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 logic=== {{further|Formal system}} Formal logic is also known as symbolic logic and is widely used in [[mathematical logic]]. It uses a [[Formal system|formal]] approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the [[logical form]] of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content.{{sfnm|1a1=MacFarlane|1y=2017|2a1=Corkum|2y=2015|2pp=753β767|3a1=Blair|3a2=Johnson|3y=2000|3pp=93β95|4a1=Magnus|4y=2005|4loc=1.6 Formal languages|4pp=12-4}} Formal logic is interested in deductively [[Validity (logic)|valid]] arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false.{{sfnm|1a1=McKeon|2a1=Craig|2y=1996|2loc=Formal and informal logic}} For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a [[rule of inference]].{{sfn|Hintikka|Sandu|2006|p=13}} For example, [[modus ponens]] is a rule of inference according to which all arguments of the form "(1) ''p'', (2) if ''p'' then ''q'', (3) therefore ''q''" are valid, independent of what the terms ''p'' and ''q'' stand for.{{sfn |Magnus |2005 |loc=Proofs, p. 102}} In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of [[logical truth]]s.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1pp=13β16|2a1=Makridis|2y=2022|2pp=1β2|3a1=Runco|3a2=Pritzker|3y=1999|3p=155}} A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all [[possible world]]s and under all [[Interpretation (logic)|interpretations]] of its non-logical terms, like the claim "either it is raining, or it is not".{{sfnm|1a1=GΓ³mez-Torrente|1y=2019|2a1=Magnus|2y=2005|2loc=1.5 Other logical notions, p. 10}} These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from ''p'' to ''q'' is deductively valid then the claim "if ''p'' then ''q''" is a logical truth.{{sfn|Hintikka|Sandu|2006|p=16}} [[File:First-order logic.png|thumb|upright=1.6|alt=Visualization of how to translate an English sentence into first-order logic|Formal logic needs to translate natural language arguments into a formal language, like first-order logic, to assess whether they are valid. In this example, the letter "c" represents Carmen while the letters "M" and "T" stand for "Mexican" and "teacher". The symbol "β§" has the meaning of "and".]] Formal logic uses [[formal language]]s to express and analyze arguments.{{sfnm|1a1=Honderich|1y=2005|1loc=logic, informal|2a1=Craig|2y=1996|2loc=Formal and informal logic|3a1=Johnson|3y=1999|3pp=265β268}} They normally have a very limited vocabulary and exact [[Syntax|syntactic rule]]s. These rules specify how their symbols can be combined to construct sentences, so-called [[well-formed formula]]s.{{sfnm|1a1=Craig|1y=1996|1loc=Formal languages and systems|2a1=Simpson|2y=2008|2p=14}} This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether a given argument is valid.{{sfn |Craig |1996 |loc=Formal languages and systems}} Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be [[Logic translation#Natural language formalization|translated into formal language]] before their validity can be assessed.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1pp=22-3|2a1=Magnus|2y=2005|2loc=1.4 Deductive validity|2pp=8β9|3a1=Johnson|3y=1999|3p=267}} The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, ''a logic'' is a logical formal system. Distinct logics differ from each other concerning the rules of inference they accept as valid and the formal languages used to express them.{{sfnm|1a1=Haack|1y=1978|1loc=Philosophy of logics|1pp=1β2, 4|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=16β17|3a1=Jacquette|3y=2006|3loc=Introduction: Philosophy of logic today, pp. 1β12}} Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic.{{sfnm|1a1=Haack|1y=1978|1loc=Philosophy of logics|1pp=1β2, 4|2a1=Jacquette|2y=2006|2loc=Introduction: Philosophy of logic today|2pp=1β12}} For example, it has been suggested that only [[Completeness (logic)|logically complete]] systems, like [[first-order logic]], qualify as logics. For such reasons, some theorists deny that [[higher-order logic]]s are logics in the strict sense.{{sfnm|1a1=Haack|1y=1978|1loc=Philosophy of logics|1pp=5β7, 9|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=31-2|3a1=Haack|3y=1996|3pp=229β30}} 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! 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