Statement Logic: part II
Sandhya Sundaresan
December 18, 2014
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Two weeks ago, we looked at truth tables for the
following logical connectives in Statement Logic:
, ^, and _.
We will now look at the truth-tables for the
remaining two connectives: conditional (Ñ) and
bi-conditional (Ø).
Conditional (Ñ)
Biconditional (Ø)
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Conditional (Ñ):
Statement
Logic: part II
Sandhya
Sundaresan
+ The complex statement (P Ñ Q) should be read as:
if P (is true), then Q (is true).
+ Naturally, this means that if P has the value True
and Q has the value True, then P Ñ Q will also be
True.
+ Similarly, if P is True and Q is False, then (P Ñ Q)
seems obviously False.
+ But strange things happen when only P is False or
when both P and Q are False.
The conditional
and
bi-conditional
Conditional (Ñ)
Biconditional (Ø)
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Truth Table for conditional (Ñ):
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Conditional (Ñ)
(1)
Biconditional (Ø)
Truth table for Ñ:
P
0
1
0
1
Q
0
0
1
1
(P Ñ Q)
1
0
1
1
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
+ The Truth Table in (1) shows that: if P alone is False
(Row 3) or both P and Q are False (Row 1), then (P
Ñ Q) is True!
Obviously, this is very different from our understanding of
‘if...then’ in natural language:
If I utter a sentence like: “If Mary is at the party,
then John’s there too”, we might very well
understand this to mean that if Mary’s not at the
party, then John’s not at the party either.
On the other hand, if neither Mary nor John is at the
party, then I would be inclined to say that I have no
way to evaluate the truth of that sentence (so it’s
neither True nor False).
Sandhya
Sundaresan
The conditional
and
bi-conditional
Conditional (Ñ)
Biconditional (Ø)
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ But remember Statement Logic is not natural
language.
+ One of the main differences is that, as a binary
system, we don’t have the option of saying “Neither
True nor False”.
+ Thus in cases where the complex statement (P Ñ Q)
is not necessarily False, we will say that it is True.
The conditional
and
bi-conditional
Conditional (Ñ)
Biconditional (Ø)
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Biconditional (Ø):
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Conditional (Ñ)
Biconditional (Ø)
+ The biconditional connective (or Ø) means: “if and
only if” (also written as “iff”).
+ The biconditional thus encodes a stricter relationship
than the simple conditional.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Truth Table for biconditional (Ø):
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Conditional (Ñ)
(2)
Biconditional (Ø)
Truth table for Ø:
P
0
1
0
1
Q
0
0
1
1
(P Ø Q)
1
0
0
1
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ We can understand the Truth Table in (2) if we
realize that (P Ø Q) is simply: ((P Ñ Q) ^ (Q Ñ
P)).
+ I.e. ‘P if and only if Q’ means: if P (is true), then Q
(is true), and if Q (is true), then P (is true).
+ In other words, if we compute the truth table for ((P
Ñ Q) ^ (Q Ñ P)), we will get exactly the same
result as we did for (P Ø Q).
The conditional
and
bi-conditional
Conditional (Ñ)
Biconditional (Ø)
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Truth tables for (more) complex statements:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
+ So far, we’ve looked at truth tables for expressions
that consist of maximally two atomic statements.
+ But a truth table provides a systematic method to
compute the truth value of any expression in
statement logic: so we can construct truth tables for
expressions that are much more complex than these.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ The number of rows in a truth table for a complex
expression are determined by the requirement that
we compute all possible truth values for every atomic
statement in that complex expression.
+ Since we are dealing with binary logic, every atomic
statement in a complex expression will have two
possible truth values: True and False.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
+ Thus, if an expression X has n atomic statements,
then the truth table forX will have 2n rows/lines.
Logical
equivalence and
logical
consequence
+ In general, we compute the truth values of the
innermost expressions before moving on to those of
the outer expressions.
Laws of
Statement
Logic
A working example:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Let us compute the truth table of a complex expression
step-by-step to see how it’s done:
(3)
ppp ^ qq Ñ p pp _ rqqq
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Algorithm:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
1
Construct columns for the atomic statements: p, q, r.
2
Construct columns for the innermost expressions:
pp ^ qq and pp _ rq.
3
4
Construct a new column for pnegpp _ rqq, reversing
the values for pp _ rq.
Then construct the truth table for the entire complex
statement in a separate column, applying the rules
for the condition (Ñ) between pp ^ qq and p pp _ rqq.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
+ Tip: We have 3 atomic statements: p, q and r, so
the truth table will have: 23 rows = 8 rows.
Computing
complex truth
tables
+ It’s a good idea to always compute this before you
start writing out the truth table, so you don’t forget
any rows.
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
Table : Truth Table for: ppp ^ qq Ñ p pp _ rqqq
p
1
1
1
1
0
0
0
0
q
1
1
0
0
1
1
0
0
r
1
0
1
0
1
0
1
0
pp ^ qq
1
1
0
0
0
0
0
0
pp _ rq
1
1
1
1
1
0
1
0
p pp _ rqq
0
0
0
0
0
1
0
1
The conditional
and
bi-conditional
Computing
truth
tables
ppp ^ qq Ñ p pp _ rqqqcomplex
0
0
1
1
1
1
1
1
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Exercise 0:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
+ Prove that (P Ø Q) = ((P Ñ Q) ^ (Q Ñ P)), by
constructing a truth table for ((P Ñ Q) ^ (Q Ñ P)).
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
These look at three different types of logical relations. We
will look at these in turn.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Logical tautology:
Statement
Logic: part II
Sandhya
Sundaresan
+ A (complex) statement is called a logical tautology iff
the final column in its truth table contains only the
values: 1/True, regardless of what the truth values of
its atomic statements are.
+ A tautological statement is true simply because of
the meaning of the logical connective(s) in it: this is
why the meanings of the individual atomic
statements in it don’t matter.
+ Another way to express this would be to say that a
tautological statement is always true.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
(4)
Example of a logical tautology: p Ñ p
p
1
0
p
1
0
pp Ñ pq
1
1
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Logical contradiction:
Statement
Logic: part II
Sandhya
Sundaresan
A logical contradiction is the opposite of a logical
tautology, in a sense:
+ A (complex) statement is called a logical
contradiction iff the final column of its truth table
only contains the values: 0{False, regardless of what
the truth values of its atomic statements are.
+ Just like with a tautology, a contradictory statement
is false simply because of the meaning of the logical
connective(s) in it: this is why the meanings of the
individual atomic statements in it don’t matter.
+ A logically contradictory statement is always false.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
(5)
Example of a logical contradiction: pp ^ p pqq
p
1
0
p pq
0
1
pp ^ p pqq
0
0
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ An important property of logical tautologies and
logical contradictions is that the nature of the atomic
statements in them simply doesn’t matter.
The conditional
and
bi-conditional
So, in Ex. (4), we could have replaced the atomic
statement p with q to yield pq Ñ qq, and it would
remain a logical tautology.
Logical
tautologies,
-contradictions,
-contingencies
Similary, in Ex. (5), we could have replaced p with x
to yield px ^ p xqq and it would remain a logical
contradiction.
Computing
complex truth
tables
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Logical contingency:
Statement
Logic: part II
Sandhya
Sundaresan
+ All other statements, with both 1{True and 0{False
in the final column of their truth table are called
logical contingencies.
+ The idea is that the truth of these statements is
contingent (i.e./ dependent) on the truth of the
atomic statement(s) contained in them.
+ Most of the examples (involving both complex and
atomic statements) we’ve seen so far have been
logical contingencies.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
You try it:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Exercise 1: Among the three complex expressions below,
one is a logical tautology, one is a logical contradiction,
and one is a logical contingency. Say which is which by
drawing truth tables for each of the expressions:
i. pp Ñ pq Ñ pqq
ii. pp _ pq
iii. p pp _ p pqqq
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
+ If the statement in question is very complex, drawing
an entire truth table to determine whether it is a
logical tautology or not can be long and cumbersome.
+ A quicker way to test this is by the process of logical
reasoning called: reductio ad absurdum.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
The reasoning works like this:
1
2
3
4
You simply assume that the statement is in fact not a
logical tautology. I.e. you assume that one of its
possible truth values is 0{False.
You then reason “backwards” from this assumption
to compute the possible values of the atomic
statements in this complex expression.
If, based on this assumption, you run into a
contradiction, then your assumption was wrong and
the statement is indeed a tautology.
If, on the other hand, you don’t run into a
contradiction, then your assumption was correct after
all, and the complex statement is not a logical
tautology.
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Working example
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Let’s look at an example to see how this works.
(6)
Task: Find out if the complex expression
pp Ñ pq Ñ pqq is a logical tautology or not:
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Steps:
Sandhya
Sundaresan
(i) Assume that pp Ñ pq Ñ pqq is false. So we write 0
under the main or “highest” connective:
(p
Ñ
0
(q
Ñ
p))
(ii) Reasoning backwards from this assumption, we know
that the only way the whole expression can be
0{False is if the antecedent is 1{True and the
consequent is 0{False, so we have:
(p
1
Ñ
0
(q
Ñ
0
p))
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Steps (Contd.):
(iii) Now we give the second p (the one after the second
“Ñ”) in the statement the value 1 (since the first p is
1, the second one must be, as well):
(p
1
Ñ
0
(q
Ñ
0
p))
1
(iv) But now we run into a problem. On the one hand,
we have just said that pq Ñ pq = 0. But on the other
hand, if p = 1, then pq Ñ pq can never be 0.
Contradiction!
(v) Our assumption that pp Ñ pq Ñ pqq = 0 led us to a
contradiction. Ergo, our assumption must be false
and pp Ñ pq Ñ pqq must never have a 0 as its truth
value – i.e. pp Ñ pq Ñ pqq must be a logical
tautology.
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
You try it:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
Exercise 2: Show, using the reasoning under reductio ad
absurdum that the complex statement pp _ p pqq is a
logical tautology.
Logical
tautologies,
-contradictions,
-contingencies
Logical Tautology
Logical
Contradiction
Logical Contingency
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
These are relations that are built on the ideas of logical
tautology, contradiction and contingency.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Logical equivalence:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
+ If a biconditional statement is a logical tautology,
then the two constituent statements on either side of
the biconditional arrow are logically equivalent.
+ To denote logical equivalence between two arbitrary
expressions P and Q, we write: P ô Q.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
E.g. if pp Ø qq is a logical tautology (for specific
values of p and q), then p and q are logically
equivalent. I.e. p ô q.
if two statements are logically equivalent, this means
they have the same truth value regardless of the
truth values of the atomic statements in them.
Of course, this property is exactly what ensures that
the biconditional connecting these statements is
always true (i.e. is a tautology).
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Consider the truth table for the logical tautology:
pp pp _ qqq Ø pp pq ^ p qqqq:
Sandhya
Sundaresan
The conditional
and
bi-conditional
p
1
1
0
0
q
1
0
1
0
pp _ qq
1
1
1
0
p pp _ qqq
0
0
0
1
p pq
0
0
1
1
p qq
0
1
0
1
pp pq ^ p qqq
0
0
0
1
In the table above, the statements on either side of the
“Ø" are logically equivalent. I.e.
p pp _ qqq ô pp pq ^ p qqq.
Notice that, regardless of the truth values of the atomic
statements p and q, p pp _ qqq and pp pq ^ p qqq always
have the same truth values: compare columns 4 and 7.
pp ppComputing
_ qqq Ø pp pq ^
complex truth
tables
1
1
1
1
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
+ One of the useful things about logically equivalent
expressions is that you can easily substitute an
expression for another one that is logically equivalent
to it.
+ This is because substitution of logically equivalent
expressions preserves truth values: i.e. it preserves
both truth and falsity.
+ I.e. since two logically equivalent statements have
exactly the same truth values in every row of the
truth table, one can easily substitute for another in a
larger expression.
+ This principle is sometimes referred to as the Rule of
Substitution.
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Thus, in the expression pp _ qq, we may easily replace
p with its logically equivalent expression pp ^ pq to
yield: ppp ^ pq _ qq.
Since p ô pp ^ pq, they will always have the same
truth values.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ Of course, we can only substitute one logically
equivalent expression for another if they are syntactic
constituents (i.e. sub-formulas) in a larger expression.
E.g. we cannot convert pp ^ pq Ñ rqq into
pq ^ pp Ñ rqq even though pp ^ qq ô pq ^ pq.
This is because: pp ^ qq is not a subformula or
constituent in the larger expression: pp ^ pq Ñ rqq
(as the bracketing shows).
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Logical consequence:
Statement
Logic: part II
Sandhya
Sundaresan
+ If a conditional statement is a logical tautology, we
say that the consequent (statement on the right side
of the “Ñ”S) is a logical consequence of the
antecedent (statement on the left side of the “Ñ”)
+ Alternatively, we say that the antecedent logically
implies the consequent.
+ I.e. If the expression pp Ñ qq is a tautology, q is a
logical consequent of p, or p logically implies q.
+ Formally, we write this as: p ñ q.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
+ In contrast to logical equivalence, the relation of
logical consequence preserves truth but doesn’t
necessarily preserve falsity.
+ I.e. if p ñ q, then if p is True, q will also be True,
but if p is False, q doesn’t necessarily also have to be
False.
+ The relation of logical consequence is quite important
because it is the basis on which we construct valid
logical arguments.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Truth table involving logical consequents:
p
1
1
0
0
q
1
0
1
0
pp Ñ qq
1
0
1
1
ppp Ñ qq ^ pq
1
0
0
0
pppp Ñ qq ^ pq Ñ qq
1
1
1
1
The rightmost column of the truth table above shows
that the conditional statement: pppp Ñ qq ^ pq Ñ qq
is always true, i.e. a tautology.
Thus, q – on the right-side of “Ñ” is a logical
consequent of ppp Ñ qq ^ pq. I.e. ppp Ñ qq ^ pq ñ q.
Notice that the truth values of these two constituents
is not always the same: compare columns 2 and 4
above.
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
+ Precisely because the truth values of the two
expressions related by logical consequence doesn’t
always have to be the same, we cannot substitute one
expression for the other.
+ I.e. if P ñ Q, we cannot substitute Q fo P (or
vice-versa) in a larger formula.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Statement
Logic: part II
Sandhya
Sundaresan
E.g. given that pp ^ p pqq ñ p we cannot conclude
that: ppp ^ p pqq Ñ qq ñ pp Ñ qq, by replacing p
with pp ^ p pqq.
In fact, if p is True and q is False, pp Ñ qq will be
True, but ppp ^ p pqq Ñ q will be False.
This again is a contrast from the logical equivalence
relation where, as we saw, an sub-formula in a
complex expression may be replaced by another one
that is logically equivalent to it.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Logical Equivalence
Logical Consequence
Laws of
Statement
Logic
Laws of statement logic:
Statement
Logic: part II
Sandhya
Sundaresan
+ Now we will look at some logical equivalences that
are useful to remember, simpy because they are so
common.
+ Some logical equivalences on this list will be
derivable from others.
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
+ “T ” stands for “logical tautology” and “F ” stands for
“logical contradiction”.
Logical
equivalence and
logical
consequence
+ Notice the similarities to the laws we came up for
set-relationships.
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Idempotent Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP _ Pq ô P.
b. pP ^ Pq ô P
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Associative Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. ppP _ Qq _ Rq ô pP _ pQ _ Rqq
b. ppP ^ Qq ^ Rq ô pP ^ pQ ^ Rqq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Commutative Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP _ Qq ô pQ _ Pq
b. pP ^ Qq ô pQ ^ Pq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Distributive Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP _ pQ ^ Rqq ô ppP _ Qq ^ pP _ Rqq
b. pP ^ pQ _ Rqq ô ppP ^ Qq _ pP ^ Rqq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Identity Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
a. pP _ F q ô P
b. pP _ T q ô T
c. pP ^ F q ô F
d. pP ^ T q ô P
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Complement Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP _ p Pqq ô T
b. pP ^ p Pqq ô F
c. p p Pqq ô P (double negation)
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
DeMorgan’s Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. p pP _ Qqq ô pp Pq ^ p Qqq
b. p pP ^ Qqq ô pp Pq _ p Qqq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Conditional Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP Ñ Qq ô pp Pq _ Qq
b. pP Ñ Qq ô pp Qq Ñ p Pqq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
BiConditional Laws:
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Computing
complex truth
tables
a. pP Ø Qq ô ppP Ñ Qq ^ pQ Ñ Pqq
b. pP Ø Qq ô ppp Pq ^ p Qqq _ pP ^ Qqq
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Statement
Logic: part II
Sandhya
Sundaresan
As mentioned earlier, some logical equivalences above can
be derived from others.
E.g. for two arbitrary statements P and Q:
1
pP Ñ Qq
2
pp Pq _ Qq (Conditional)
3
pQ _ p Pqq (Commutative)
4
pp p Qqq _ p Pqq (Complementary)
5
pp Qq Ñ p Pqq (Conditional)
The conditional
and
bi-conditional
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws
Statement
Logic: part II
Sandhya
Sundaresan
The conditional
and
bi-conditional
Exercise 3: Verify, using a truth table, the logical equivalence given under the Conditional Law: pP Ñ Qq ô
pp Pq _ Qq for two arbitrary atomic statements x and
y. I.e. verify that: px Ñ yq ô pp xq _ yq using a truth
table.
Computing
complex truth
tables
Logical
tautologies,
-contradictions,
-contingencies
Logical
equivalence and
logical
consequence
Laws of
Statement
Logic
Idempotent Laws
Associative Laws
Commutative Laws
Distributive Laws
Identity Laws
Complement Laws
DeMorgan’s Laws
Conditional Laws
BiConditional Laws