Editing Philosophical logic (section)

=== Higher-order ===
[[Higher-order logics]] extend first-order logic by including new forms of [[Quantifier (logic)|quantification]].<ref name="Cambridge"/><ref name="Väänänen"/><ref name="Ketland">{{cite book |last1=Ketland |first1=Jeffrey |title=Encyclopedia of Philosophy |date=2005 |url=https://www.encyclopedia.com/humanities/encyclopedias-almanacs-transcripts-and-maps/second-order-logic |chapter=Second Order Logic}}</ref><ref name="Computing">{{cite book |title=A Dictionary of Computing |chapter=predicate calculus |url=https://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/predicate-calculus}}</ref> In first-order logic, quantification is restricted to singular terms. It can be used to talk about whether a predicate has an extension at all or whether its extension includes the whole domain. This way, propositions like {{nowrap|"<math>\exists x (Apple(x) \land Sweet(x))</math>"}} (''there are some'' apples that are sweet) can be expressed. In higher-order logics, quantification is allowed not just over individual terms but also over predicates. This way, it is possible to express, for example, whether certain individuals share some or all of their predicates, as in {{nowrap|"<math>\exists Q (Q(mary) \land Q(john))</math>"}} (''there are some'' qualities that Mary and John share).<ref name="Cambridge"/><ref name="Väänänen"/><ref name="Ketland"/><ref name="Computing"/> Because of these changes, higher-order logics have more expressive power than first-order logic. This can be helpful for mathematics in various ways since different mathematical theories have a much simpler expression in higher-order logic than in first-order logic.<ref name="Cambridge"/> For example, [[Peano arithmetic]] and [[Zermelo-Fraenkel set theory]] need an infinite number of axioms to be expressed in first-order logic. But they can be expressed in second-order logic with only a few axioms.<ref name="Cambridge"/>

But despite this advantage, first-order logic is still much more widely used than higher-order logic. One reason for this is that higher-order logic is [[Completeness (logic)|incomplete]].<ref name="Cambridge"/> This means that, for theories formulated in higher-order logic, it is not possible to prove every true sentence pertaining to the theory in question.<ref name="Hintikka"/> Another disadvantage is connected to the additional ontological commitments of higher-order logics. It is often held that the usage of the existential quantifier brings with it an ontological commitment to the entities over which this quantifier ranges.<ref name="Britannica"/><ref name="Schaffer">{{cite journal |last1=Schaffer |first1=Jonathan |title=On What Grounds What |journal=Metametaphysics: New Essays on the Foundations of Ontology |date=2009 |pages=347–383 |url=https://philpapers.org/rec/SCHOWG |access-date=23 November 2021 |publisher=Oxford University Press}}</ref><ref name="Bricker">{{cite web |last1=Bricker |first1=Phillip |title=Ontological Commitment |url=https://plato.stanford.edu/entries/ontological-commitment/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=23 November 2021 |date=2016}}</ref><ref name="Quine">{{cite journal |last1=Quine |first1=Willard Van Orman |title=On What There Is |journal=Review of Metaphysics |date=1948 |volume=2 |issue=5 |pages=21–38 |url=https://philpapers.org/rec/QUIOWT-7}}</ref> In first-order logic, this concerns only individuals, which is usually seen as an unproblematic ontological commitment. In higher-order logic, quantification concerns also properties and relations.<ref name="Britannica"/><ref name="Väänänen"/><ref name="HaackLogics1"/> This is often interpreted as meaning that higher-order logic brings with it a form of [[Platonism]], i.e. the view that [[Universal (metaphysics)|universal]] properties and relations exist in addition to individuals.<ref name="Cambridge"/><ref name="Ketland"/>