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! ==Areas of research== Logic is studied in various fields. In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science.{{sfnm|1a1=Hintikka|1y=2019|1loc=§Logic and other disciplines|2a1=Haack|2y=1978|2loc=Philosophy of logics|2pp=1–10}} In other cases, logic itself is made the subject of research in another discipline. This can happen in diverse ways. For instance, it can involve investigating the philosophical assumptions linked to the basic concepts used by logicians. Other ways include interpreting and analyzing logic through mathematical structures as well as studying and comparing abstract properties of formal logical systems.{{sfnm|1a1=Hintikka|1y=2019|1loc=lead section, §Features and problems of logic|2a1=Gödel|2y=1984|2pp=447–469|2loc=Russell's mathematical logic|3a1=Monk|3y=1976|3pp=1–9|3loc=Introduction}} ===Philosophy of logic and philosophical logic=== {{main | Philosophy of logic | Philosophical logic}} ''Philosophy of logic'' is the philosophical discipline studying the scope and nature of logic.{{sfnm|1a1=Hintikka|1y=2019|1loc=lead section, §Nature and varieties of logic|2a1=Audi|2loc=Philosophy of logic|2y=1999b}} It examines many presuppositions implicit in logic, like how to define its basic concepts or the metaphysical assumptions associated with them.{{sfn |Jacquette |2006 |loc=Introduction: Philosophy of logic today |pp=1–12}} It is also concerned with how to classify logical systems and considers the [[ontological]] commitments they incur.{{sfn |Hintikka |2019 |loc=§Problems of ontology}} ''Philosophical logic'' is one of the areas within the philosophy of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics, ethics, and epistemology.{{sfnm|1a1=Jacquette|1y=2006|1loc=Introduction: Philosophy of logic today|1pp=1–12|2a1=Burgess|2y=2009|2loc=1. Classical logic}} This application usually happens in the form of [[#Extended|extended]] or [[#Deviant|deviant logical systems]].{{sfnm|1a1=Goble|1y=2001|1loc=Introduction|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=31–32}} ===Metalogic=== {{main|Metalogic}} Metalogic is the field of inquiry studying the properties of formal logical systems. For example, when a new formal system is developed, metalogicians may study it to determine which formulas can be proven in it. They may also study whether an [[algorithm]] could be developed to find a proof for each formula and whether every provable formula in it is a tautology. Finally, they may compare it to other logical systems to understand its distinctive features. A key issue in metalogic concerns the relation between syntax and semantics. The syntactic rules of a formal system determine how to deduce conclusions from premises, i.e. how to formulate proofs. The semantics of a formal system governs which sentences are true and which ones are false. This determines the validity of arguments since, for valid arguments, it is impossible for the premises to be true and the conclusion to be false. The relation between syntax and semantics concerns issues like whether every valid argument is provable and whether every provable argument is valid. Metalogicians also study whether logical systems are complete, sound, and [[consistency|consistent]]. They are interested in whether the systems are [[decidability (logic)|decidable]] and what [[expressive power (computer science)|expressive power]] they have. Metalogicians usually rely heavily on abstract mathematical reasoning when examining and formulating metalogical proofs. This way, they aim to arrive at precise and general conclusions on these topics.{{sfnm|1a1=Gensler|1y=2006|1pp=xliii–xliv|2a1=Sider|2y=2010|2pp=4–6|3a1=Schagrin}} ===Mathematical logic=== {{main|Mathematical logic}} [[File:Bertrand Russell 1949.jpg|thumb|left|alt=Photograph of Bertrand Russell|Bertrand Russell made various contributions to mathematical logic.{{sfn |Irvine |2022}}]] The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include [[model theory]], [[proof theory]], [[set theory]], and [[computability theory]].{{sfnm|1a1=Li|1y=2010|1p=ix|2a1=Rautenberg|2y=2010|2p=15|3a1=Quine|3y=1981|3p=1|4a1=Stolyar|4y=1984|4p=2}} Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based [[foundations of mathematics]].{{sfn |Stolyar |1984 |pp=3–6}} The latter was a major concern in early 20th-century mathematical logic, which pursued the program of [[logicism]] pioneered by philosopher-logicians such as Gottlob Frege, [[Alfred North Whitehead]], and [[Bertrand Russell]]. Mathematical theories were supposed to be logical [[tautology (logic)|tautologies]], and their program was to show this by means of a reduction of mathematics to logic. Many attempts to realize this program failed, from the crippling of Frege's project in his ''Grundgesetze'' by [[Russell's paradox]], to the defeat of [[Hilbert's program]] by [[Gödel's incompleteness theorem]]s.{{sfnm|1a1=Hintikka|1a2=Spade|1loc=[https://www.britannica.com/topic/history-of-logic/Godels-incompleteness-theorems Gödel's incompleteness theorems]|2a1=Linsky|2y=2011|2p=4|3a1=Richardson|3y=1998|3p=15}} Set theory originated in the study of the infinite by [[Georg Cantor]], and it has been the source of many of the most challenging and important issues in mathematical logic. They include [[Cantor's theorem]], the status of the [[Axiom of Choice]], the question of the independence of the [[continuum hypothesis]], and the modern debate on [[large cardinal]] axioms.{{sfnm|1a1=Bagaria|1y=2021|2a1=Cunningham}} Computability theory is the branch of mathematical logic that studies effective procedures to solve calculation problems. One of its main goals is to understand whether it is possible to solve a given problem using an algorithm. For instance, given a certain claim about the positive integers, it examines whether an algorithm can be found to determine if this claim is true. Computability theory uses various theoretical tools and models, such as [[Turing machines]], to explore this type of issue.{{sfnm|1a1=Borchert|1y=2006a|1loc=Computability Theory|2a1=Leary|2a2=Kristiansen|2y=2015|2p=195}} ===Computational logic=== {{main|Computational logic|Logic in computer science}} [[File:TransistorANDgate.png|thumb|alt=Diagram of an AND gate using transistors|Conjunction (AND) is one of the basic operations of Boolean logic. It can be electronically implemented in several ways, for example, by using two [[transistor]]s.]] Computational logic is the branch of logic and [[computer science]] that studies how to implement mathematical reasoning and logical formalisms using computers. This includes, for example, [[automatic theorem prover]]s, which employ rules of inference to construct a proof step by step from a set of premises to the intended conclusion without human intervention.{{sfnm|1a1=Paulson|1y=2018|1pp=1–14|2a1=Castaño|2y=2018|2p=2|3a1=Wile|3a2=Goss|3a3=Roesner|3y=2005|3p=447}} [[Logic programming]] languages are designed specifically to express facts using logical formulas and to draw inferences from these facts. For example, [[Prolog]] is a [[logic programming]] language based on predicate logic.{{sfnm|1a1=Clocksin|1a2=Mellish|1y=2003|1pp=237–238, 252–255, 257|1loc=The Relation of Prolog to Logic|2a1=Daintith|2a2=Wright|2y=2008|2loc=[https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/logic-programming-languages Logic Programming Languages]}} Computer scientists also apply concepts from logic to problems in computing. The works of [[Claude Shannon]] were influential in this regard. He showed how [[Boolean logic]] can be used to understand and implement computer circuits.{{sfnm|1a1=O'Regan|1y=2016|1p=49|2a1=Calderbank|2a2=Sloane|2y=2001|2pp=768}} This can be achieved using electronic [[logic gates]], i.e. electronic circuits with one or more inputs and usually one output. The truth values of propositions are represented by voltage levels. This way, logic functions can be simulated by applying the corresponding voltages to the inputs of the circuit and determining the value of the function by measuring the voltage of the output.{{sfn |Daintith |Wright |2008 |loc=[https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/logic-gate Logic Gate]}} ===Formal semantics of natural language=== {{main|Formal semantics (natural language)}} Formal semantics is a subfield of logic, [[linguistics]], and the [[philosophy of language]]. The discipline of [[semantics]] studies the meaning of language. Formal semantics uses formal tools from the fields of symbolic logic and mathematics to give precise theories of the meaning of [[natural language]] expressions. It understands meaning usually in relation to [[truth condition]]s, i.e. it examines in which situations a sentence would be true or false. One of its central methodological assumptions is the [[principle of compositionality]]. It states that the meaning of a complex expression is determined by the meanings of its parts and how they are combined. For example, the meaning of the verb phrase "walk and sing" depends on the meanings of the individual expressions "walk" and "sing". Many theories in formal semantics rely on model theory. This means that they employ set theory to construct a model and then interpret the meanings of expression in relation to the elements in this model. For example, the term "walk" may be interpreted as the set of all individuals in the model that share the property of walking. Early influential theorists in this field were [[Richard Montague]] and [[Barbara Partee]], who focused their analysis on the English language.{{sfnm|1a1=Janssen|1a2=Zimmermann|1y=2021|1pp=3–4|2a1=Partee|2y=2016|3a1=King|3y=2009|3pp=557–8|4a1=Aloni|4a2=Dekker|4y=2016|4pp=[https://books.google.com/books?id=ltSgDAAAQBAJ&pg=PT22 22–23]}} ===Epistemology of logic=== The epistemology of logic studies how one knows that an argument is valid or that a proposition is logically true.{{sfnm|1a1=Warren|1y=2020|1loc=6. The Epistemology of Logic|2a1=Schechter}} This includes questions like how to justify that modus ponens is a valid rule of inference or that contradictions are false.{{sfn |Warren |2020 |loc=6. The Epistemology of Logic}} The traditionally dominant view is that this form of logical understanding belongs to knowledge [[A priori and a posteriori|a priori]].{{sfn |Schechter}} In this regard, it is often argued that the [[mind]] has a special faculty to examine relations between pure ideas and that this faculty is also responsible for apprehending logical truths.{{sfn |Gómez-Torrente |2019}} A similar approach understands the rules of logic in terms of [[Conventionalism|linguistic conventions]]. On this view, the laws of logic are trivial since they are true by definition: they just express the meanings of the logical vocabulary.{{sfnm|1a1=Warren|1y=2020|1loc=6. The Epistemology of Logic|2a1=Gómez-Torrente|2y=2019|3a1=Warren|3y=2020|3loc=1. What is Conventionalism}} Some theorists, like [[Hilary Putnam]] and [[Penelope Maddy]], object to the view that logic is knowable a priori. They hold instead that logical truths depend on the [[empirical]] world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world. According to this view, they may be explored by studying general patterns of the [[fundamental sciences]]. For example, it has been argued that certain insights of [[quantum mechanics]] refute the [[principle of distributivity]] in classical logic, which states that the formula <math>A \land (B \lor C)</math> is equivalent to <math>(A \land B) \lor (A \land C)</math>. This claim can be used as an empirical argument for the thesis that [[quantum logic]] is the correct logical system and should replace classical logic.{{sfnm|1a1=Chua|1y=2017|1pp=631–636|2a1=Wilce|2y=2021|3a1=Putnam|3y=1969|3pp=216–241}} 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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