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! ==Basic concepts== ===Premises, conclusions, and truth=== ====Premises and conclusions==== {{Main|Premise|Logical consequence}} ''Premises'' and ''conclusions'' are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Honderich|2y=2005|2loc=philosophical logic}} For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars is a red planet". For most types of logic, it is accepted that premises and conclusions have to be [[truth-bearer]]s.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Honderich|2y=2005|2loc=philosophical logic}}{{efn|However, there are some forms of logic, like [[imperative logic]], where this may not be the case.{{sfn |Haack |1974 |p=51}}}} This means that they have a [[truth value]]: they are either true or false. Contemporary philosophy generally sees them either as ''[[proposition]]s'' or as ''[[Sentence (linguistics)|sentences]]''.{{sfn |Audi |loc=Philosophy of logic |1999b}} Propositions are the [[denotation]]s of sentences and are usually seen as [[abstract object]]s.{{sfnm|1a1=Falguera|1a2=Martínez-Vidal|1a3=Rosen|1y=2021|2a1=Tondl|2y=2012|2p=111}} For example, the English sentence "the tree is green" is different from the German sentence "der Baum ist grün" but both express the same proposition.{{sfn|Olkowski|Pirovolakis|2019|pp=[https://books.google.com/books?id=FhaGDwAAQBAJ&pg=PT65 65–66]}} Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, [[Naturalism (philosophy)|philosophical naturalists]] usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions.{{sfn |Audi |loc=Philosophy of logic |1999b}} These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like the symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Pietroski|2y=2021}} Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as [[psychologism]]. It was discussed at length around the turn of the 20th century but it is not widely accepted today.{{sfnm|1a1=Audi|1loc=Philosophy of logic|1y=1999b|2a1=Kusch|2y=2020|3a1=Rush|3y=2014|3pp=1–10, 189–190}} ====Internal structure==== Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex.{{sfnm|1a1=King|1y=2019|2a1=Pickel|2y=2020|2pp=2991–3006}} A complex proposition has other propositions as its constituents, which are linked to each other through [[Logical connective|propositional connectives]] like "and" or "if...then". Simple propositions, on the other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like [[singular term]]s and [[Predicate (grammar)|predicates]].{{sfn |Honderich |2005 |loc=philosophical logic}}{{sfnm|1a1=King|1y=2019|2a1=Pickel|2y=2020|2pp=2991–3006}} For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by the propositional connective "and".{{sfn |Honderich |2005 |loc=philosophical logic}} Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using [[Truth function|truth-functional]] propositional connectives, their truth only depends on the truth values of their parts.{{sfn |Honderich |2005 |loc=philosophical logic}}{{sfn |Pickel |2020 |pp=2991–3006}} But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects.{{sfnm|1a1=Honderich|1y=2005|1loc=philosophical logic|2a1=Craig|2y=1996|2loc=Philosophy of logic|3a1=Michaelson|3a2=Reimer|3y=2019}} Whether the simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by [[theories of reference]].{{sfn |Michaelson |Reimer |2019}} ====Logical truth==== {{Main|Logical truth}} Some complex propositions are true independently of the substantive meanings of their parts.{{sfnm|1a1=Hintikka|1y=2019|1loc=§Nature and varieties of logic|2a1=MacFarlane|2y=2017}} In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like the simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it.{{sfnm|1a1=Gómez-Torrente|1y=2019|2a1=MacFarlane|2y=2017|3a1=Honderich|3y=2005|3loc=philosophical logic}} This means that it is true under all interpretations of its non-logical terms. In some [[modal logic]]s, this means that the proposition is true in all possible worlds.{{sfnm|1a1=Gómez-Torrente|1y=2019|2a1=Jago|2y=2014|2p=41}} Some theorists define logic as the study of logical truths.{{sfn|Hintikka|Sandu|2006|p=16}} ====Truth tables==== [[Truth table]]s can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for the truth values "true" and "false".{{sfnm|1a1=Magnus|1y=2005|1loc=3. Truth tables|1pp=35–38|2a1=Angell|2y=1964|2p=164|3a1=Hall|3a2=O'Donnell|3y=2000|3p=[https://books.google.com/books?id=yP4MJ36C4ZgC&pg=PA48 48]}} The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression {{nowrap|"<math>p \land q</math>"}} uses the logical connective <math>\land</math> ([[Logical conjunction|and]]). It could be used to express a sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, <math>p</math> ("yesterday was Sunday") and <math>q</math> ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are <math>\lnot</math> ([[Negation|not]]), <math>\lor</math> ([[Logical disjunction|or]]), <math>\to</math> ([[Material conditional|if...then]]), and <math>\uparrow</math> ([[Sheffer stroke]]).{{sfnm|1a1=Magnus|1y=2005|1loc=3. Truth tables|1pp=35–45|2a1=Angell|2y=1964|2p=164}} Given the conditional proposition {{nowrap|<math>p \to q</math>}}, one can form truth tables of its [[Converse (logic)|converse]] {{nowrap|<math>q \to p</math>}}, its [[Inverse (logic)|inverse]] {{nowrap|(<math>\lnot p \to \lnot q</math>)}}, and its [[contrapositive (logic)|contrapositive]] {{nowrap|(<math>\lnot q \to \lnot p</math>)}}. Truth tables can also be defined for more complex expressions that use several propositional connectives.{{sfn |Tarski |1994 |p=40}} {| class="wikitable" style="margin:1em; text-align:center;" |+ Truth table of various expressions |- ! style="width:15%" | ''p'' ! style="width:15%" | ''q'' ! style="width:15%" | ''p'' ∧ ''q'' ! style="width:15%" | ''p'' ∨ ''q'' ! style="width:15%" | ''p'' → ''q'' ! style="width:15%" | ''¬p'' → ''¬q'' ! style="width:15%" | ''p'' <math>\uparrow</math> ''q'' |- | T || T || T || T || T || T || style="background:papayawhip" | F |- | T || style="background:papayawhip" | F || style="background:papayawhip" | F || T || style="background:papayawhip" | F || T || T |- | style="background:papayawhip" | F || T || style="background:papayawhip" | F || T || T || style="background:papayawhip" | F || T |- | style="background:papayawhip" | F || style="background:papayawhip" | F || style="background:papayawhip" | F || style="background:papayawhip" | F || T || T || T |} ===Arguments and inferences=== {{Main|Argument|inference}} Logic is commonly defined in terms of arguments or inferences as the study of their correctness.{{sfnm|1a1=Hintikka|1y=2019|1loc=lead section, §Nature and varieties of logic|2a1=Audi|2loc=Philosophy of logic|2y=1999b}} An ''argument'' is a set of premises together with a conclusion.{{sfnm|1a1=Blackburn|1y=2008|1loc=argument|2a1=Stairs|2y=2017|2p=343}} An ''inference'' is the process of reasoning from these premises to the conclusion.{{sfn |Audi |loc=Philosophy of logic |1999b}} But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on the other hand, are true or false depending on whether they are in accord with reality. In formal logic, a [[Soundness (logic)|sound]] argument is an argument that is both correct and has only true premises.{{sfn |Copi |Cohen |Rodych |2019 |p=[https://books.google.com/books?id=38bADwAAQBAJ&pg=PA30 30]}} Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For a complex argument to be successful, each link of the chain has to be successful.{{sfn |Audi |loc=Philosophy of logic |1999b}} [[File:Argument_terminology.svg|thumb|upright=1.8|right|alt=Diagram of argument terminology used in logic|[[Argument]] terminology used in logic]] Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different [[method of reasoning|types of reasoning]].{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=20|2a1=Backmann|2y=2019|2pp=235–255|3a1=IEP Staff}} The strongest form of support corresponds to [[deductive reasoning]]. But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases, the term ''ampliative'' or ''inductive reasoning'' is used.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=16|2a1=Backmann|2y=2019|2pp=235–255|3a1=IEP Staff}} Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic.{{sfnm|1a1=Groarke|1y=2021|1loc=1.1 Formal and Informal Logic|2a1=Weddle|2y=2011|2loc=36. Informal logic and the eductive-inductive distinction|2pp=383–8|3a1=van Eemeren|3a2=Garssen|3y=2009|3p=191}} ====Deductive==== A deductively valid argument is one whose premises guarantee the truth of its conclusion.{{sfnm|1a1=McKeon|2a1=Craig|2y=1996|2loc=Formal and informal logic}} For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs" is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises.{{sfn |Evans |2005 |loc=8. Deductive Reasoning, [https://books.google.com/books?id=znbkHaC8QeMC&pg=PA169 p. 169]}} According to an influential view by [[Alfred Tarski]], deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by [[logical necessity]] for the given propositions, independent of any other circumstances.{{sfn |McKeon}} Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference.{{sfn|Hintikka|Sandu|2006|pp=13–4}} Rules of inference specify the form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1pp=13-4|2a1=Blackburn|2y=2016|2loc=rule of inference}} The modus ponens is a prominent rule of inference. It has the form "''p''; if ''p'', then ''q''; therefore ''q''".{{sfn |Blackburn |2016 |loc=rule of inference}} Knowing that it has just rained (<math>p</math>) and that after rain the streets are wet (<math>p \to q</math>), one can use modus ponens to deduce that the streets are wet (<math>q</math>).{{sfn |Dick |Müller |2017 |p=157}} The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=13|2a1=Backmann|2y=2019|2pp=235–255|3a1=Douven|3y=2021}} Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in the premises.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=14|2a1=D'Agostino|2a2=Floridi|2y=2009|2pp=271–315}} But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on the depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=14|2a1=Sagüillo|2y=2014|2pp=75–88|3a1=Hintikka|3y=1970|3pp=135–152}} ====Ampliative==== Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with a certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth.{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1pp=13-6|2a1=Backmann|2y=2019|2pp=235–255|3a1=IEP Staff}} This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to ''[[Non-monotonic logic|non-monotonicity]]'' and ''[[Defeasible reasoning|defeasibility]]'': it may be necessary to retract an earlier conclusion upon receiving new information or in the light of new inferences drawn.{{sfnm|1a1=Rocci|1y=2017|1p=26|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=13, 16|3a1=Douven|3y=2021}} Ampliative reasoning plays a central role for many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments is blurry in some cases, as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between.{{sfnm|1a1=IEP Staff|2a1=Douven|2y=2021|3a1=Hawthorne|3y=2021}} The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term "[[inductive reasoning|induction]]" to cover all forms of non-deductive arguments.{{sfnm|1a1=IEP Staff|2a1=Hawthorne|2y=2021|3a1=Wilbanks|3y=2010|3pp=107–124}} But in a more narrow sense, ''induction'' is only one type of ampliative argument alongside ''[[abductive reasoning|abductive arguments]]''.{{sfn |Douven |2021}} Some philosophers, like Leo Groarke, also allow ''conductive arguments''{{efn|Conductive arguments present reasons in favor of a conclusion without claiming that the reasons are strong enough to decisively support the conclusion.}} as one more type.{{sfnm|1a1=Groarke|1y=2021|1loc=4.1 AV Criteria|2a1=Possin|2y=2016|2pp=563–593}} In this narrow sense, induction is often defined as a form of statistical generalization.{{sfnm|1a1=Scott|1a2=Marshall|1y=2009|1loc=analytic induction|2a1=Houde|2a2=Camacho|2loc=Induction|2y=2003}} In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains.{{sfn |Borchert |2006b |loc=Induction}} In this sense, one may infer that "all elephants are gray" based on one's past observations of the color of elephants.{{sfn |Douven |2021}} A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray.{{sfn |Borchert |2006b |loc=Induction}} Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference.{{sfn |Douven |2021}} Abductive inference may or may not take statistical observations into consideration. In either case, the premises offer support for the conclusion because the conclusion is the best [[explanation]] of why the premises are true.{{sfnm|1a1=Douven|1y=2021|2a1=Koslowski|2y=2017|2loc=[https://www.taylorfrancis.com/locs/edit/10.4324/9781315725697-20/abductive-reasoning-explanation-barbara-koslowski Abductive reasoning and explanation]}} In this sense, abduction is also called the ''inference to the best explanation''.{{sfn |Cummings |2010 |loc=Abduction, p. 1}} For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean the table. This conclusion is justified because it is the best explanation of the current state of the kitchen.{{sfn |Douven |2021}} For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it is not the best or most likely explanation.{{sfnm|1a1=Douven|1y=2021|2a1=Koslowski|2y=2017|2loc=[https://www.taylorfrancis.com/locs/edit/10.4324/9781315725697-20/abductive-reasoning-explanation-barbara-koslowski Abductive reasoning and explanation]}}{{sfn |Cummings |2010 |loc=Abduction, p. 1}} ===Fallacies=== Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as [[Fallacy|fallacies]]. Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion.{{sfnm|1a1=Hansen|1y=2020|2a1=Chatfield|2y=2017|2p=194}} So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though the conclusion is true. Some theorists, like [[John Stuart Mill]], give a more restrictive definition of fallacies by additionally requiring that they appear to be correct.{{sfnm|1a1=Walton|1y=1987|1loc=1. A new model of argument|1pp=7|2a1=Hansen|2y=2020}} This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them.{{sfn |Hansen |2020}} However, this reference to appearances is controversial because it belongs to the field of [[psychology]], not logic, and because appearances may be different for different people.{{sfnm|1a1=Hansen|1y=2020|2a1=Walton|2y=1987|2loc=3. Logic of propositions|2pp=63}} [[File:Young America's dilemma - Dalrymple. LCCN2010651418.jpg|thumb|upright=1.5|alt=Poster from 1901|Young America's dilemma: Shall I be wise and great, or rich and powerful? (poster from 1901) This is an example of a [[false dilemma]]: an [[informal fallacy]] using a disjunctive premise that excludes viable alternatives.]] Fallacies are usually divided into [[formal fallacy|formal]] and informal fallacies.{{sfnm|1a1=Vleet|1y=2010|1loc=Introduction|1pp=ix–x|2a1=Dowden|3a1=Stump}} For formal fallacies, the source of the error is found in the ''form'' of the argument. For example, [[denying the antecedent]] is one type of formal fallacy, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore Othello is not male".{{sfnm|1a1=Sternberg|2a1=Stone|2y=2012|2pp=327–356}} But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the ''content'' or the ''context'' of the argument.{{sfnm|1a1=Walton|1y=1987|1loc=1. A new model of argument|1pp=2–4|2a1=Dowden|3a1=Hansen|3y=2020}} Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity, the ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark".{{sfnm|1a1=Engel|1y=1982|1loc=2. The medium of language|1pp=59–92|2a1=Mackie|2y=1967|3a1=Stump}} Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise.{{sfnm|1a1=Stump|2a1=Engel|2y=1982|2loc=4. Fallacies of presumption|2pp=143–212}} In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it.{{sfnm|1a1=Stump|2a1=Mackie|2y=1967}} ===Definitory and strategic rules=== The main focus of most logicians is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as ''rules of inference''.{{sfn|Hintikka|Sandu|2006|p=20}} They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach a given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In [[chess]], for example, the definitory rules dictate that [[Bishop (chess)|bishops]] may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's [[King (chess)|king]].{{sfnm|1a1=Hintikka|1a2=Sandu|1y=2006|1p=20|2a1=Pedemonte|2y=2018|2pp=1–17|3a1=Hintikka|3y=2023}} It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning.{{sfn|Hintikka|Sandu|2006|p=20}} ===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=Flotyński|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=1–10|2a1=Hintikka|2a2=Sandu|2y=2006|2pp=31–32|3a1=Jacquette|3y=2006|3loc=Introduction: Philosophy of logic today|3pp=1–12}} 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 12–13]}} 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=145–6|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=34–42|2a1=Shapiro|2a2=Kouri Kissel|2y=2022|3a1=Bimbo|3y=2016|3pp=8–9}} 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 131–146]|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