# OrderLogic ‑ Min & Max Limits

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### OrderLogic Set up & Use

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### Prenex Normal Form | First Order Logic

so now we've looked at some logical,equivalences what we're going to do in,this video is look at how to convert,sentences into what we call,pre-next normal form okay now this is a,type of normal form just like,disjunctive and conjunctive in,propositional logic,so it's another way you can write a,sentence okay and this is one of the,theorems of first order logic that ev,for every sentence,for every sentence,okay there is an equivalence,an equivalence,in pre-next normal form okay and i'm,gonna write that as p,n,f okay,so for every sentence that we have in,first order logic we also have a,sentence that is equivalent to this,sentence in prenux normal form and all,prenix normal form is,is a sentence is in pre-next normal form,when,we read the sentence,and,the,quantifiers occur before any connective,or atomic sentence okay,so let's take an example it's easy just,to look at an example of this okay so,let's say we have,for some y,brackets f y,if f y than,for all z gz,okay,so this is a perfectly valid first order,sentence however this is not in prenux,normal form for this to be in prenux,normal form,this,quantifier here would have to be before,this here so we'd have to move,it out and put it here where the,where the existential is,and this is really easy to do in this,example because we've known about how to,move them any about how to move things,anyway from the last video all we do is,we just move it okay and we don't have,to do anything else to the logic of the,sentence,so if we have,for all,y and sorry for some y and all z,if f y,then,g z,and this is exactly is equivalent to,this and it is in prenex normal form now,this is a really easy example and this,is really what you'll find in most cases,okay,this is why pre-next normal form is is,particularly interesting and quite easy,to understand and quite easy to do,okay now this is,really,where we start to find the difficulties,with prenex normal form,okay,now because there's one thing that can,happen if you just do this in prenux,normal form with more complicated,sentence and that is you end up um,you end up,you must to to go into pre-next normal,form you must avoid something called a,variable capture,variable,capture,okay,and this is where,the variable on the outside okay applies,to more things than it did when it was,not in prenux normal form so if we take,this example again,if we just go back to the original,example so for some y,f y,if and if then um,for all z,g z okay,now this is quite simple,because we know the binding of these,quantifiers we know that this quantifier,binds to this variable here and we know,that this quantifier binds to this,variable here and even if we do move,this quantifier out okay and we end up,with this again we still know that this,quantifier binds to this and this,quantifier binds to this okay,but what if we had a more complicated uh,example okay what if we had an example a,little bit like this so for all x,brackets,if ffx,then gx,okay,or,for,some x,okay,gx,then fx,okay,so th

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### Convert english sentence into FOL(first order logic) in Hindi | Artificial intelligence Series

hey friends welcome to last moon,tuitions as on the keys sanctum Cisco,fol mckessie convert Keerthi how to,convert a sentence from sentence to fo,at first order logic the mapa be secon,superior can you allow or aqua basic,with a novel of the classical name to,sub simple again her thing of civilities,the clothes are all aya so Ethne either,making implication was sent as mäkinen a,to aya somewhere a job' all I have S&S,MacLean any implication ayah or job,there exists just some yeah anyone,Yakushi aya to whopping and I who are,again all wheel Adam elack they or their,I exist as some good cheese are no one,even though me elected jaha all are my,implication or jawed Aristotle then and,are they a guru sentence but out the,palace and is Nick thing a bill either,student to some cans licking it,please simple a bill bracket Milliken or,barn licking it this is true Nabisco,reading a second the bill is a student,easy this is AMA targeted scoop a seam,of a ragged bill is a student to sir I,our arm is a professor to kayak around,and Araya bar gaya professor again,the upside intersect a takes,analyst either analysis or geometry just,go Pegasus looking a bit ragged may be,barn looking it takes fear is a common,leaking analysis which Corddry Jessica,rebuilt takes analysis or bill takes,form either geometry,some da Bay takes analysis or built,extremity ii dissenters have built its,analysis on geometry the yaga yaga and,the universe is aya bill takes analysis,if an namely if you will not take,geometry up with bill takes analysis no,geometry but not both at the same time,those two vasilich night so she throws,in a bill Kabila crow takes analysis if,then only if bill bill does not take,geometry ii am on the way agar bill,analysis siddhartha be lega there was AA,material geometry Lehrer thorough,analysis Malaika the yes second level,are answerable even but I have now fol,to convert Knigge the novelty record or,Vacarro bill takes analysis if and only,if bill does not think this was not the,symbol again octopod arigato does not,take geometry again so lick this some,student cloves bill throat ascendancy,apni some students close will do some,shooting killing Alec there exists X dou,X college student exit the register exit,students loss so we have an X's are,those organic term a delicti loves 2x,ex-con student loves bin showcasing I,can get the resident students and,student,loves bin so there is exist some,students who loves bill so that there,exists I AMA an tiger,well liquor the all store loves will,pick up a temple that all X,love speed that exists Austrians we're,best students loves bin any all showing,those bill but now all I know my,implicates implication effect or JA,there exists a the Buddha and other so,see there as an also in love speed up,this second all students are smart with,directs for all X X corner student X it,just might be a bra that is it all Japan,organic the implication yeah smartest,all shown X implies smartest among,examine Duchess I use correctly a

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### First order logic

assalamu alaikum and peace be upon you,okay this is the third video lecture of,logic representation,uh so the learning outcome again i hope,you will know what is logic presentation,and this time it is first order logic at,least you know what is the difference or,the additional of first order logic,compared to the previous two logic that,is propositional logic and practical,logic,and i hope you will know how to,translate suitable knowledge into logic,representation,and you can critically think the,advantage and disadvantages of each type,of representation so as an edit,from what i have listed we have listed,here,so,now it's logic presentation we have,looked into production rules semantic,network object attribute values as well,as frames,okay so first order logic,so first order logic try to model the,word in terms of object properties,relations and functions,okay so they want to represent by,dividing the world or things around us,into objects,properties of the objects relationship,between the objects,as well as the,function,okay where there is only one value for,any given,input,okay so if a propositional logic all,sentences isn't,atoms okay so this time everything,around us is divided into four things,and they try to,model the relationship between the,object properties relation and functions,let's take an example,now to represent a car,okay,to represent a car,so the object is car,uh,and then the properties is blue,okay the relationship,maybe this car is owned by okay this car,is used,to visit someone okay bigger than what,okay part of what has color of what okay,so that's uh the relationship,and the function,okay um,drive to school for example,okay um,what else a function of a car,owned by is a relationship,maybe,functions,to pick up kids,all right or go to work,okay so that is a function of occur so,this is an example how first audiologic,try to,explain about car,okay so the user will need to provide a,constant symbol okay logic remember they,use symbols,so,we have symbols and we have a function,symbol so function symbol is uh,further off mary so remember when you,learn maths you have x and then you have,f,and then you have x inside the bracket,right so same with this father of mary,is equal to john so this is a,function,and then predicate symbol okay,the outside for example this one greater,okay or green or color could be the,relation,function,or properties,okay,the one inside,is object,so for example this sentence five is,greater than three,grass is green,so you can also write here,a shirt okay inside the bracket so grass,and shirt are greens,okay,if the output if the outside is color,okay so this is the relation okay,properties,so grass,color is green,okay so this is to map uh individuals to,truth values okay what is the values of,each individual objects,so what is the addition in first order,logic,okay they have a,quantifier,okay so we will look at this,the universal uh a,or existing existential,at the same time uh f o l also have,sorry,okay f

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### Transition to Advanced Math: 06 Predicate Calculus 39 min

hello this is professor roman let's,continue with the lecture series,transition to advanced mathematics,we have finished our discussion of the,propositional calculus,and so now we're going to turn to the,predicate predicate calculus,as i mentioned before the propositional,calculus is simply not,powerful enough to serve as a logical,foundation,for mathematics,there are a number of reasons why one of,which,is that there is no way to express,within the propositional calculus two,very important,concepts that we use a lot in,mathematics one,is for all objects,something happens and the other is there,exists an object for which something,happens,these are quite common in mathematics in,fact sometimes they're,hidden for instance if you,were to say the square of a real number,is not negative,what you are really saying is that for,all,real numbers say x we have,x squared bigger than or equal to zero,so there's an implicit mention of,for all in this case,so to incorporate the ideas of for all,and there exists,we use what are called quantifiers,and in both cases,quantifiers require a,set or class which i call,you known as the universe,of discourse or simply the universe,for the quantifiers so here is,the formal definition let u be,a universe of discourse so you is either,a set,or a class a universal quantifier over,you is an expression that looks like,this,and this is read for all x in,u if the universe is clearly,specified beforehand,you can simply write for all x,the variable x is called the subject,of the universal quantifier,an existential quantifier over universe,u,is an expression that looks like this,and it is read there exists an x,in u and again if the universe has been,clearly specified we can just write,there exists x,and notice that everything is enclosed,in parentheses for reasons it'll become,apparent,as we continue,also in this case x is called the,subject of the existential quantifier,now the intended use of quantifiers of,course is in,that they be incorporated into symbolic,expressions into well-formed formulas,so here are a couple of examples for all,x in r,x is greater than zero and x is less,than,five there exists an x and q,such that x squared equals four,a well-formed formula that looks like,this is read,for all x and u p of x,or p of x holds or p of x is true,something like that,and in this case there exists an x in u,such that p of x holds,since the subjects of a quantifier,range over a specified,universe the choice of that universe has,a profound effect,on truth value for example the statement,there exists an x in,r such that x squared equals 2 is,true but if we reduce the universe,there exists an x and q such that x,squared equals 2.,this is not true because,no rational number has the property that,it's square,is equal to 2 which is something by the,way we'll prove when we get to proof,techniques,so again if the universe of discourse is,clearly specified,it's common to omit explicit,mention of the universe within the,quantifiers,so

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### 4c First-Order Logic

this is bill farmer again welcome back,to mcmaster university course,computing and software 701,logic and discrete mathematics,we're talking about first order logic,now,and today we're going to talk about,axomatic theories,so an xmap theory is a one of the most,important notions,in logic so we're going to define a,first order axomatic theory,which we're probably from now on we'll,just call a theory for short,it's a pair of a language,and a set of formulas the language is a,first order language,and the set of formulas are closed and,we call these formulas,axioms so it's a language,and a set of axioms and then,an interpretation is a model of our,theory,if that interpretation is a model of the,axis,and a formula is valid in a theory,if that formula is a logical consequence,of the axioms,and a theory is satisfiable,if the axioms are satisfiable,now it turns out i forgot the definition,of a,of a set of formulas being satisfiable,so i added that to the slides,it's right here,but it's what you would expect a set of,formulas satisfiable,if there is a some interpretation,and some assignment such that,that assignment satisfies,the formulas in the set in the,interpretation,okay so so when we know now what a first,order x back theory is,we know what its models are we know what,it means for formulas to be valid in,in the theory we know what it means for,the theory to be satisfied,now the really important thing about,theories which i have written and read,here,is that a theory can be viewed as a,specification of its models,and there's many examples of theories,we're going to look at,some of them but there's theories of,orders lattices,boolean algebras for instance of a,theory of,of weak partial orders all its models,would be essentially,partial orders,weak partial orders and,remember a model is a structure plus a,mapping,of the symbols in the language to that,structure,anyway the continue we can have theories,of monoids,groups and rings fields these are,algebraic,theories of algebraic structures another,important theory is pressburg arithmetic,this is the theory,of a,natural number arithmetic with zero,successor and plus but not times,we have first order piano arithmetic,this is a theory of,of arithmetic with zero successor plus,and times the theory of real close,fields this is basically the theory of,the algebra,of the real numbers and we're going to,take uh,example very specific example the theory,of monoids,so this theory has a language right here,it is,and we have one,constant symbol e,one function symbol mol which is,binary and one predicate symbol,equals with is also binary and it has,these axioms now,we also implicitly have the axioms for,equality being a congruence or,representing a congruence relation,equality is not actually a relation it's,a predicate but,every predicate represents a relation so,it represents,the congruence relation so we haven't,put down those axioms they're,implicit okay so what what,are they what do they axioms of you say,they sa

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### 10 8 10 8 First Order Proofs 5 min

fitch system for first-order logic or,first order fetch is similar to that for,her Brown logic we have the same logical,rules and the same rules for quantifiers,there are some new rules to handle,equations and some rules are mess,discard namely domain closure and,induction in the Fitch system for,first-order logic we include all ten of,the rules of inference for propositional,logic and the ten logical rules for her,Brown logic we have negation,introduction and negation elimination,implication introduction and implication,elimination by conditional introduction,and by conditional elimination and we,have introduction in elimination rules,for conjunctions and disjunctions we,also include the four rules of inference,for quantifiers from her brawn logic we,have universal introduction and,universal elimination and we have,existential introduction and existential,elimination our first new rule of,inference the rule for equality is,called quality introduction Qi according,to this rule we can write down any,equation in which the first term and the,second term are identical for example,without any premises whatsoever we're,going to write down equations like a,equals a F of a equals F of a and f of X,equals f of X our second equality rule,is equality elimination or QE equality,elimination tells the step when we have,an equation and a sentence containing,one or more occurrences of one of the,terms in the equation then we can deduce,a version of sentence in which that term,has been replaced by the other term in,the equation in order to avoid,unintended capture of variables kiwi,requires that the replacement must be,substitutable for the term being,replaced in the sentence is the same,substitutability condition that adorns,the universe elimination rule of,inference note that the equation in the,Equality elimination rule can be used in,either direction,that is an occurrence of tau 1 can be,replaced by tau 2 or an occurrence of,tau 2 can be replaced by tau 1 for,example if we have the equation x equals,B and the sentence hates of X X we can,infer hates of X B we going to defer,hates of BX or we can even infer hates,of BB and that's it for equality to,finish off the proof system we need to,deal with the main closure and induction,well these rules are sound for her Brown,logic they are not sound for first-order,logic simply guaranteeing that a,sentence fee is true of every ground,term does not necessarily mean that it,is true of everything in the universe of,discourse hence these rules no longer,work the fix is simple we just drop the,two rules no domain closure and no,induction of any sort okay that's it to,summarize fitch for first-order logic is,similar to fit for her Brown logic and,that we include our logical rules of,inference and our rules for quantifiers,there are just these two differences we,include two new rules of it for equality,and we eliminate our domain closure and,induction rules okay now let's see we,can say about this proof procedure,remember th

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### Unification-in-first-order-logic | Artificial Intelligence #unification#FirstOrderLogic#AIBasics

hello friends,so after having a look on influencing,first order logic today we will proceed,with unification in a while,so unification so unification is making,of two different logical expression,or two different atomic expression,identical by doing a substitution,now substitution care has to suppose two,expressions given here like this p,of x f of y and p of a and,f of z of b so here we can replace,x by a and f of y by f of z,of b,we can write f of z of b so substitution,variables,are called most general unifier,or mgu so my example,there like two sentence given as a,sentence one a king of x and sentence,substitution possible so we can,replace this x by both,so this is our substitution so we will,see some more examples regarding,substitution,so again though expressions given a p of,x y and p of,a and f of z so yes we can easily,replace,this because x can be replaced by a,and y can be replaced by f of,z now second expression,p of x and f of y,and p of a and f of z so again,x ac replace hojar no problem and f of y,and f of z,again replace ojaga no problem because,function jo same hair,similar now and third expression,x to ac replace no problem,but here the function is h and here the,function is f so,this expression so is though expressions,by unification possible,now let us see some conditions or rules,for unification,so first condition predicate,the second condition is number of,arguments in both,expressions must be identical that means,he suppose,expression though argument say one and,two,and if and here,also though arguments present in one and,two so,unification possible here but suppose,for okay,second expression maintain arguments,given,so we cannot unify this type of,expressions now a third and final,condition here,unification will fail if there are two,similar variables present in the same,expression,so suppose you have to present,similar variable x for x a,he expression unification will,definitely fail,so now in next video we will see,algorithm and implementation of,unification,with some more examples

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### Inference in First Order Logic (FOL) and Unification

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