Second-order logic: Difference between revisions

Revision as of 21:28, 30 May 2020

Revision as of 23:24, 30 May 2020

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==Examples==

In first-order logic, one can quantify over valuesindividuals, but not over properties. That is, in FOL we can take an atomic sentence like Cube(b) and obtain a quantified sentence by replacing the name with a variable and attaching a quantifier<ref>Professor Marc Cohen lecture notes https://faculty.washington.edu/smcohen/120/SecondOrder.pdf</ref>:

∃x Cube(x)

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And this will come out true in exactly when a and b are either both cubes, both tetrahedra, or both dodecahedra. So in second-order logic we can express the idea of ''same shape'' using identity and the second-order predicate Shape; we can do without the special predicate SameShape<ref>Professor Marc Cohen lecture notes https://faculty.washington.edu/smcohen/120/SecondOrder.pdf</ref>.

Similarly, we can express the claim that no object has every shape in a way that brings out the quantifier in ''every shape'':

¬∃x ∀P(Shape(P) → P(x))