Gottlob Frege's Begriffschrift
introduced the notion of quantifier in
a graphical notation, which here
represents the judgment that
 is true.
mathematics, it does not include logical vocabulary relevant to many other topics of philosophical
importance. Examples of concepts it overlooks are the contrast between necessity and possibility and the
problem of ethical obligation and permission. Similarly, it does not address the relations between past,
present, and future.[119] Such issues are addressed by extended logics. They build on the basic intuitions
of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical
approach is applied to fields like ethics or epistemology that lie beyond the scope of mathematics.[120]
Propositional logic comprises formal systems in which formulae are built from atomic propositions using
logical connectives. For instance, propositional logic represents the conjunction of two atomic
propositions  and  as the complex formula . Unlike predicate logic where terms and predicates
are the smallest units, propositional logic takes full propositions with truth values as its most basic
component.[121] Thus, propositional logics can only represent logical relationships that arise from the
way complex propositions are built from simpler ones. But it cannot represent inferences that result from
the inner structure of a proposition.[122]
First-order logic includes the same propositional connectives as
propositional logic but differs from it because it articulates the
internal structure of propositions. This happens through devices
such as singular terms, which refer to particular objects,
predicates, which refer to properties and relations, and quantifiers,
which treat notions like "some" and "all".[123] For example, to
express the proposition "this raven is black", one may use the
predicate  for the property "black" and the singular term 
referring to the raven to form the expression . To express that
some objects are black, the existential quantifier  is combined
with the variable  to form the proposition . First-order logic contains various rules of inference
that determine how expressions articulated this way can form valid arguments, for example, that one may
infer  from .[124]
Extended logics are logical systems that accept the basic principles of classical logic. They introduce
additional symbols and principles to apply it to fields like metaphysics, ethics, and epistemology.[125]
Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal
logic", it introduces two new symbols:  expresses that something is possible while  expresses that
something is necessary.[126] For example, if the formula  stands for the sentence "Socrates is a
banker" then the formula  articulates the sentence "It is possible that Socrates is a banker".[127] To
include these symbols in the logical formalism, modal logic introduces new rules of inference that govern
Propositional logic
First-order logic
Extended
Modal logic