Editing Semantics (section)

=== Logic ===
{{main|Semantics of logic}}

Logicians study correct [[Logical reasoning|reasoning]] and often develop [[formal languages]] to express arguments and assess their correctness.<ref>{{multiref | {{harvnb|Riemer|2010|pp=173–174}} | {{harvnb|Jaakko|Sandu|2006|pp=13–14}} | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead Section, § 2. Language}} }}</ref> One part of this process is to provide a semantics for a formal language to precisely define what its terms mean. A semantics of a formal language is a set of rules, usually expressed as a [[mathematical function]], that assigns meanings to formal language expressions.<ref>{{multiref | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead Section, § 4. Semantics}} | {{harvnb|Jansana|2022|loc=§ 5. Algebraic Semantics}} | {{harvnb|Jaakko|Sandu|2006|pp=17–18}} }}</ref> For example, the language of first-order logic uses lowercase letters for [[individual constant]]s and uppercase letters for [[Predicate (mathematical logic)|predicates]]. To express the sentence "Bertie is a dog", the formula <math>D(b)</math> can be used where <math>b</math> is an individual constant for Bertie and <math>D</math> is a predicate for dog. Classical model-theoretic semantics assigns meaning to these terms by defining an [[Interpretation (logic)|interpretation function]] that maps individual constants to specific objects and predicates to [[Set (mathematics)|sets]] of objects or [[tuple]]s. The function maps <math>b</math> to Bertie and <math>D</math> to the set of all dogs. This way, it is possible to calculate the truth value of the sentence: it is true if Bertie is a member of the set of dogs and false otherwise.<ref>{{multiref | {{harvnb|Grimm|2009|pp=[https://books.google.com/books?id=QwyS2rZKnB0C&pg=PA116 116–117]}} | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead Section, § 4. Semantics}} | {{harvnb|Magnus|Button|Thomas-Bolduc|Zach|2021|pp=193–195}} }}</ref>

Formal logic aims to determine whether arguments are [[deductively valid]], that is, whether the premises entail the conclusion.<ref>{{multiref | {{harvnb|Riemer|2010|pp=173–174}} | {{harvnb|Jaakko|Sandu|2006|pp=13–14}} | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead Section}} | {{harvnb|Gregory|2017|p=[https://books.google.com/books?id=9heXDQAAQBAJ&pg=PA82 82]}} }}</ref> Entailment can be defined in terms of syntax or in terms of semantics. Syntactic entailment, expressed with the symbol <math>\vdash</math>, relies on [[rules of inference]], which can be understood as procedures to transform premises and arrive at a conclusion. These procedures only take the [[logical form]] of the premises on the level of syntax into account and ignore what meaning they express. Semantic entailment, expressed with the symbol <math>\vDash</math>, looks at the meaning of the premises, in particular, at their truth value. A conclusion follows semantically from a set of premises if the truth of the premises ensures the truth of the conclusion, that is, if any semantic interpretation function that assigns the premises the value ''true'' also assigns the conclusion the value ''true''.<ref>{{multiref | {{harvnb|Forster|2003|pp=[https://books.google.com/books?id=mVeTuaRwWssC&pg=PA74 74–75]}} | {{harvnb|Johnstone|1987|p=[https://books.google.com/books?id=_hlsBpVA3qkC&pg=PA23 23]}} | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead Section, § 4. Semantics}} | {{harvnb|Jaakko|Sandu|2006|pp=17–20}} }}</ref>