Welcome back in the last lecture we discussed
in detail about one of the important methods
which is considered as a decision procedure
method which is called as a truth table method
truth table method is considered to be the
most simplistic method especially in this
course introduction to logic it is simplistic
in the sense that as long as the number of
propositional variables are less in number
that means two or three for example if you
have two prepositional variables you have
four entries in your truth table.
And if you have three propositional variables
like P Q R etcetera representing some kind
of prepositions we have eight entries in the
truth table but the problem is, is that it
is very difficult for us to manage for example
when you have more than five or six variables
propositional variables if you have six variables
that means two to the power of n entries will
be there in the truth table that means two
to the power of six may be 64entries.
You need to inspect to find out whether group
of statements are consistent or whether a
particular kind of conclusion follows from
the premises that means the validity etcetera
for that you know you need to check all the64
entries know that means you need to inspect
each and row each and every row of your truth
table meticulously that means 64 rows are
there and all the rows you need to inspect.
I mean those rows in which whether or not
you have true premises in a false conclusion
if you have a true premises and a false conclusion
then the argument is obviously considered
to be in value so instead of inspecting these
64 rows which will be difficult for us and
there are some other better methods one such
method which we will be discussing in this
class that is semantic tab locks method.
So this is also called as analytic tab locks
method or it is also called as tree method
then one of the same so this tree method is
a very useful kind of method which is originated
in the works of l addition bit b the in the
year 1955 Beth lived from 1980 to 1982 1964
is considered to be a Dutch philosopher and
in the history of logic it seems that this
method has originated in the works of Beth
later z Raymond's bouillon has formulated
in his book first order logic his own trees.
And all which is a little bit simpler than
what Beth has proposed in his analytic tab
Locke's method so then simultaneously around
the same year in Tikal also developed independently
the same kind of method it seems that somehow
tike seems to have proposed this method around
the same year but there is no evidence that
whether or not Raymond's bouillon has borrowed
something from in tike.
So there is a controversial debate in who,
who has actually formulated this method first
so that is not of interest to us but what
is of interest to us is to understand this
particular kind of method especially in understanding
in understanding the validity of a given formulas
are when two groups of statements are consistent
to each other or one can even show whether
a given well-formed formula is a tautology
or a contradiction or a contingent statement
using this tabular method.
And you can also show when two logical formula
to well form formulas are logically equivalent
to each other using the same kind of method
that is semantics the flux method so it has
these applications in atomic automated theorem
proving and it is also applications in the
logic of programs etc and all which we will
not go into the details of these things but
we will try to introduce what we what exactly
this method is all about we will introduce.
This method and then we will try to show with
some examples that given group of statements
are consistent to you whether or not they
are consistent to each third or when a conclusion
follows from the premises that means the cause
that is the validity of a given argument etcetera
so what is a semantic trick before we begin
it is also a kind of constructive method so
what we will be doing is given a well-formed
formula we will be constructing a corresponding
tree diagram for this particular kind of thing
usually trees will have trunks.
And you have leaves except I know branches
etc so the same way we have we usually here
in this case we have upside-down kind of tree
usually trees will be down in the knobs up
will have branches except why not but here
we have some kind of upside-down kind of tree
which you will find it here for each and every
formula you will be constructing a corresponding
tree and then we will try to evaluate whether
the fall falling formula is a tautology etc.
So in a semantic tree is considered to be
a device for displaying all the valuations
on which the formula or set of formulas are
going to be true so one of the basic an important
the essence of this method is this that you
know we will be constructing some kind of
counterexample so the essence of this one
is like this it consists of finding some kind
of counter examples where what is considered
to be a counter example.
Suppose if an argument is considered to be
invalid especially when your premises are
true and the conclusion is false so if you
can construct one kind of one particular kind
of counter example in which your premises
are true and the conclusion is false then
obviously the argument is invalid so that
means you have we are said to have constructed
a counter example so the basic idea of this
method is this that an inference is considered
to be valid if.
And only if there exists there exists a counterexample
otherwise the inference is going to be valid
if and only if there are no counter examples
so that is there is no situation in which
the premises hold and the conclusion is false
so this also involves some kind of rule-based
construction which we are going to talk about
in a while from now and using those rules
we will construct trees and then we are going
to show that.
If there are no counter examples then obviously
the formula is going to be valid otherwise
there are any counter examples we could construct
a counter example then obviously the argument
is considered to be invalid so each step of
the construction is given in account of some
kind of tree like structure which is also
called as a table so usually this table closes
when there is a conflicting information computing
information is since then suppose if we have
a formula X.
And not X then obviously the branch closed
because you have a conflicting information
suppose if a or some information like it is
raining and simultaneously you say that it
is no training then that is a conflicting
kind of information which one to believe you
will be in some kind of dilemma so it is in
your in conflict so in that case the branch
closes a literal and it is negation appears
in a branch then obviously the tree closes
that particular branch closes.
So no counterexamples can be constructed for
it if the branch is not open so if the other
branch is closed that means it implies that
there is no counter examples exist so now
this tree method is based on some kind of
rules so now what we will be doing in another
20 to 25 minutes is this that we will be talking
about these rules and then we will construct
this tree diagrams for various kinds of well-formed
formulas then we are going to show that when
a given well form formula is a tautology contradiction
or contingent statement.
And second we will talk about when a given
well form formula is a tautology and third
we will talk about when two groups of statements
are inconsistent to each other and fourth
we will talk about when two groups of statements
are or two well-formed formulas are considered
to be logically equivalent to each other so
all these things which we will be trying to
talk about in terms of this particular kind
of semantic tab locks method. And then in
the process we will also be trying to talk
about some of the important strategies that
will be following while adopting this particular
kind of method.
So the rules are like this first to be in
we have propositional variables etc P QR etc
so these are all propositional variables.
And then we have these symbols one which is
always true is represented in this way usually
with the symbol T and then this is represented
as God one is always something which is always
false is represented in this way and then
we have some other things like parenthesis
exit lying on and apart from that you have
this logical connectives negation arc and
implies and if and only if and then the other
symbol which we will be using is this one
that means the when the branch closes we will
put this mark cross mark.
So that means the branch process so now what
we are trying to do simply is, is that so
we will be ascending some kind of truth values
to these propositional variables that means
we are interpreting this formulas so now there
are some kind of rules which we need to understand
before applying this semantic tables method
so to begin with so there are something called
root and then nose so you need to understand
this thing little bit late we will talk about
this thing it is a bit later.
So first we will talk about some kind of rules
with which you know you can say whether a
given formula is valid or not these are the
construction three construction routes so
suppose if you come across a simple formula
like this negation of P then you simply write
it like this only the construction of this
? P is same as this one so now if you have
a formula like this p RQ the construction
three for this one is same now for a compound
formula P and Q then the construction tree
will be like this.
So usually a tree will be like this so these
are all branches and this is considered to
be the root in so this is a formula that we
are trying to begin with and now these formula
is reduced it is reduced into some kind of
atomic prepositions at all as you will see
in all these rules the things which are there
at the nodes are considered to be only atomic
sentences at the PRQ or may be negation of
that one etcetera.
So whenever you have a formula P and Q you
just write it like this it is a trunk it is
an upside-down kind of tree usually the tree
will be like this now you have to reverse
it a little bit and then you will see these
things so now P implies Q so the definition
of P implies Q is ? P or Q so that means it
is not P or Q so the branch suggests that
there is a distinction so that is either ? P
or Q so this is exactly in alliance with the
semantics that we have talked about in the
last few classes.
So that means P R Q is going to be false only
when both P R Q are false in all other cases
it is going to be true in the same way P and
Q is going to be true only when P Q's are
true in all other cases it is going to be
false so that is the semantics of propositional
logic in the same way P implies Q is going
to be false only when P is T and Q is false
in all of the cases it is going to be true
so based on that kind of information we are
just trying to come up with some kind of constructive
method.
And then what we are trying to do is for simple
formulas like this we are trying to construct
trees tree diagrams are easily a picture says
thousand words so given your formula we are
trying to construct trees like this so now
this is called as a branch and this is called
as a trunk of a tree etcetera so now the only
logical connective which is left now here
is this one so we will write it here if and
only if Q.
So this is either PQ is the case R 0 B 0 Q
you can write 0B 0 Q here itself P and Q can
shift it to the other side it does not make
any big difference it is one of the same so
these are the rules which we have for each
and every logical connective for naught this
is the thing for our the tree appears to be
like this for P and Q it appears to be like
this so these are considered to be a rules
so now in this a rules.
So there are some rules which are considered
to be branching rules that means wherever
you find a branch this is consider to be a
branching rule and wherever you do not find
the branching kind of thing it is called as
non branching rule 0 so why we are talking
about branching and non-branching rules because
so while adopting this particular kind of
technique or method so there are some kind
of strategies that one will be following so
the one, one of the important strategies is
that always apply non-branching rules first.
So once you exhaust with the non-branching
rules you enter into branching rules so now
so these are this is non branching rule it
is not leading into any branch so non-branching
rule so now all these things are branching
kind of rules so given suppose if you are
supposed to apply this method you have to
ensure that first you apply the non branching
rule and then apply all these rules so now
this list is called as a rules usually it
is considered to be positive kind of rules
and all.
So now we will be writing rules so beta rules
are exactly the negations of these things
so now if you come across a formula like this
that is negation of negation of P it is it
is not the case that it is not the case that
it is raining that means it is raining so
now if you have a formula like this you simply
substituted with this particular kind of formula
P so now using demerger’s laws it's quite
simple so now negation of P and P are Q if
we push this negation inside then it will
become negation of P and the negation of this
function will become conjunction.
So that is why we need to write it in this
format so now as you clearly see this is enough
this is a formula and then once you apply
this rules and all at the end you will find
only atomic propositions what is an atomic
proposition an atomic proposition is a one
which it which cannot be further reduced into
any other kind of proposition P R Q scan be
reduced into PE q except I naught but PQ R
etcetera they are propositional variables
they are the most simplistic kind of sentences
which cannot be further reduced into other
thing.
So that is why they are called as atomic sentences
so now this is the rule for this one negation
of P and in the same way negation of P and
Q using the de Morgan's law it leads to a
branch it is negation you push it inside it
becomes negation of P and negation of conjunction
will become disjunction that is why this Junction
will always have a branch so this is the form
that we have so now negation of P ?Q is simply
P and ? Q because P P ?Q is ? P or Q
And negation of ? P R Q is ? P that is P and
? Q so now negation of P ? Q is a branch again
each and ? Q and ? Q ? P so these are the
only things that we have these are the rules
which we will be applying for judging whether
a given well form formula is a tautology it
is a contradiction or contingent statement
on the one hand or when two groups of statements
are consistent to each other except triangle.
So now using these rules we will be trying
to talk about whether are not given formula
is valid or invalid etc so now to start with
we start with some simple examples.
So we want to see whether this particular
kind of argument is valid or not we start
with some simple examples in the beginning
and then we will move on to some other things
usually when you write to P ? Q Q ? R P ? talks
and usually you say that it is a valid argument
by virtue of transitive property obviously
you know P ? Q ? R and obviously P in place
R.
So now let us do not we do not talk about
the valid argument obviously valid arguments
which we know so instead of this what we do
is rightly change this thing P ? Q you instead
of this thing R ? Q and then be in place apart
let us assume that this is a weather or not
this P ? R follows from these two things are
not so now these two are considered to be
usually premises and this is usually called
as conclusion so now how do we know that P
? R follows from these two statements.
Now P ? Q and R ? Q whether it is it follows
then it is valid otherwise it is invalid so
how do we check that this particular kind
of formula that is P inverse R follows from
these two things there are various methods
one method which we have already discussed
that is the truth table method and since there
are three variables eight interests will be
therein the truth table so that is also a
little bit easy to do but.
So what we are trying to do is we are trying
to see whether team placer follows from these
are not so now the very essence of symmetric
tab locks method is this that we are trying
to construct a counterexample if you fail
to construct a counter example that means
the version of the conclusion is going to
be valid so what we will be doing is we will
begin with the same thing we will restore
these things P ? Q and R ? Q so these are
premises and we have a conclusion in place
R.
So now what we will be doing is we will be
negative conclusion so the idea here is, is
that negation of the conclusion leads to the
closure of branch that means is unsatisfiable
then obviously your negation of conclusion
is going to be false that means all the branch
closes it is going to be false that means
X has to be a total is if X is a tautology
obviously the formula is going to be valid
because all tautologies are considered to
be valid formulas.
So now we are trying to check whether P ? R
follows from these two or not it is denial
of conclusion that means we denied the conclusion
all and we are trying to construct a counter
example if you fail in the process then obviously
this is the actual conclusion which V that
follows so now so now these are the compound
formulas in all so now we will be applying
a ß rules exactly.
So now one of the important strategies is
that we need to apply non-branching rules
first so which formula you take into consideration
is commander non-branching roots non-branching
rules are this is the one and then of course
this is another so these are the two non-branching
rules that you are finding because there is
no branch here the only trunk, trunk of a
tree so now this looks like this one knot
of P ? Q so instead of Q you are so.
Now this reduces to e and ? R so this is three
simplification three is this formula and we
simplified it then this leads to this one
so we applied ß rule here beta rule is talking
about the negation of this formula so now
this is the one so now once you apply this
particular kind of road you need to see whether
there is any conflicting information in your
branch in your tree so right now we do not
have since we have checked this formula then
we put this tick mark.
So that you know you will not use it again
and again otherwise you will confuse and we
will use it again so once it is got exhausted
then you would put a check mark here let me
see in it how to use this one again so now
these are the formulas which are left so now
we use this particular kind of thing P ? Q
that means you apply this tree you are constructing
a tree for this one means you have to use
this.
So it is ? P or Q so you draw a diagram like
this under this you put ? P and Q you need
to pause a second and you need to see whether
there is any completing information in your
in your branch so now this branch is going
like this all the way till here it is a trunk
and this is a branch now, now this is going
like this so one is going like this another
one other branch is like this so now you have
P here.
And you or ? P here so that means there is
a conflicting information so the branch closes
here itself that means it stops there itself
there is no question of any construction of
any counter example possible here in this
case since you have conflicting information
there is a contradictory information it closes
so now this branch is still open in all so
now since this is a formula that we checked
out already.
So that is why we had put tick mark here so
now what is the formula which is left here
is this one R ? Q so now again apply the same
rule alpha rule and it becomes ? R Q so now
this is also over now we checked our checked
all the formulas you know and then obviously
at the end of all the branches and all you
have only atomic propositions not a few etc
not so now you have to inspect this particular
kind of branch.
So now this branch is open and even this branch
is also open that mean seven after the denial
of the conclusion it could still construct
a counterexample that satisfies this formula
so now that means it is possible for the premises
to be true and the conclusion to be false
that is the reason why we have at least one
or two open branches so now we can study these
open branches and we can talk about this particular
kind of thing.
So now 0 R means R is false I means R is true
that means R has to be false and Q is true
Q is T and then of course ? are we have taken
into consideration and P so that means assigning
these values are false Q T and P T this satisfies
this particular kind of formula this particular
kind of formula that means P ? Q R implies
Q ? P? R this is an assignment which said
which makes this formula true in our open
branch means and all. So this is going to
make these formulas true.
So this is one model which we have that means
so whenever you give our F Q T and P T that
satisfies this particular kind f former what
is satisfying this formula that means if you
have true premises and a false conclusion
that means this is a counter example for this
one and another counter example is there here
whenever Q takes value P that means Q T are
false ? R is true means R is false and then
PEP so this is another counter example enough.
So what is that we have achieved with this
particular kind of thing denial of the conclusion
does not lead to branch closure that means
you already said to have constructed a counter
example whenever you are said to have constructed
a counter example then obviously the argument
is invalid so it is possible that your premises
are true but it your conclusion is false how
the conclusion is going to be false it is
going to be false in two different ways these
two different ways are considered to be the
two different open branches.
So especially when R is a of Q is tphd that
is going to make this thing true and when
Q is TR is equal to F P = T then also your
premises are true and the conclusion is false
even if at least one branch is open by denying
the conclusion and obviously the argument
is considered to be invalid so this is based
on some kind of falsifiability kind of instead
of looking for things which are true we'll
be looking for things which are false that
means you are looking for always so you are
looking for a counterexample.
So it is like you have 100 for example in
a bag which consists of hundred Tomatoes even
if you have one rotten tomato and all you
will say that not all tomatoes are can start
to being good in this basket one tomato will
spoil the entire thing so this is based on
some kind of falsifiability a kind of method
as long as it is not falsifiable the formula
is going to be true it is like as long as
you do not find a white crow it is going to
be or you are going to accept it.
As statement that you know you are going to
accept the statement that all crows are going
to are going to be black so now this is the
way to show that this argument is invalid
so what about saw the arguments which are
valid will take up some kind of arguments
which are obviously valid so this is the one
P ?Q and P thank you this is obviously the
modus ponens rule which is considered to be
obviously valid so how do we show that this
Q follows from these two things P ? Q and
P except you will store these premises.
As it is these are premises and then what
you do here is you negate the conclusion so
now you need to write here denial of conclusion
so now once you deny the conclusion we need
to check whether it leads to the closure of
a branch or not so now we need to apply is
alpha or ß roots depending upon this thing
there is no branching rule non branching rule
which we can apply here so any rule which
you can use.
So now we need to apply this one this is already
an atomic sentence nothing needs to be done
so now P?Q is simply ? P or Q so now you have
P here and you are not be here this branch
closes and ? Q and Q here it is a conflicting
information this branch also closes so what
did we get negation of the conclusion leads
to the closure of a branch that means it is
false usually represented as this one.
But is a symbol that we use so that means
X has to be obviously true since exist say
X is true means X is the tautology X is a
tautology means it is a valid formula all
tautologies in prepositional logics are obviously
all valid formulas that is reason why logicians
will be insisting on tautologies there is
a there are special kinds of statements which
are obviously are always considered to be
true they are like god-given kind of truths
which are always.
And there are some other groups of statements
a group of formulas which are obviously false
it is like two plus two is equal to five which
is obviously false and two plus two is equal
four is always true so now this is the way
which you can show the validity of a given
formula for validity what you need to do is
you do have to deny the conclusion and then
you need to see whether the branch closes
on all the branches close them.
If at least one branch is open that means
the open branch is considered to be it has
to be analyzed in detail so that means open
branch indicates such that there is a counter
example what is the counter example you have
true premises and a false conclusion so that
is going to be the case if we can construct
one counter example that means it is obviously
going to be any invalid argument.
So what else one can do with the help with
the help of this particular semantic tables
method so now we are trying to see whether
a given group of statements are consistent
to each other or not so for that what you
will be doing is like this let us consider
an example and we will try to see whether
this particular kind of group of statements
are consistent to each other be peace the
first statement and the second statement is
not K some statements which does not matter
whatever you take into consideration.
And three not be B or B implies not forget
about what this C's V is X cetera and all
they all correspond to some kind of statement
it can be it is raining or it may be something
like there is some kind of statements which
it expresses enough so now given English language
sentences we have converted into the language
of propositional logic and now we are going
to see whether these are consistent to each
other or not that means they simultaneously
they can be true or not that is what you are
trying to explore.
So now here is obviously there is no conclusion
you never have to look into the negation of
the conclusion and seethe branch closes except
I naught so we will keep it as it is at all
so now we are trying to find out a model in
which trying to interpret is formal this preposition
variables that means are sending with some
kind of values to it so that all these things
will turn out to be true enough that means
you need to inspect the open branches here
so now so as usual we need to apply non-branching
rules.
And all here there is no scope for any non-branching
rolling on so you can open up any formula
and then construct tree for these things so
now let us consider the second one you can
choose the first one also so this is not KRG
so that's why we applied this particular kind
of rule PRQ means either P is the case of
Q is the case so now this is over you tick
mark this one now the second formula you can
take any formula into consideration so now
we will open up the first one it is either
C B and I have be.
And the same kind of information you need
to pass it to the other branch also so C and
then B and not so now this formula is also
open so now each time you apply this aß roof
etcetera you need to see whether any branch
closes or not so this branch remains open
so now this you can write it in this way B
and ? B which can be written in that form
of trunk and all that is B and not B so this
is also B and ? B.
So now you have a conflicting kind of information
P and not be this branch closes and here also
is ranch closes now on the branches that are
open are these two so now whatever has remained
the formula needs to be applied to this Oh
needs to be written just below this open branches
so now what is left now this is the one so
now this can be written as this is X and this
is y and now X implies Y is ? X or Y that
means not of ? B and ? G remains the same
thing.
And now this branch is these two branches
are already closed in order to worry much
so now this is ? B are the same information
you put it in all the open branches that you
will see here so now this is not now you further
simplify it using de Morgan's laws etcetera
or maybe you can apply any one of this rules
there then not of this one you push the negation
inside this becomes B and.
Then not of negation of disjunction will become
conjunction that is why we are writing one
below the other so this is like this formula
and then this one and not G remains same and
then this becomes B and ? B not that B is
B and not of distinction is connection that
is why we are writing just below this one
and it is ? B so now all the things are exhausted
so now we need to see this kind of thing your
be here and not be here is branch closes and
C 0 K in all this match remains open and this
obviously closes.
Because of conflicting information B and not
B and then this branch also remains open,
open branches are the ones which you need
to inspect so these are the branches that
makes these formulas to that means each open
branch will serve as a model some kind of
in each branch is considered to be satisfiable
you know for example when you give when you
assign value ? K T that means K is false and
C C as T and.
Then not G as Steamiest these false under
this particular kind of assignment of truth
values these three statements are going to
be satisfiable that means that is going to
be true together in the same way another open
branch that you will find it here in this
tree diagram is this when you give valuation
this is one kind of interpretation under which
these formulas are going to be true together
or satisfiable the other one is the one which
you need to see not G = P that means Z is
false the first one.
And then C is true if you are saying some
kind of true value T to see the only values
that you can ascend to see it as a TR FN o
and G is ? G not G we already have this information
and on so you have this particular sign of
information that also satisfies this particular
kind of formula so these are the things so
interpretations which satisfies this formula
that means these formulas are going to be
consistent especially when you interpret in
this particular kind of way.
So list out all the formulas one after another
construct a tree diagram and you find any
open branch an open branch corresponds to
the satisfiability distancing so that means
the formulas can be true together especially
when you have when you assign some kind of
values like this so that means these three
formulas are simultaneously said to be consistent
to each other so this is another way of showing
that these formulas are going to be consistent
now suppose if you have used truth table method
so now the number of variables are 1 2 3 4
4 and 4 variables are set that means there
are 16 entries you need to inspect.
So that means each row leads to true or not
so all the rows under the final column or
whatever value that is the value that you
are going to get is T those rows you need
to inspect so instead of doing all these things
this is the most simplistic kind of method
is easy to use based on some simple kind of
rules the Melons rules will help us so with
the help of which we can easily see that you
know here are these particular kinds of things
which satisfy this formula that means under.
This particular kind of ins interpretation
is going to be true all the formulas are going
to be true so now we can also show that the
given formulas are considered to be inconsistent
to each of them so obviously these three formulas
are said to be consistent example if you have
formula P and Q and I have written deliberately
choose in this example just to show that these
two formulas are inconsistent to each other
so now first you state all these things.
So we store these things with some numbers
and then apply non-branching rules first P
and Q is a non branching rule so that is why
you apply this first and then you apply branching
rules at all how was both are branching rules
one being when you have to worry much so any
rule which you can apply so this is not P
and not of design chain is conjunction so
this will become not so now you will see here
Angus negation is this.
So we start the premises father this restored
these formulas and then constructed three
if all the branches closes then that is said
to be unsatisfiable or it also called as inconsistent
so P and Q is inconsistent with ? P and Q
that is obviously the case because P and Q
is exactly ? p RQ not of p RQ is a country
contradictory to P and Q obviously to statement
which are in contradictory to each other or
Wisconsin' inconsistent or unsatisfied.
So this is the way in which you can show that
a given formula formulas are consistent or
inconsistent to each other or satisfy Palermo
so now there are other things which you can
do with the help of this semantic tab locks
method so that is you can show whether given
formulas are logically identical to each other
and later we will see whether this is going
to be statement is going to ß otology at
that also one can do with the help of semantic
time flux method so now we are trying to see
whether.
So these are the two formulas now we are trying
to see whether they are logically identical
to each other like it here so now this one
can do it in several ways and all again using
truth table method if the truth values of
this one exactly matches with the truth table
truth values of this one under the main the
main logical connective then these two are
said to be logically identical to each other
for example P and Q and Q and you know P D
Q and P and Q and Q.
So you know two variables satisfy right two
T's and to here alternate to T and if and
this formula is going to be false going to
be true only in this case in all other cases
it becomes false exactly Q and P also the
same thing so this formula Q and P is going
to be true when both P Q are true in all other
cases obviously it becomes false so now according
to the truth table I mean this exactly matches
with the other one I know for example when
P and Q takes value t even Q and P also takes
the value T.
So this matches with this, this matches with
this and this matches with this one in that
sense P and Q and Q and P are logically identical
to each of them so that is one way of for
showing it the other way of showing it is
using some other kind of thing so that is
this one suppose X is a formula and Y is another
formula so when do we say that these two are
lies logically identical to each other these
two are logically identical to each other
especially when X if and only if Y please
consider to be a tautology.
If we can show that X if and only if Y is
a tautology then x and y are said to be identical
to each other this is logically equal in to
each other so now what we are trying to see
here is this thing so now we are trying to
see whether this formula is logically occurring
to this or not so now for that what we'll
be doing is we will be putting this particular
kind of sign and then we are trying to see
whether this formula is going to be a tautology
or not so now this is the formula which is
there now.
So now the idea of symmetric tab blocks method
is simple that you know you start with the
counter example first that means you negate
the conclusion negation of X is this one not
of the entire thing which you need to write
so P in place R the first formula and then
the second formula is being placed heart if
the negation of this formula leads to the
closure of branch and all that means negation
of X is false that means X else to be T if
X is T and obviously this is considered to
be valid enough.
So now if X if and only if Y is a tautology
then obviously these two are said to be logically
identical to each of them so now this you
treat it as X and this is y so now you need
to apply negation of P?Q so now you need to
apply this particular kind of Rho beta rule
so this is this is X P and Q ? R the first
one that is the X part and then ? P? Q is
our little bit big can on here so that is
the first one X and ? Y.
And then ? P and Q ? R it is not x and y y
is same as brackets needs to be put properly
now so now we need to further simplify this
thing then it will be like this we need to
expand this branch and on so once you apply
branching rule here so that is this becomes
? P and Q and then this becomes path so now
this is over now this is the thing so now
you further expand this thing it becomes ? P
? Q and this as it is.
So now you apply a rule again here ? P?Q ? R
so this is so here what you need to do here
is better to use non branching rule first
so here your play first rule for this one
so this is B and not of Q in place afar so
now we apply for this one first so always
the strategy is this that first you open up
a bra for Miller which leads to non-branching
kind of for formula now so how do you expand
this one rather this one so now this becomes
this so now this further reduces to Q and
not because ? Q ? R is Q ?.
So now you apply this one here so this is
? P and Q and then so now you see here are
and ? R is at this branch closest so now this
further expands to ? P and ? Q so now you
have P here and all the way down here you
are not P this branch closes and you are not
Q here and Q here this branch also closes
so the left hand side all the branches closes
so now you have to see the right hand side
of this so now first you open a formula which
leads to some kind of non-branching non branching
rule is the one which you need to apply first.
So this is X and this is y e so that is why
not offer this one leaves two non-branching
kind of route so that is P and Q and then
not so this is the formula which we have used
yeah this is the one ? P ? Q is P and ? Q
so not Q instead of not kill how are here
so now this is over so now we apply branching
rules it does not matter so now this is ? P
and Q ? R so now P and Q this P and Q can
be written as P and ? P and Q one after another
this is the rule which we use P and Q means
you can write P Q as a trunk this reflects
that trance.
So now you have P here and you are not free
here this closes now you expand it further
this becomes ? Q and R now you are not are
here and are here this closes and Q here this
is hidden here Q here and ? Q this also closes
so now none of the branches remains open so
that leads to negation of the given well-formed
formula leads to the closure of further branch
that means what we are showed is this particular
kind of T0 of X is false that means X is a
tautology.
So that means these two P and Q ? Rand P ? Q
? R are said to be logically identical to
each other so we have established that actually
this is the thing which we need to use mistake
here so by imply Le x so in this way you can
show that to given logical formulas well form
formulas are said to be logically equivalent
to each other or the other way of showing
it is, is that how do we show that a given
L form formula is a tautology especially when
you deny the well form formula that means
negate the well form formula if all the branches
closes.
Then that means negation of the given formula
X is false that is a contradiction that means
X has to be a tautology so these are some
of the things one can do with the help of
semantic tablets method so you can show that
a given formula is a tautology you can show
that two groups of statements are consistent
to each other are satisfiable.
Or you can even show that to given logical
formulas or consist equivalent to each other
etcetera all these things are considered to
be some kind of decision procedure methods.
So the advantage of this simulator blocks
method is that it conducts a direct search
for models the models in a sense that whenever
you find a open branch consider me a model
that means under these particular kind of
assignments a given formula is going to be
true for example P and Q is going to be true
especially when both are both P and Q are
going to be true in all other cases is going
to be false.
So that means when you are saying truth values
P P Q T and that will serve as they come some
kind of model all the woven parts of the tree
that you are seeing in this all this example
corresponding to satisfiability of conjunctions
of formulas at the node suppose if for all
the branches closes then it means unsatisfiability
so now traditional approaches such as constructing
truth table etc it can take two to the power
of n steps for any given n for example n stands
for some number of variables.
That exist in a given formula if n is too
big in all realm is large the truth table
method is difficult for us to handle because
the number of revolution variables are large
if it is more than 6 it becomes 64entries
of the entries you need to inspect to find
out whether a is there a given well form formula
is valid or not.
So these are some of the definitions that
will be funny we have just discussed in a
very informal way about this particular kind
of method there are some definitions which
will be following while constructing the truth
this method semantic tab locks method.
So the first definition is about the path
a path of a tree in any stage of construction
wherever in the left hand side you will see
a tree diagram for this particular kind of
thing so a path of a tree is a complete column
of formulas from top to the bottom of the
tree or not for example in this case so this
is considered to be one path and this is one
path and is going like this and ending like
this, this is path number one and there is
one more path exactly the same thing.
And it is going like this and it ends here
another path to that means it starts all the
way from the root and ends with some kind
of atomic propositions so that is the register
to be the path it is already we have discussed
in somewhere other but you know we are discussing
in and explicitly what do we mean by path
of a trick so this is what is considered to
be path definition of a finished path that
means the path is there is no way in which
you can progress further.
That means then there is a conflicting information
there is no way in which you can go further
why we are not able to go further because
whenever you have an inconsistent information
you can derive anything so that is the reason
why we want to avoid this conflicting kind
of information so this is the way in which
one can show that suppose if you have a conflicting
information like P.
And not to begin with you have a conflicting
information like this it is raining and it
is not raining so now you just state these
things like this Omega P and not P so now
fourth one not P this is one thing cliff occasion
and one simplification you will get this one
so now you can safely add another kind of
proposition without disturbing the truth value
of this one if suppose you assume that not
P is already true then whatever you add after
this one Q is always going to be true enough.
Because of the semantics is like this that
and this is obviously true irrespective of
whether it is Q is T Q is false this is going
to be true only that means P Q and Q so obviously
so this, this becomes false only in this case
in all other cases it becomes TF if in all
other case is going to be true so whatever
you add after this one this formula you retain
the same thing in all it is also going to
be true.
So that is why you can say if we add any kind
of strange kind of prepositions suppose if
we P stands for it is not raining you can
safely add another kind of preposition that
fix, fix flies you know that is a strange
thing about this particular kind so now what
we are trying to show is that whenever you
have comforting information you can be there
anything so this is what is called as law
of addition this is a truth preserving kind
of law which I which you commonly see in logic
what are the important laws of logic.
So that means you help y and you can easily
safely add PR q without disturbing the truth
value of this one so now quickly what you
can see is 3 & 4sorry 2 and for this thank
you seller is I am will lead to Q so that
is the reason why we are not going further
and then whenever you have a conflicting information
we stop but in the same way.
So we have proved to is the case Q is any
kind of strange kind of preposition which,
which is which comes as a consequence of this
computing kind of information exactly in the
same way you follow some other kind of steps
you can even prove not Q also or maybe some
other kind of propositional so that means
if we start with the contradictions inconsistencies
etc you can derive anything so that is the
reason why whenever you have come across a
closed Rancher now you will stop there itself.
So a path is considerably finished if it is
so and if the only unchecked formula it contains
are only propositional variables and that
means there is no way in which you can go
beyond it or the negation of the propositional
variables so that no more rules can apply
on this one no more all for beta rules can
apply on that one so that means a tree is
said to be closed or finished if all of its
paths are closed.
And an open path is a path that has no that
has been marked with X and a closed path the
mark with the tick mark and the closed path
is marked with some kind of cross okay in
this class we just introduced the semantic
tab locks method as one of the important decision
procedure method and we have seen with some
examples that how the semantic tab locks method
can be used to decide you know whether a given
well form formula is a tautology.
Suppose if you can show that given well form
formula is a tautology obviously the formula
is going to be valid or you can even show
that when two groups of statements are consistent
to each of them that is what you can show
and then you can also show with the help of
semantic dialogues method is a constructive
method that when to given logical formulas
or prepositional well form formulas are going
to be logically equivalent to each other so
what we will be doing in the next class is
that we will be applying the semantic tables
method particularly in solving some of the
important logical puzzles as well as you know.
Once we translate the English language sentence
in appropriately into the language of propositional
logic then we can see whether a conclusion
follows from the premises are not again by
using the semantics tablets method the semantics
tables method has an edge over the truth table
method especially in the sense that when the
number of variables are more than four or
five seminary tablets method is easy to use.
So that is why it has an edge over the truth
table method so it depends upon our convenience
which method we will be using in the next
class we will be seeing some of the important
logical puzzles interesting logical puzzles
that can be solved with the help of semantic
gap flux method.
