From researchgate: https:\/\/www.researchgate.net\/post\/What_is_the_actual_difference_between_1st_order_and_higher_order_logic<\/p>\n
What is the actual difference between 1st order and higher order logic?
\nYes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can’t we define the domain X’ = X u 2^X and for elements of x in X’ define predicates:
\nSET(x)
\nELEMENT(x)
\nBELONGS_TO(x, y) – undefined (or false) when ELEMENT(y)
\netc.
\nNow, we can express sentences about subsets of X in the 1st-order logic!
\nSimilarly we can define FUNCTION(x), etc. and… we can express all 2nd-order sentences in the 1st order logic!
\nI’m obviously overlooking something, but what actually? Where have I made a mistake?<\/p>\n
My answer:<\/p>\n
In many cases one can reduce a higher order formalization to a first-order, but it will come at the price of complexity of the formalization.<\/p>\n
For instance, formalize the follow argument in both first order and second order logic:
\nAll things with personal properties are persons. Being kind is a personal property. Peter is kind. Therefore, Peter is a person.<\/p>\n
One can do this with either first or second order, but it is easier in second-order.<\/p>\n
First-order formalization:
\n1. (\u2200x)(PersonalProperty(x)\u2192((\u2200y)(HasProperty(y,x)\u2192Person(y)))
\n2. PersonalProperty(kind)
\n3. HasProperty(peter,kind)
\n\u22a2 4. Person(peter)<\/p>\n
Second-order formalization
\n1. (\u2200\u03a6)(PersonalProperty(\u03a6)\u2192(\u2200x)(\u03a6x\u2192Person(x)))
\n2. PersonalProperty(IsKind)
\n3. IsKind(peter)
\n\u22a2 4. Person(peter)<\/p>\n
where \u03a6 is a second-order variable. Basically, whenever one uses first order to formalize arguments like this, one has to use a predicate like “HasProperty(x,y)” so that one can treat variables as properties indirectly. This is unnecessary in second-order logics.<\/p>\n","protected":false},"excerpt":{"rendered":"
From researchgate: https:\/\/www.researchgate.net\/post\/What_is_the_actual_difference_between_1st_order_and_higher_order_logic What is the actual difference between 1st order and higher order logic? Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can’t we define the domain X’ = X u 2^X and […]<\/p>\n","protected":false},"author":17,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11],"tags":[1995,1996],"class_list":["post-4189","post","type-post","status-publish","format-standard","hentry","category-logic-philosophy","tag-first-order-logic","tag-second-order-logic","entry"],"_links":{"self":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/4189","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/users\/17"}],"replies":[{"embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/comments?post=4189"}],"version-history":[{"count":2,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/4189\/revisions"}],"predecessor-version":[{"id":4342,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/posts\/4189\/revisions\/4342"}],"wp:attachment":[{"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/media?parent=4189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/categories?post=4189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/emilkirkegaard.dk\/en\/wp-json\/wp\/v2\/tags?post=4189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}