In recent years second-order logic has made something of a recovery, buoyed by George Boolos’ interpretation of second-order quantification as plural quantification over the same domain of objects as first-order quantification (Boolos 1984). Boolos furthermore points to the claimed nonfirstorderizability of sentences such as “Some critics admire only each other” and “Some of Fianchetto’s men went into the warehouse unaccompanied by anyone else” which he argues can only be expressed by the full force of second-order quantification. However, generalized quantification and partially ordered, or branching, quantification may suffice to express a certain class of purportedly nonfirstorderizable sentences as well and it does not appeal to second-order quantification.
I fail to see how this is evidence of Nonsecondorderizability of some possible sentences.
There is no known trick to encode all sentences expressible in higher-order logics into first-order logic, but there is such a trick to encode all sentences expressible in higher-order logics in second-order logic.
The trick in question is described in the SEP article. Doesn’t that suffice as a reference and starting point for studying the notion that second-order logic can encode higher-order logics?
https://en.wikipedia.org/wiki/Nonfirstorderizability
I fail to see how this is evidence of Nonsecondorderizability of some possible sentences.
There is no known trick to encode all sentences expressible in higher-order logics into first-order logic, but there is such a trick to encode all sentences expressible in higher-order logics in second-order logic.
The trick in question is described in the SEP article. Doesn’t that suffice as a reference and starting point for studying the notion that second-order logic can encode higher-order logics?
i misread it XD trhanks for your help