Can someone please explain the difference between first/second/third order logic, using examples?

Question

I am currently struggling to get my head around how to recognise the difference between first/second/third order logical arguments.

I feel that the easiest way for me to understand the difference would be if someone could give me an example of each

In plain English
In terms of predicate logic

Any help would be much appreciated.

First order logic: "every men is good", i.e. $\forall x \ (Man(x) \to Good(x))$. Here the variable to be quantified ranges over individuals and the propoerties expressed by the predicates are properties of individuals. — Mauro ALLEGRANZA, May 22 '17 at 10:57
Second order logic: "every virtue is possessed by some man", i.e. $\forall V \exists x (Man(x) \to V(x))$. Here $x$ is an individual variable but $V$ is not: it is a second-order variable, ranging over "properties" of individuals. — Mauro ALLEGRANZA, May 22 '17 at 11:03
@MauroALLEGRANZA shouldn't the translation be $\forall V \exists x (Man(x) & V(x))$ ? — Mirko, May 22 '17 at 11:47
See Second-order and Higher-order Logic for mathematical examples. — Mauro ALLEGRANZA, May 22 '17 at 11:47
The following answer on a related question may be of help: https://math.stackexchange.com/a/3018585/620978 — digilante, Nov 29 '18 at 13:03

score 3 · Answer 1 · answered May 22 '17 at 12:08

First-order, second-order and third-order logic are all logical languages with universal and existential quantifiers. The difference lies in what quantifiers speak about.

In first-order logic one can quantify over individuals. An example of a first-order formula is the commutativity axiom for a group $(G,*,e)$:

$$\forall x. \forall y. x * y = y * x$$

In second-order logic one can quantify over sets of individuals. An example of a second-order formula is the axiom defining the existence of least upper bounds for a complete partial order $(D,\sqsubseteq)$, which says that every subset $Y$ has a least upper bound:

$$\forall Y. Y \subseteq D \Rightarrow (\exists z. (\forall y. y \in Y \Rightarrow y \sqsubseteq z) \wedge (\forall x. (\forall y. y \in Y \Rightarrow y \sqsubseteq x) \Rightarrow z \sqsubseteq x))$$

In third-order logic one can quantify over sets of sets of individuals. An example of a third-order formula is the axiom for topological spaces which states that the union of a family of open sets is an open set.

Can someone please explain the difference between first/second/third order logic, using examples?

1 Answers1