In today’s lecture we will start discussing
fundamentals of logic. Now in logic we will
be dealing with statements rather than numbers.
Now a particular type of statements are called
propositions.
Let us define propositions nor formally 
any declarative sentence 
to which it is meaningful 
to assign one and only one 
truth value 
true or false is said to be a proposition.
Now we have to be careful that each and every
sentence is not a proposition. The key feature
of proposition is that it has to be either
true or false. And if a proposition is not
true and then it is false, and if it not false
then it is true.
Let us look at one example of a proposition.
We have written down two propositions, so
first proposition is denoted by P, and the
second proposition is denoted by Q. and the
proposition P states that there are seven
days in a week. Now this is of course a sentence
and this is true. So we know that this cannot
be true and false at the same time, and we
know that it is meaningful to say that it
is true.
Thus, P is a proposition, the next sentence
is Q which states that a week has more number
of days than a month. Now this sentence is
obviously false, but is a proposition because
it is a declarative sentence which has a truth
value which in this case is false. Let us
look at another sentence we denote by R which
is an interrogation what is your name? Now
this sentence cannot have a truth value, because
that is meaningless.
And hence, this sentence is not a proposition.
Therefore, we see that all propositions are
sentences, but all sentences are not propositions.
Another thing that we notice here is hat we
can write propositions, we can denote propositions
by symbols like P, Q and R. We generalize
this fact and introduce propositional variables.
A variable which can take propositions as
values 
are called propositional variables.
We will denote the propositional variables
by small letters P, Q, R and so on. Now if
we have some basic propositions we can connect
these propositions by using the so called
logical connectives and build up compound
propositions. The basic connectives are five
in number.We will note down these five basic
connectives are 5 in number.
We will note down he this 5 basic connectives
the first connective is called conjunction
it is also called and denoted by a ^ the 2nd
connective is called disjunction 
it is also called or and denoted by v the
3rd connective is not or negation 
and it is denoted by either an over line or
a ~ or symbol like this prefixed before a
propositional variable 4 conditional 
this is also known as if – then – it is
denoted by an arrow like this or sometimes
an arrow like this the 5th one is bi conditional
this is also known as if and only if or just
if with 2 f ‘S it is denoted by a both sided
arrow or a both sided arrow like this.
Given now one by one check the effect of these
connectives the prepositions one conjunction
suppose p and q are 2 propositions 
the conjunction 
of p, q is the statement p and q if we denote
this compound statement or this compound proposition
as p ^ q now since which is a proposition
we must know definitely when it is true and
when it is false p ^ q is true 
only when both p and q are true.
Otherwise p ^ q is false 
now we can translate this things to a table
which is called a true table and which is
very useful in understanding these connectives
and more complicated compound propositions
if we write the propositional variables p,
q and also like the statement p and q the
possible values of p, q are FF that is false,
false TF, FT and TT.
P conjunction q or p and q will have the truth
values FF and T here T means true and F means
false this table is called a truth table and
this specifics the truth values of the compound
proposition p and q next we move on disjunction.
Again we take 2 propositions p and q and p
or q is called disjunction of p, q or simply
p disjunction q which is denoted by p v q
now this statement is true only when at least
one of p or q is true, so we write now we
go to that truth table of p or q the third
connective is called negation this is the
invalid apposition that is it involves only
one the position and variable suppose p is
a preposition 
negation of p or cp not of p denoted by either
go for line or ~p or this is a preposition
which is true when p is false and false when
p is true, the corresponding truth table will
look like this where T false and true and
negation of p true when p is false.
And false when p is true 
next we have conditional now this is called
in conditional language as if then, so if
something, then something else 
the preposition p implies q denoted by p arrow
q is true 
if q is true whenever p is true 
that truth table of p ? q is like this, now
when p is true and q is true that means that
p ? q is true now when p is false then I cannot
prove that p ? q is false because if we have
to that p implies q is false we have to show
one is stands where p is true but q is false.
Since we cannot prove that p implies q is
false if we have the truth value true so in
case p is false in both these cases p ? q
is true whereas if p is true 
and q is false that means that p ? q is not
true because p ? q forces q to be true when
p is true therefore we will write false over
here and of course when p is true and q is
true p implies this the last conjunction last
connective is called by conditional.
Now if p and q are prepositions p if and only
if q denoted by p is called the bio conditional
and p if an only q is essentially congestion
of p implying q and q implies p, now if we
consider the truth table of bio conditional
we will have p q taking all the possible values
and p implying q is true f and true q implying
p is true f q and q and therefore the bio
conditional which is congestion of this two
prepositions is true f f and t.
Now next we will take preposition are variables
and use this logical connectives to build
up compound statements or compound prepositions.
Now let us see examples of that, suppose we
have prepositional variables pqr consider
a compound preposition f p q r p and q are
not of r, now these types of expressions will
be called prepositional functions. We can
find out the truth table of these prepositional
functions for example for the one that I have
written just now we can build up the truth
table in this way we write the prepositional
variables then we start writing the terms
p and q and r0.
Now all the possibility list down all the
possible truth values of pqr which are f f
f, f f t, f t f, f t t, t f f, t f t, t t
f, t t t, now first column at the right hand
side is p and q which is true if an only if
both p and q are true therefore here it will
have f the next row also f then f f f f t
t and r0 is t f t f t f t f. now in the last
column we will calculate the function f pqr
which is equal to p and q are 0 of r this
is t f t f t f t and t.
Once we have discussed the prepositional functions
which are also called compound statement or
compound prepositions then we introduced the
idea of equivalence of prepositional functions.
Two proportional are functions 
are logically equivalent if a have the same
truth table to be example we start with two
proportional variables p and q 
and consider two proportional functions pq
and not of p or p 
so p ? q so we have already seen that it 
is first row but here not if not p is t so
the so r will be t this one is false because
lot of p is going to be false and q is false
and lastly it is going to be p therefore we
see that p ? q.
And both p and q have truth table therefore
these two statements are equivalent we have
two statements as p ? q equivalent to three
pair of deadlines equivalent not of p and
q for that we observe another point let me
consider the bi conditional between these
two equivalent statements that is p ? q by
conditional not p or q we will see that it
is always true 
now introduce one more notion that is of equivalent.
If there is a proportional function which
is true with respective of the values of the
original variables involved in it then it
is called a tautology from what we have discussed
it is not difficult to see if two proportional
functions are equivalent then if it for another
function by connecting these two proportional
functions by conditional and the resulting
function is going to be the tautology 
now if we have a proportional function which
is never true and it is called the continuation
that usual proportional which are sometimes
true sometimes false are called contingents
by this we come to an end of this lecture
thank you.
