief|believing]] it to be the case.{{sfn |Rendsvig |Symons |2021}} ====Higher order logic==== {{Main|Higher-order logic}} [[Higher-order logic|Higher-order logics]] extend classical logic not by using modal operators but by introducing new forms of quantification.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Väänänen|2y=2021|3a1=Ketland|3y=2005|3loc=Second Order Logic}} Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula {{nowrap|"<math>\exists x (Apple(x) \land Sweet(x))</math>"}} (''some'' apples are sweet) is an example of the [[Existential quantification|existential quantifier]] {{nowrap|"<math>\exists</math>"}} applied to the individual variable {{nowrap|"<math>x</math>"}}. In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula {{nowrap|"<math>\exists Q (Q(Mary) \land Q(John))</math>"}}. In this case, the existential quantifier is applied to the predicate variable {{nowrap|"<math>Q</math>"}}.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Väänänen|2y=2021|3a1=Daintith|3a2=Wright|3y=2008|3loc=[https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/predicate-calculus Predicate calculus]}} The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories.{{sfn |Audi |loc=Philosophy of logic |1999b}} But it has drawbacks in regard to its meta-logical properties and ontological implications, which is why first-order logic is still more commonly used.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Ketland|2y=2005|2loc=Second Order Logic}} ===Deviant=== {{main|Deviant logic}} [[Deviant logic]]s are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue.{{sfnm|1a1=Haack|1y=1996|1loc=1. 'Alternative' in 'Alternative Logic'|2a1=Wolf|2y=1978|2pp=327–340}} [[Intuitionistic logic]] is a restricted version of classical logic.{{sfnm|1a1=Moschovakis|1y=2022|2a1=Borchert|2y=2006c|2loc=Logic, Non-Classical}} It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that <math>A</math> follows from <math>\lnot \lnot A</math>. This is a valid rule of inference in classical logic but it is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the [[law of excluded middle]]. It states that for every sentence, either it or its negation is true. This means that every proposition of the form <math>A \lor \lnot A</math> is true.{{sfnm|1a1=Moschovakis|1y=2022|2a1=Borchert|2y=2006c|2loc=Logic, Non-Classical}} These deviations from classical logic are based on the idea that truth is established by verification using a proof. Intuitionistic logic is especially prominent in the field of [[Constructivism (philosophy of mathematics)|constructive mathematics]], which emphasizes the need to find or construct a specific example to prove its existence.{{sfnm|1a1=Borchert|1y=2006c|1loc=Logic, Non-Classical|2a1=Bridges|2a2=Ishihara|2a3=Rathjen|2a4=Schwichtenberg|2y=2023|2pp=73–74|3a1=Friend|3y=2014|3p=101}} [[Multi-valued logics]] depart from classicality by rejecting the [[principle of bivalence]], which requires all propositions to be either true or false. For instance, [[Jan Łukasiewicz]] and [[Stephen Cole Kleene]] both proposed [[ternary logic]]s which have a third truth value representing that a statement's truth value is indeterminate.{{sfnm|1a1=Sider|1y=2010|1loc=Chapter 3.4|2a1=Gamut|2y=1991|2loc=5.5|3a1=Zegarelli|3p=30|3y=2010}} These logics have been applied in the field of linguistics. [[Fuzzy logics]] are multivalued logics that have an infinite number of "degrees of truth", represented by a [[real number]] between 0 and 1.{{sfn|Hájek|2006}} [[Paraconsistent logic]]s are logical systems that can deal with contradictions. They are formulated to avoid the principle of explosion: for them, it is not the case that anything follows from a contradiction.{{sfnm|1a1=Borchert|1y=2006c|1loc=Logic, Non-Classical|2a1=Priest|2a2=Tanaka|2a3=Weber|2y=2018|3a1=Weber}} They are often motivated by [[dialetheism]], the view that contradictions are real or that reality itself is contradictory. [[Graham Priest]] is an influential contemporary proponent of this position and similar views have been ascribed to [[Georg Wilhelm Friedrich Hegel]].{{sfnm|1a1=Priest|1a2=Tanaka|1a3=Weber|1y=2018|2a1=Weber|3a1=Haack|3y=1996|3loc=Introduction}} ===Informal=== {{main|Informal logic}} [[Informal logic]] is usually carried out in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation. Nonetheless, some frameworks of informal logic have also been presented that try to provide a systematic characterization of the correctness of arguments.{{sfnm|1a1=Hansen|1y=2020|2a1=Korb|2y=2004|2pp=41–42, 48|3a1=Ritola|3y=2008|3p=335|4a=Goarke|4y=2021|4loc=lead section; 2. Systems of Informal Logic; 4.2 Fallacy Theory}} The ''pragmatic'' or ''dialogical approach'' to informal logic sees arguments as [[speech act]]s and not merely as a set of premises together with a conclusion.{{sfnm|1a1=Hansen|1y=2020|2a1=Korb|2y=2004|2pp=43–44|3a1=Ritola|3y=2008|3p=335}} As speech acts, they occur in a certain context, like a [[dialogue]], which affects the standards of right and wrong arguments.{{sfnm|1a1=Walton|1y=1987|1loc=1. A new model of argument|1pp=2–3|2a1=Ritola|2y=2008|2p=335}} A prominent version by [[Douglas N. Walton]] understands a dialogue as a game between two players. The initial position of each player is characterized by the propositions to which they are committed and the conclusion they intend to prove. Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion.{{sfn |Walton |1987 |loc=1. A new model of argument |pp=3–4, 18–22 }} This is achieved by making arguments: arguments are the moves of the game.{{sfnm|1a1=Walton|1y=1987|1loc=1. A new model of argument|1pp=3–4, 11, 18|2a1=Ritola|2y=2008|2p=335}} They affect to which propositions the players are committed. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them. This is usually not possible straight away. For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one's intended conclusion. Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent's victory by denying their conclusion.{{sfn |Walton |1987 |loc=1. A new model of argument |pp=3–4, 18–22 }} Whether an argument is correct depends on whether it promotes the progress of the dialogue. Fallacies, on the other hand, are violations of the standards of proper argumentative rules.{{sfnm|1a1=Hansen|1y=2020|2a1=Walton|2y=1987|2loc=3. Logic of propositions|2pp=3–4, 18–22}} These standards also depend on the type of dialogue. For example, the standards governing the scientific discourse differ from the standards in business negotiations.{{sfn |Ritola |2008 |p=335}} The ''epistemic approach'' to informal logic, on the other hand, focuses on the epistemic role of arguments.{{sfnm|1a1=Hansen|1y=2020|2a1=Korb|2y=2004|2pp=43, 54–55}} It is based on the idea that arguments aim to increase our knowledge. They achieve this by linking justified beliefs to beliefs that are not yet justified.{{sfn |Siegel |Biro |1997 |pp=277–292}} Correct arguments succeed at expanding knowledge while fallacies are epistemic failures: they do not justify the belief in their conclusion.{{sfnm|1a1=Hansen|1y=2020|2a1=Korb|2y=2004|2pp=41–70}} For example, the [[fallacy of begging the question]] is a ''fallacy'' because it fails to provide independent justification for its conclusion, even though it is deductively valid.{{sfnm|1a1=Mackie|1y=1967|2a1=Siegel|2a2=Biro|2y=1997|2pp=277–292}} In this sense, logical normativity consists in epistemic success or rationality.{{sfn |Siegel |Biro |1997 |pp=277–292}} The [[Bayesian epistemology|Bayesian approach]] is one example of an epistemic approach.{{sfnm |1a1=Hansen |1y=2020 |2a1=Moore |2a2=Cromby |2y=2016 |2p=60}} Central to Bayesianism is not just whether the agent believes something but the degree to which they believe it, the so-called ''credence''. Degrees of belief are seen as [[subjective probability|subjective probabilities]] in the believed proposition, i.e. how certain the agent is that the proposition is true.{{sfnm|1a1=Olsson|1y=2018|1pp=431–442|1loc=Bayesian Epistemology|2a1=Hájek|2a2=Lin|2y=2017|2pp=207–232|3a1=Hartmann|3a2=Sprenger|3y=2010|3pp=609–620|3loc=Bayesian Epistemology}} On this view, reasoning can be interpreted as a process of changing one's credences, often in reaction to new incoming information.{{sfn|Shermer|2022|p=136}} Correct reasoning and the arguments it is based on follow the laws of probability, for example, the [[Bayesian epistemology#Principle of conditionalization|principle of conditionalization]]. Bad or irrational reasoning, on the other hand, violates these laws.{{sfnm|1a1=Korb|1y=2004|1pp=41–42, 44–46|2a1=Hájek|2a2=Lin|2y=2017|2pp=207–232|3a1=Talbott|3y=2016}} Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. 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