Examine individual changes

'In [[mathematical logic]], a '''ground term''' of a [[formal system]] is a [[term (logic)|term]] that does not contain any [[free variables]]. Similarly, a '''ground formula''' is a [[well formed formula|formula]] that does not contain any free variables. In [[First-order logic#Equality and its axioms|first-order logic with identity]], the sentence {{all}}&nbsp;''x''&nbsp;(''x''=''x'') is a ground formula. A '''ground expression''' is a ground term or ground formula. == Examples == Consider the following expressions from [[first order logic]] over a [[signature (mathematical logic)|signature]] containing a constant symbol 0 for the number 0, a unary function symbol ''s'' for the successor function and a binary function symbol + for addition. * ''s''(0), ''s''(''s''(0)), ''s''(''s''(''s''(0))) ... are ground terms; * 0+1, 0+1+1, ... are ground terms. * '''x'''+''s''(1) and ''s''('''x''') are terms, but not ground terms; * ''s''(0)=1 and 0+0=0 are ground formulae; * ''s''(1) and ∀'''x''':&nbsp;(''s''('''x''')+1=''s''(''s''('''x'''))) are ground expressions. == Formal definition == What follows is a formal definition for [[first-order language]]s. Let a first-order language be given, with <math>C</math> the set of constant symbols, <math>V</math> the set of (individual) variables, <math>F</math> the set of functional operators, and <math>P</math> the set of predicate symbols. === Ground terms === Ground terms are [[term (logic)|terms]] that contain no variables. They may be defined by logical recursion (formula-recursion): # elements of C are ground terms; # If ''f''∈''F'' is an ''n''-ary function symbol and α₁, α₂, ..., α_n are ground terms, then ''f''(α₁, α₂, ..., α_n) is a ground term. # Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms). Roughly speaking, the [[Herbrand universe]] is the set of all ground terms. === Ground atom === A '''ground predicate''' or '''ground atom''' or '''ground literal''' is an [[atomic formula]] all of whose argument terms are ground terms. If ''p''∈''P'' is an ''n''-ary predicate symbol and α₁, α₂, ..., α_n are ground terms, then ''p''(α₁, α₂, ..., α_n) is a ground predicate or ground atom. Roughly speaking, the [[Herbrand base]] is the set of all ground atoms, while a [[Herbrand interpretation]] assigns a [[truth value]] to each ground atom in the base. ≈×—≈266546546546456.76.264462.654.6574..45/657 95797957.5/46/657978978 808.8.08 9054/./56579 0890.0070 8.07 8/0564.5787 9897000 .0000 77 7576. mllmml. m llnkb NB, ml.j b NGC hg bbl njhv chfdx x cg 🍞🎇🎓🍛🎓^3^):🗼🗾🌌X_X➿™X_X<3🌌™🎇<3X_X™🍞🍛🍞🎓™🎇🎓🍛🗻™🎇🍛🌌X_X➿🍜🗾🗾➿):~_~➿🍜X_X<3~_~X_X^o^~_~O.O:/:/~_~^o^O.O~_~<3O.O~_~<3:-O>"<~_~:-O>"<~_~U_U>:(:-O>:(U_U:-O:P:-P:(:):-):D^_~:D:-):D:D^_^;-)^_^:P:P^_^:-):-P^0^:-P:P^0^:-D^0^:-):P:-P:(:-D:(:-D^0^:(;P:P>:(:(:-P:(:-OU_UO.O:[>:(:[:-O:-O>"<~_~~_~O.O^3^<3^o^:/<3^o^X_X:-S^3^=/⛄❄☁✨❄⭕⭕🚤🚧🚙🚒🚓🚇🚅🚇🚅🚒⭕🚉🚃🚅🚧🚃🚅🚌🚅🚇🚒⭕🚑🚇🚅🚏🚌X_X^3^^3^:-S=/=/:-x:-S:-()O.O:-S):U_U;P^0^:D:-);):)^_~^_~;):):P^_^:):-P:P>:(U_U>:(:-O:-OO.O<3<3^o^:/<3^o^:/:-S^3^X_X^3^:-SX_X^o^:-()X_X^o^:-():-x:-S=/:-S:-S=/:-():-x)::-S>"<:[O.O:-P:-DU_U^_^:-)^0^^_~;-)^0^;-)^_~:)^_^:-):D^_^:-):-P:-P:-)^0^:P^0^:-):-P^0^:-D:-PU_U:-D>:(:-P;P:-P>"<;PO.O:[>"<U_U:[:(:P:-P>"<U_UO.OO.O>"<:-()^3^^o^^o^^3^^o^^3^:/^3^X_X:-S^3^:-x=/🚪⭕🚗🚓🚗🚃🚅🚌🚗🚃🚅🚄🚄🚀🚗🚀🚉🚄🚢🚑🚅🚧🚌🚑🚅🚕🚕🚒🚓⛔🚥🚢🚪🚤🚨🚧🔰⛔🚪🚪🚨🅿🔰🚫🚭🚩🅿🚬🚬🅿🚩🚭🚬🅿🚨🚩⛔🚤🚭🚧🚩🚙🚪🚚🚧🚗🚒🚏🚨🚓🚕🚅🚤🚉🚇🚗🚅🚄🚃🚗🚄🚀🚃🚀🚃🚉🚇🚅🚄🚧🚑🚉🚌🚏🅿🚕🚙🚚🚥🚗🚪🔰🚤🚧🚪🚭🅿🚨🔰🚫🚬🚩🚬🚨🚪🚢🚢🚢⭕🚓🚒🚗🚤🚇🚅🚌🚗🚄🚀🚃🚉🚃🚀🚢🚅🚑🚉🚇🚧🚅🚑🅿🚉🚏🚏🚑⭕🚢🚥🚪⛔🚧🚤⛔🔰🔰🚨🚨🔰🅿🚨🚬🅿🚧🚭🚬🚙🚢🚥🚒🚨🚗🚏🚇🚤🚅🚄🚗🚃🚀🚗✈🚀🚄🚀✈🚗🚄🚅🚤🚇🚇🚌🚒🚗🅿⛔🔰🗾🚪🚩➿:-O🚇⛔🚀➿🚑~_~~_~🚩🚇🚑🚒🚉➿🚉➿🚉🗾:-O🚒🚀🚬🚧):🗾🗾^o^^o^X_X^o^🚪^3^^3^=/):X_X🚀<3):<3🚀<3🚅🚄🚪🚧🚌🚪🚬🚪🚒🚀🔰:-O⛔🚇~_~➿🚉~_~:-O~_~🚩🚉🚒:-O🚀🚒⛔🚒🚬~_~🚬🚧🚀X_X):🚗🚬X_X):🚗^o^🚬🚗X_X<3)::-OX_X:-O🚨🚗🚬🚗🚗X_X🚄🅿🚢~_~🚧🚢🚌🚪🗾🚢~_~🚒🚇🚑🚩🚩🚇~_~✈🚉🚒~_~🚉✈:-O🚩🚩🚒🚒🚅<3🚀^3^O.O🚪^3^^o^=/^3^O.O^o^🗾<3~_~):X_X~_~🚄🚪🚧🚄🗾🎐🎓💍🍥🍱🍩🍬🍨🍬🍰🍱🍪🍨🍬🍱🍨🍯🍪🍭🍨🍰🍧🍮🍫🍟🍡🍔🍖🍡🍗🍗🍕🍔🍤🍗🍗🍘🍨🍜🍞🍝🍝🍫🍣🍦🍤🍥🍩🍪🍯🍬🍰🍭🍪🍱🍭🍱🍧🍤🍧🍫🍢🍢🍖🍚🍕🍔🍗🍚🍔🍥🍙🍛🍜🚢🚢🚙💑💍🎓🎑👹💐💋💏👹💍💒💍💋💐👸🎓💑💋🎇🎀🎏🎂🎎🎈🎂🎀🎁🎂🎈🎅🎃🎄💋🎇🎍🎊🎇🎊🎋🎍🎌🎋🎎🎏💌💏💑💍💐👸💑💏💒👹💏💋👹💒🎓💐👹🎓👹🎑🎓💍💑🎏😜😔😈😖😌😁😍😄😞😍😃😉😞😊😄😉😂😄😇😂😔😇😇😌😏😍😔😘😝😖😔😝😣😖😖😠😤😚😣😤😔😝😤😢😖😝😒😠😡😚😔😡😔😚😜😐😏😈😒😅😇😂😍😊😍😉😃😊😃😞😉😉😃😞😄😂😉😔rjesdxkccf,tu.bnpok??'nlmkl'nmokpmoklionjihvb,vvhu.tyccyguy.vrfkweahhezezr😐😄😊😉😂😇😚😋😈😍😎😏😤😝😒😚😣😖😠😚😤😘😔😣😤😚😠😖😡😚😔😢😖😘😚😤😤😚😔😠😠😚😒😖😡😚😒😔😐😋😓😓😡😎😏😓😎😤😍😓😓😏😍😋😜😏😌😋😌😔😇😆😇😆😜😈😖😆😅😇🗾🚗😖<3~_~<3😇😤😋😤🚪😏🗾😅😋😍😈🗾😋😆😉🚪😠😈😅😆🚪😌😉😆🚪😠<3😤😜😇~_~😤🚗😜😔🗾😖):^o^X_X^3^^3^X_X^3^😏🚗^3^~_~😜=/):🚗~_~😚~_~😜<3🚪😍🚪😇🗾😈🗾😌🗾🚒😅😋🚒😈🚒😋😈🗾😉😌😅🗾🚒🚒😌😈😝😉😠😆😉🚪😅😤😜😇<3~_~😤🗾😔):rrktdxkrexkdctrccr,yctv,y,yu v.ibjbno.jpk.o'nkopn.ukhbuv,vugcntdrjdrkrct,tucv..iublonk.ju.hi,tyfcejrszrzdx,drcc.yujnk'n'kl.knop.k.huvitfjtwjezrlx):🗾😤cfcyu.bjipkm kn,vhb vu,hvjouuvytrmcX_X=/~_~😅X_X~_~😠):=/X_X😠~_~<3😍😍😈😆😡😍😈😆😍😈😜😎😏😎😓😤😔😒😎😚😡😚😎😓😠😐😓😠😎😔😓😣😍😓😎😡😋😏😌😈😜😉😐😂😄😉😞😉😂😉😚😇😇😅😆😠😇😏😐😝😓😔😠😔😘😡😡😠😔😝😒😓😣😝😒😘😔😤😓😌😋😜😈😔😉😄😂😂😇😚😁😇😅😜😆🗾😓🚒😇😓😉😠😓😁😉😜👹🎏🎎🎋😡😏😎😓😐😚😤😔😠😒😚😔😐😎😝😤😒😚😝😚😒😤😝😔😔😐😘😎😤😏😍😡😓😋😋😇😖😌😄😄😔😁😂😄😚😆😄😇😠🗾😇):😏=/🚪<3😌😆<3😅😜😍😌😅🚗😅😜😍😌🚗<3🚗😜🚪😌🚪🚗😎🚪🚗😜😌😅😜<3🚗<3😆<3😏😆😓😠😚^o^O.O😆😔😔🚗X_X😚X_X😤😚😏😤😤🗾😏😚🗾😉😇😇😌🗾😌😇😈~_~😌):~_~~_~<3😜😜😍🚗😋🚗😅😍😌😅😌😜🚪😖😄😋😍😎😋😅🚗😌😆🚪<3😌):😈=/😏😠😇):😆😓😚^o^😉😔O.O^o^😉😔O.O😆😤😚X_X😓^o^^o^O.O😓😚😤X_XX_X😏😇🗾😏😠):~_~=/<3😖😎😍😎😖😄😄🚗😜😍😄😜🚪🚗=/=/😌😉~_~😉~_~😈😓🚗😚😓O.O😆😚😓😓😋😤😚😤😋😤X_X😤🚗X_X😤O.O😚O.O😆😚O.O^o^😆😚😤😜X_X😚O.O😚😆O.O🚗🗾😇X_X😏😇😇🗾):😌😈~_~~_~):😈😉😆=/😋😖🚪😋😍🚗😅😄😜🚗😄<3🚪😆😌😆<3🚪😏~_~):🗾😈X_X😠😚🚗O.O😚😓😆😆O.O😓😔😤😋😚😤X_X😏😚X_X🗾😌X_X😉<3😈=/😋😍🚪🚒😋😄🚒🚗😄😅<3😆😜😄😌😈<3=/~_~😉~_~):😇X_X😠😋😆O.O😓😚😚O.O😆😓😚O.O😆^o^😆^3^O.O😔😔😤^o^😤😜😔X_XX_X😚😋😚😤X_XX_X😚🗾😏X_X😈):~_~):<3😆<3😍=/🌙🌔🌠🌕✨🌛🌠🌛☔✨🌛☀☔✳✳☔☔☀✴❄🌠☔🌛❄🌔🌙☁🌔🌉🌆🌅🌑🌁🌄🌂🌀🌊🌁🌀🌃🌕🌁🌟🌁🌃🌀🌟🌁🌆🌁☁🌃🌄🌉🌆^3^<3~_~😇<3~_~🗾😇😌🚪😄😍=/🚪😈😅🌆🚪😈😅🌆😏🌆😈😇😌😌😋~_~😜😄🗾😉<3😔😆chf😈😋:-O🌔🌌🌊😍😐😔🌌🌑🌑😐😔😎🌌🌔🌊🚓🚗🚢😍😋😔🚒🚌🚌🚅🚃🍞🍛🍙 fzdbk km' === Ground formula === A ground formula or ground clause is a formula without free variables. Formulas with free variables may be defined by syntactic recursion as follows: # The free variables of an unground atom are all variables occurring in it. # The free variables of ¬''p'' are the same as those of ''p''. The free variables of ''p''∨''q'', ''p''∧''q'', ''p''→''q'' are those free variables of ''p'' or free variables of ''q''. # The free variables of {{all}}&nbsp;''x''&nbsp;''p'' and {{exist}}&nbsp;''x''&nbsp;''p'' are the free variables of ''p'' except ''x''. == References == * {{Citation | title = Handbook of discrete and combinatorial mathematics | contribution = Logic-based computer programming paradigms | year = 2000 | editor1-last = Rosen | editor1-first = K.H. | editor2-last = Michaels | editor2-first = J.G. | last = Dalal | first = M. | page = 68 }} * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-58713-6 | year=1997}} * [http://web.engr.oregonstate.edu/~afern/classes/cs532/notes/fo-ss.pdf First-Order Logic: Syntax and Semantics] <!-- these references are essentially random; Hodges is a standard reference but does not define all the terms used in this article --> [[Category:Mathematical logic]] [[Category:Logical expressions]]'