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proof by contradictionnow for forward chaining and backward chaining both have drawbacks but another approach is to think about proving theorems by
backward chainingin generally given that we are only interested in constructing the path whether we can set our initial state to be the theorem
forward chainingnow we have suppose we have a set of axioms that we know are true statements about the world whether we set these to each be an
chains of inferencenow we have to look at how to get an agent to prove a given theorem using various search strategies thus we have noted in previous
existential introductionnow if we have any sentence as a and variable v that does not occur in a so then for any ground term g such occurs in a than
existential elimination now we have a sentence a is with an existentially quantified variable v so then just for every constant symbol k that it
universal eliminationhere for any sentence there is a containing a universally quantified variable v just for any ground term g so we can substitute
ground substitutionhere the act of performing an instantiation is a function like there is only one possible outcome means we can write it as a
first-order inference ruleshere now we have a clear definition of a first-order model is that we can define soundness for first-order inference rules
propositional truth tablesthere xt is a substitution that replaces all occurances of variable x with a term representing an object t as forallx a
quantifiers and variables - propositional modelthere is one question is arrives that what do sentences containing variables mean in other way of
predicates in propositional modelthe predicates take a number of arguments in which for now we assume are ground terms and represent a relationship
terms in propositional modelthere in first-order logic allows us to talking about properties of objects that the first job for our model delta theta
propositional modelhence a propositional model was simply an assignments of truth values to propositions in distinguish a first-order model is a pair
first-order modelshere if we proposed first-order logic as a good knowledge representation language than propositional logic is just because there is
unit resolutionby assuming that we knew the sentence as tony blair is prime minister or may the moon is made of blue cheese is true or we later found
or-introduction thus if we know about one thing is true and also we know that a sentence when there thing is in a disjunction is true here if we
and-introductionin generally english says that if we know that a lot of things are true so we know that the conjunction of all of them is true then
and-elimination rulein generally english says that if you know that lots of things are all true so you know like any one of them is also true because
implication connective - modus ponens rulewe notice that this is a trivial example so it highlights how we use truth tables as the first line is the
propositional inference rulespropositional inference rules equivalence rules are mostly useful because of the vice-versa aspect that means like we
eequivalences rulesthis conveys a meaning that is actually much simpler so than you would think on first inspection hence we can justify this by
equivalencesin this following miscellaneous equivalence rules are often useful during rewriting sessions so there the first two allow us to
double negation all parents are forever correcting their children for the find of double negatives there we have to be very alert with them in
associativity of connectives here brackets are very important in order to tell us where to perform calculations in arithmetic and logic by using