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! ===Formal systems=== {{main|Formal system}} A formal system of logic consists of a formal language together with a set of [[axiom]]s and a [[proof system]] used to draw inferences from these axioms.{{sfnm|1a1=Boris|1a2=Alexander|1y=2017|1p=74|2a1=Cook|2y=2009|2p=124}} In logic, axioms are statements that are accepted without proof. They are used to justify other statements.{{sfnm|1a1=FlotyΕski|1y=2020|1p=[https://books.google.com/books?id=EC4NEAAAQBAJ&pg=PA39 39] |2a1=Berlemann|2a2=Mangold|2y=2009|2p=[https://books.google.com/books?id=XUGN9tKTIiYC&pg=PA194 194]}} Some theorists also include a [[Semantics of logic|semantics]] that specifies how the expressions of the formal language relate to real objects.{{sfnm|1a1=Gensler|1y=2006|1p=xliii|2a1=Font|2a2=Jansana|2y=2017|2p=8}} Starting in the late 19th century, many new formal systems have been proposed.{{sfnm|1a1=Haack|1y=1978|1loc=Philosophy of logics|1pp=1β10|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=31β32|3a1=Jacquette|3y=2006|3loc=Introduction: Philosophy of logic today|3pp=1β12}} A ''formal language'' consists of an ''alphabet'' and syntactic rules. The alphabet is the set of basic symbols used in [[expression (mathematics)|expressions]]. The syntactic rules determine how these symbols may be arranged to result in well-formed formulas.{{sfnm|1a1=Moore|1a2=Carling|1y=1982|1p=53|2a1=Enderton|2y=2001|2loc=Sentential Logic|2pp=[https://books.google.com/books?id=dVncCl_EtUkC&pg=PA12 12β13]}} For instance, the syntactic rules of [[propositional logic]] determine that {{nowrap|"<math>P \land Q</math>"}} is a well-formed formula but {{nowrap|"<math>\land Q</math>"}} is not since the logical conjunction <math>\land</math> requires terms on both sides.{{sfn |Lepore |Cumming |2012 |p=5}} A ''proof system'' is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of [[conjunction introduction]] states that <math>P \land Q</math> follows from the premises <math>P</math> and <math>Q</math>. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including [[natural deduction]] and [[sequent calculus|sequent calculi]].{{sfnm|1a1=Wasilewska|1y=2018|1pp=145β6|2a1=Rathjen|2a2=Sieg|2y=2022}} A ''semantics'' is a system for [[map (mathematics)|mapping]] expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance, the semantics for [[classical logic|classical]] propositional logic assigns the formula <math>P \land Q </math> the denotation "true" whenever <math>P</math> and <math>Q </math> are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true.{{sfnm|1a1=Sider|1y=2010|1pp=34β42|2a1=Shapiro|2a2=Kouri Kissel|2y=2022|3a1=Bimbo|3y=2016|3pp=8β9}} A system of logic is [[Soundness|sound]] when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly.{{sfnm|1a1=Restall|1a2=Standefer|1y=2023|1pp=91|2a1=Enderton|2y=2001|2loc= Chapter 2.5 |2pp=[https://books.google.com/books?id=dVncCl_EtUkC&pg=PA131 131β146]|3a1=van Dalen|3y=1994|3loc=Chapter 1.5}} 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. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page