CmSc 175 Discrete Mathematics
Lesson 04: Valid and Invalid Arguments. Syllogisms
1. Definitions
Definition: An argument is a sequence of statements, ending in a conclusion. All the statements
but the final one (the conclusion) are called premises(or assumptions, hypotheses)
Verbal form of an argument:
(1) If Socrates is a human being then Socrates is mortal.
(2) Socrates is a human being
Therefore
(3) Socrates is mortal
Here (1) and (2) are the assumptions, and (3) is the conclusion.
Abstract (logical) form of an argument - using variables:
(1) If P then Q
(2) P
Therefore
(3) Q
Another way to write the above argument:
P→Q
P
 Q
Definition: A logical argument is valid, if the conclusion is true whenever the assumptions are
true. An argument is invalid if it is not valid.
2. Testing an argument for its validity
Three ways to test an argument for validity:
A. Critical rows
1. Identify the assumptions and the conclusion and assign variables to them.
2. Construct a truth table showing all possible truth values of the assumptions and the
conclusion.
3. Find the critical rows - rows in which all assumptions are true
4. For each critical row determine whether the conclusion is also true.
a. If the conclusion is true in all critical rows, then the argument is valid
b. If there is at least one row where the assumptions are true, but the conclusion is
false, then the argument is invalid
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B. Using tautologies
The argument is true if the conclusion is true whenever the assumptions are true.
This means: If all assumptions are true, then the conclusion is true.
"All assumptions" means the conjunction of all the assumptions.
Thus, let A1, A2, … An be the assumptions, and B - the conclusion.
For the argument to be valid, the statement
If (A1 Λ A2 Λ… Λ An) then B must be a tautology - true for all assignments of values to
its variables, i.e. its column in the truth table must contain only T
i.e.
(A1 Λ A2 Λ… Λ An) → B ≡ T
C. Using contradictions
If the argument is valid, then we have (A1 Λ A2 Λ… Λ An) → B ≡ T
This means that the negation of (A1 Λ A2 Λ… Λ An) → B should be a contradiction containing only F in its truth table
In order to find the negation we have first to represent the conditional statement as a
disjunction and then to apply the laws of De Morgan
(A1 Λ A2 Λ… Λ An) → B ≡ ~( A1 Λ A2 Λ… Λ An) V B ≡
~A1 V ~A2 V …. V ~An V B.
The negation is:
~((A1 Λ A2 Λ… Λ An) → B) ≡ ~(~A1 V ~A2 V …. V ~An V B)
≡ A1 Λ A2 Λ …. Λ An Λ ~B
The argument is valid if A1 Λ A2 Λ …. Λ An Λ ~B ≡ F
There are two ways to show that a logical form is a tautology or a contradiction:
a. by constructing the truth table
b. by logical transformations applying the logical equivalences (logical identities)
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Examples:
1. Consider the argument:
P→Q
P
 Q
Testing its validity:
a. by examining the truth table:
P
Q
P→ Q
---------------------------T
T
T
T
F
F
F
T
T
F
F
T
There is only one critical row - the first one, where both the premises ( P and P→ Q) are true. In
that row the value of Q is true, hence the argument is a valid argument.
b. By showing that the statement 'If all premises then the conclusion" is a tautology:
The premises are P and P→ Q. The statement to be considered is:
(P Λ (P→ Q)) → Q
We shall show that it is a tautology by using the following identity laws:
(1) P→ Q ≡ ~P V Q
(2) (P V Q) V R ≡ P V (Q V R)
commutative laws
(P Λ Q ) Λ R ≡ P Λ (Q Λ R)
(3) (P Λ Q) V R ≡ (P V R) Λ (Q V R)
distributive law
(4) P Λ ~P ≡ F
(5) P V ~P ≡ T
(6) P V F ≡ P
(7) P V T ≡ T
(8) P Λ T ≡ P
(9) P Λ F ≡ F
(10) ~(P Λ Q) ≡ ~P V ~Q De Morgan's Laws
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(P Λ (P→ Q)) → Q
by (1)
≡
~(P Λ (P→ Q)) V Q
by (10)
≡
( ~P V ~(P→ Q) ) V Q
by (1)
≡
( ~P V ~(~P V Q)) V Q
by (10)
≡
(~P V (P Λ ~Q)) V Q
by (3)
≡
((~P V P) Λ (~P V ~Q)) V Q
by (5)
≡
(T Λ (~P V ~Q)) V Q
by (8)
≡
(~P V ~Q) V Q
by (2)
≡
~P V (~Q V Q)
by (5)
≡
~P V T
by(7)
≡
T
c. We can prove the argument also by showing that the negation of the conclusion and the
assumptions are contradictory, i.e. the conjunction of all assumptions and the negation of
the conclusion is a contradiction:
(P Λ (P→ Q)) Λ ~ Q
by (1)
≡
(P Λ (~P V Q)) Λ ~ Q
by (3)
≡
((P Λ ~P) V (P Λ Q)) Λ ~ Q
by (4)
≡
(F V (P Λ Q)) Λ ~ Q
by (6)
≡
(P Λ Q) Λ ~ Q
by (2)
≡
P Λ (Q Λ ~ Q)
by (4)
≡
PΛF
by (9)
≡
F
Crucial fact about a valid argument: the truth of its conclusion follows necessarily from the
logical form alone and the truth of the assumptions
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Thus in the argument
(1) If P then Q
(2) P
Therefore
(3) Q
Q is true whenever (1) and (2) are true, no matter what is the nature of the statements P and Q.
2. Consider the argument
P→Q
Q
 P
We shall show that this argument is invalid by examining the truth tables of the assumptions and
the conclusion. The critical rows are in boldface.
P
Q
P→ Q
---------------------------T
T
T
T
F
F
F
T
T
F
F
here the assumptions are true, however the
conclusion is false
T
Exercise:
Show the validity of the argument:
1. P V Q
2. ~Q
(premise)
(premise)
Therefore P
(conclusion)
a. by using critical rows
b. by contradiction using logical identities
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Solution:
a. by critical rows
conclusion
P
T
T
F
F
Q
T
F
T
F
Premises
PVQ
T
T
T
F
~Q
F
T
F
T
Critical row
b. By contradiction using identities
((P V Q) Λ ~Q ) Λ ~P ≡
((P Λ ~Q ) V (Q Λ ~Q )) Λ ~P ≡
((P Λ ~Q ) V F) Λ ~P ≡
(P Λ ~Q ) Λ ~P ≡
P Λ ~P Λ ~Q ≡ F Λ ~Q ≡ F
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Syllogisms (patterns of arguments, inference rules)
Aristotle
(384 – 322 B.C) was the first to study patterns of arguments, which he called syllogisms.
.
Syllogisms are inference rules, rules to make valid arguments, rules for deductive reasoning.
1. Modus Ponens and Modus Tollens
Modus Ponens (method of affirming)
This is the well known already argument
(1) If P then Q
(2) P
Therefore (3) Q
Modus ponens uses a conditional statement: P → Q, i.e. if P is true, then Q is true.
The second assumption in the argument states that P is true.
Hence we conclude that Q is true.
Modus Tollens (method of denying)
(1) If P then Q
(2) ~Q
Therefore (3) ~P
Modus tollens is based on ~Q → ~P and this is the contrapositive of P → Q.
If ~Q is true, the ~P is true.
The second assumption states that ~Q is true.
Hence we conclude that ~P is true (i.e. P is false)
Examples:
1.
Modus ponens
If today is Monday, tomorrow is Tuesday.
Today is Monday.
Therefore tomorrow is Tuesday.
If it is Sunday we go fishing.
It is Sunday
Therefore we go fishing
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2.
Modus tollens
If today is Monday, tomorrow is Tuesday.
Tomorrow is not Tuesday.
Therefore today is not Monday.
If it is Sunday we go fishing
We do not go fishing
Therefore it is not Sunday
Examples of invalid arguments
Inverse error
If P then Q
~P
Therefore ~Q
If it is Sunday we go fishing
It is not Sunday
We do not go fishing
The argument would be valid if the inverse of the conditional If P then Q had been
used as an assumption. (The inverse of "If P then Q" is "If ~P then ~Q")
Converse error
If P then Q
Q
Therefore P
If it is Sunday we go fishing
We go fishing
It is Sunday
The argument would be valid if the converse of the conditional If P then Q had
been used as an assumption. (The converse of "If P then Q" is "If Q then P")
To show that the argument is invalid we use truth tables:
Let
P = It is Sunday
Q = We go fishing
P
Q
P → Q
---------------T
T
T
T
F
F
F
T
T
F
F
T
The first and the third rows are the critical rows.
In the third row however the conclusion P is false
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3.
Disjunctive Syllogism
(1) P V Q
(2) ~P
Therefore (3) Q
Example:
During the weekend we either go fishing or we play cards
This weekend we did not go fishing
Therefore, this weekend we were playing cards
4.
Hypothetical Syllogism
(1) P → Q
(2) Q → R
Therefore (3) P → R
Example:
If we win the game we will get much money.
If we have money we will go on a trip to China.
Therefore, if we win the game we will go on a trip to China
In the truth table below the critical rows are in boldface.
P
Q
R
P→Q Q→R P→R
------------------------------------------------------------------T
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
T
T
T
F
F
F
T
F
F
F
F
F
T
T
F
F
T
F
T
F
T
T
T
T
T
F
T
T
T
T
T
T
The value of the conclusion in the critical rows is T
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Inference rules
A → B, A , therefore B
Modus ponens
A → B, ~B, therefore ~A
Modus tollens
A V B, ~A, therefore B
Disjunctive syllogism
A V B, ~B, therefore A
A → B, B → C, therefore A → C
Hypothetical syllogism
A, therefore A V B
Disjunctive addition
A, B, therefore A Λ B
Conjunctive addition
A Λ B, therefore A
A Λ B, therefore B
Conjunctive simplification
A V B, A → R, B → R, therefore R
Dilemma, proof by division into cases
A → B, ~A → B, therefore B
~P → F, therefore P
Law of contradiction
A  B, therefore A → B, B → A
Equivalence elimination
A → B, B → A, therefore A  B
Equivalence introduction
A, ~A, therefore B
Inconsistency law
Fallacies
a.
Using incorrect syllogism, incorrect argument.
Examples are: converse error, inverse error.
If I read a book, I need my glasses
I am not reading a book
Therefore I don't need my glasses
Where is the error?
b.
The argument is correct, however the premises are false.
If you are a college student, you don't need to study.
You are a college student
Therefore you don't need to study.
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Summary
1.
Arguments
An argument is a sequence of statements. All but the final one are called premises
the last one is the conclusion.
2.
Syllogisms
Syllogisms are arguments with two premises only.
Two important syllogisms based on the conditional statement and its contrapositive are
Modus Ponens and Modus Tollens.
3.
Valid and invalid arguments
An argument is a valid argument if the conclusion is true whenever the premises are true.
Otherwise the argument is invalid.
4.
Test for validity


We can show that an argument is valid by:
a.
examining the critical rows in the truth table of the premises
and the conclusion. The conclusion must be true in all rows where all the
premises are true.
b.
showing that the expression "If premises then conclusion" is
a tautology:
 by constructing its truth table
 by transforming to "T"
c.
showing that the conjunction of all premises and the negation
of the conclusion is a contradiction:
 by constructing its truth table
 by transforming to "F"
We can show that an argument is invalid by examining the critical rows
in the truth table of the premises and the conclusion.
The conclusion must be false in at least one critical row.
Note, that we cannot prove that an argument is invalid by a tautology or a
contradiction, because in some critical rows the conclusion may be true, in other it
may be false.
Fallacies
Fallacies are either invalid arguments, or valid arguments based on false premises.
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