moves may be used to win a game, for instance, by controlling the center and by defending one's king.[94]
It has been argued that logicians should give more emphasis to strategic rules since they are highly
relevant for effective reasoning.[93]
A formal system of logic consists of a formal language together with a set of axioms and a proof system
used to draw inferences from these axioms.[95] In logic, axioms are statements that are accepted without
proof. They are used to justify other statements.[96] Some theorists also include a semantics that specifies
how the expressions of the formal language relate to real objects.[97] Starting in the late 19th century,
many new formal systems have been proposed.[98]
A formal language consists of an alphabet and syntactic rules. The alphabet is the set of basic symbols
used in expressions. The syntactic rules determine how these symbols may be arranged to result in well-
formed formulas.[99] For instance, the syntactic rules of propositional logic determine that " " is a
well-formed formula but " " is not since the logical conjunction  requires terms on both sides.[100]
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  follows from the premises  and . 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 calculi.[101]
A semantics is a system for mapping expressions of a formal language to their denotations. In many
systems of logic, denotations are truth values. For instance, the semantics for classical propositional logic
assigns the formula  the denotation "true" whenever  and  are true. From the semantic point of
view, a premise entails a conclusion if the conclusion is true whenever the premise is true.[102]
A system of logic is 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.[103]
Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments.
For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western
world,[104] but modern developments in this field have led to a vast proliferation of logical systems.[105]
One prominent categorization divides modern formal logical systems into classical logic, extended logics,
and deviant logics.[106]
Formal systems
Systems of logic
Aristotelian