Chapter 2

Propositional Logic: Valid Arguments

and Fallacies

2.1 Arguments
Definition.
An argument is a compound proposition of the form

(p1 ∧ p2 ∧ . . . ∧ pn ) → q.

The propositions p1 , p2 , . . . , pn are the premises of the argument, and q is the conclusion.
Arguments can be written in propositional form, as above, or in column or standard
form:
p1
p2
..
.
pn
∴q

The premises of an argument are intended to act as reasons to establish the validity or accept-
ability of the conclusion. We will make this statement more precise later on.

EXAMPLE 1. Explain why the following set of propositions is an argument.

If General Antonio Luna is a national hero, then he died at the hands of the Americans in 1899.
General Antonio Luna is a national hero.
Therefore, General Luna died at the hands of the Americans in 1899.

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Solution. The set of propositions is an argument. Its premises are the propositions “If General
Antonio Luna is a national hero, then he died at the hands of the Americans in 1899,” and “General
Antonio Luna is a national hero.” The conclusion, which is flagged by the word “therefore”, is the
proposition “General Luna died at the hands of the Americans in 1899.”
Is the argument valid? At the end of the lesson, you will be able to answer these questions.

EXAMPLE 2. Write the following argument presented in the introduction in propositional

form and in standard form.

If there is limited freshwater supply, then we should conserve water.

There is limited freshwater supply.
Therefore, we should conserve water.

Solution. The premises of this argument are

p1 : If there is limited freshwater supply, then we should conserve water.
p2 : There is limited freshwater supply.
and its conclusion is
q : We should conserve water.
In symbols, we can write the whole argument in propositional form

(p1 ∧ p2 ) → q,

and in standard form

p1
p2
∴q

Valid Arguments
An argument is valid if whenever ALL the premises are true, then the conclusion MUST
also be true. If an argument is not valid, it is called a fallacy.

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 How to prove that an argument is valid
Often, there are three ways to prove the validity of the argument (p1 ∧ p2 ∧ . . . ∧ on ) → q.

1. Show that (p1 ∧ p2 ∧ . . . ∧ on ) → q is a tautology.

2. Show that whenever ALL the premises p1 , p2 , . . . , pn are true, then q is also true.

3. Use Rules of Inferences (these are rules that can be proved valid using any of the two
methods above).

EXAMPLE 3. Is the following argument valid?

p→q If my alarm sounds, then I will wake up.
p My alarm sounded.
∴q Therefore, I woke up.
Solution. .
The propositional form of this argument is ((p → q) ∧ p) → q.
Method 1: Show that ((p → q) ∧ p) → q is a tautology. We can do this using a truth table:
p q p→q (p → q) ∧ p ((p → q) ∧ p) → q
T T T T T
T F F F T
F T T F T
F F T F T
The last column of the truth table shows that ((p → q) ∧ p) → q.
Method 2: Show that whenever both premises p → q and p are true, then the conclusion q is
false.
Suppose that the premises p → q and p are both true. The truth table below shows that the
only row where both premises p → q and p are true is the first row. In this row, q is also true.
Premise Premise Conclusion
p q p→q p q
T T T T T
T F F T F
F T T F T
F F T F F

Hence, YES it is logically impossible for the premises to be true and the conclusion to be false.
Therefore the argument is valid.
Method 3: Use Rules of Inference. This will be discussed later.

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 How to prove that an argument is NOT valid
Often, there are three ways to prove the validity of the argument (p1 ∧ p2 ∧ . . . ∧ on ) → q.

1. Show that (p1 ∧ p2 ∧ . . . ∧ on ) → q is NOT a tautology.

2. Use a truth table to show that it is possible for p1 , p2 , . . . , pn to be ALL true, but the
conclusion q is false.

3. Give a specific set of truth values wherein all premises are true, but the conclusion is
false.

EXAMPLE 4. Show that the following argument is NOT valid.

p→q If my alarm sounds, then I will wake up.

q I woke up.
∴p Therefore, my alarm sounded.

Solution. .
The propositional form of this argument is ((p → q) ∧ q) → p.
Method 1: Show that ((p → q) ∧ q) → p. is a NOT tautology. We can do this using a truth
table:
p q p→q (p → q) ∧ q ((p → q) ∧ q) → p
T T T T T
T F F F T
F T T T F
F F T F T
The last column of the truth table shows that ((p → q) ∧ q) → p is NOT a tautology. Therefore,
the argument is not valid.
Method 2: Show that it is possible for both premises p → q and q to be true, but the conclusion
p is false.
The truth table below shows that the only rows where both premises p → q and q are true are
the first and third rows. There are two possibilities for both premises to be true. However, only
one of these possibilities guarantees that the conclusion p is also true. Therefore, the argument is
NOT valid.
Premise Premise Conclusion
p q p→q q p
T T T T T
T F F F T
F T T T F
F F T F F

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 Method 3: Give a specific set of truth values wherein all premises are true, but the conclusion
is false.
To prove that the argument is invalid, then the conclusion has to be false. Therefore, p must
be false.
Both premises must also be true. One premise is q. So q must be true.
Notice that if p is false and q is true, then the other premise p → q is also true.
To summarize, we have:
Premise Premise Conclusion
p q p→q q p
F T T T F

This may actually be viewed as a short version of Method 2.

Note on Examples 3 and 4

We just proved symbolically that the argument in Example 3 is valid while the argument
in Example 4 is invalid. But what does this mean in practical terms?
In Example 3, the given was that if my alarm sounds, then I will wake up. Also my alarm
sounded. Therefore, it is logical to conclude that I woke up.
In Example 4, the given was that if my alarm sounds, then I will wake up. Also, I woke
up. Based on those two statements alone, does this automatically mean that my alarm
sounded? No! Because there could be a reason why I woke up; perhaps I had a bad dream.
Therefore, it is not logically valid to conclude that my alarm sounded based on the given
statements alone.

2.2 Rules of Inference

In the previous section, you learned about three methods to prove that an argument is valid (see
page 3). The first two methods involve a truth table. The third method is based on rules of
inference, which is the focus of this section.
In Example 3, we already proved that the argument
p→q
p
∴q

or ((p → q) ∧ p) → q is valid. This argument is called Modus Ponens.

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 Modus Ponens
How do we know that Modus Ponens is valid? Because we already proved its validity
through truth tables, as seen in Example 3.
If we encounter any structurally similar argument, then we can say that the argument is
valid due to Modus Ponens.

EXAMPLE 5. Prove that the following argument is valid.

s∨ ∼ r
r
∴s

Solution. .
Proposition Reason
1. s∨ ∼ r Premise (Given)
2. ∼r∨s Commutative
3. r→s Switcheroo
4. r Premise (Given)
5. s Modus Ponens, 3,4

Modus Ponens is just one of several Rules of Inference. The following table shows some other
rules. All of these can be proved to be valid using any of the two methods described
on page 3.

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Rules of Inference
Let p, q, and r be propositions.
Propositional Form Standard Form
p∧q
Rule of Simplification (p ∧ q) → p
∴p
p
Rule of Addition p → (p ∨ q)
∴p∨q
p
Rule of Conjunction (p ∧ q) → (p ∧ q) q
∴p∧q
p→q
Modus Ponens ((p → q) ∧ p) → q p
∴q
p→q
Modus Tollens ((p → q) ∧ (∼ q)) →∼ p ∼q
∴∼ p
p→q
Law of Syllogism ((p → q) ∧ (q → r)) → (p → r) q→r
∴p→r
p∨q
Rule of Disjunctive Syllogism ((p ∨ q) ∧ (∼ p)) → q ∼p
∴q
(∼ p) → φ
Rule of Contradiction ((∼ p) → φ) → p
∴p
p→r
Rule of Proof by Cases ((p → r) ∧ (q → r)) → ((p ∨ q) → r) q→r
∴ (p ∨ q) → r

It is very difficult to memorize these rules!

Try to make sense out of the rules; do not just memorize them.
For example, in Simplification, it is already given that p ∧ q is true. Therefore by definition
of ∧, it already means that p must also be true. It is impossible that p is false and yet p ∧ q
is true.
Another example, in Modus Tollens, it says that p → q and ∼ q are true (given). Then of
course, it must be the case that ∼ p. Why? Because if p is true, then q must be true as
well (due to p → q). But it is given that ∼ q is true. Therefore ∼ p must be true.

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 EXAMPLE 6. Prove that the Rule of Simplification is a valid argument.
Solution. We need to establish that (p ∧ q) → p is a tautology. Indeed, this is so, as shown by the
following table.

p q p∧q (p ∧ q) → p
T T T T
T F F T
F T F T
F F F T

Valid arguments are not necessarily true!

If an argument is valid, it does NOT mean that the conclusions are necessarily true. It
simply means that the conclusion logically follows from the premises.

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NAME:
Verify the following rules taken from the Table of Rules of Inference.
Show the following logical equivalences (a) using Method 1 and (b) using Method 2, as discussed
in Example 3 on page 3.
1. Modus Tollens: ((p → q) ∧ (∼ q)) →∼ p
Method 1: Method 2:

. .
2. Law of Syllogism: ((p → q) ∧ (q → r)) → (p → r)
Method 1: Method 2:

. .
3. Rule of Proof by Cases: ((p → r) ∧ (q → r)) → ((p ∨ q) → r)
Method 1: Method 2:

. .

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NAME:

Show that the following arguments are valid by first converting them either to propositional or
standard form, and then identifying the Rule of Inference that makes the argument valid.

1. If Antonio and Jose are friends, then they are Facebook friends.
Antonio and Jose are not Facebook friends.
Therefore, they are not friends.

2. Antonio Luna and Jose Rizal like Nelly Boustead.

Therefore, Antonio Luna likes Nelly Boustead.

3. Antonio Luna is a scientist.

Therefore, either Antonio Luna or Jose Rizal is a scientist.

4. If the Spaniards imprison Antonio Luna, then he not join the revolution.
If Antonio Luna does not join the revolution, then he will go to Belgium to study the art of
war.
Therefore, if the Spaniards imprison Antonio Luna, then he will go to Belgium to study the
art of war.

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2.3 Proving the Validity of Arguments
Using the Rules of Inference is one way to prove that an argument is valid. However, there are
times (such as Example 5) when a statement is not yet structurally similar to one of the Rules of
Inference, so some form of simplification must still be done.
More examples will be provided in this section.
EXAMPLE 7. Prove that the following argument is valid.

p∧q
r → (∼ p)
∴∼ r

Solution. Strategy: Before you proceed to a solution, you should first try to get an idea how to
reach the goal (to prove ∼ r). If you look at the second premise, it seems that we should use Modus
Tollens. However, to apply Modus Tollens here, we need to know that p is true. Is it?
We are also given that p ∧ q is true. Then of course, p must also be true (by Simplification).
We can now formalize the proof as follows.
.
Proposition Reason
1. r → (∼ p) Premise (Given)
2. p∧q Premise
3. p Simplification, 2
4. ∼ (∼ p) Double negation, 3
5. ∼r Modus Tollens, 3,4

Writing your proof

Notice that in the previous example, all reasons were either (1) a premise, or (2) a rule
of inference, based on a previous statement, whose number is explicitly indicated. Please
follow this system, to help ensure that all statements you write are logically justified.

EXAMPLE 8. Prove that the following argument is valid.

p → (q ∨ r)
p
∼r
∴q

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Solution. Strategy: Look at the goal (to prove q). The only place where you can see q is in the
first premise. But how can we “pull out” this q? For this, we need p and apply Modus Ponens to
obtain q ∨ r.
Now how do we get q alone? For this, we need the third premise ∼ r then apply Disjunctive
Syllogism.
The formalized proof is given below.
Proposition Reason
1. p → (q ∨ r) Premise
2. p Premise
3. q∨r Modus Ponens, 1,2
4. ∼r Premise
5. q Disjunctive Syllogism, 3,4

Writing a Proof Requires Thinking!

Although the final proof can be organized in a two-column table, you need to do some
thinking beforehand. And you may get stuck. That is normal. You probably need a lot of
SCRATCH PAPER before you can finalize your proof. So do not get discouraged easily!
Think, think, think!

EXAMPLE 9. Prove that the following argument is valid.

a→q
b→q
∴ (a ∨ b) → q

Solution. The formalized proof is shown below.

Proposition Reason
1. a→q Premise
2. b→q Premise
3. ∼a∨q Switcheroo, 1
4. ∼b∨q Switcheroo, 2
5. (∼ a ∨ q) ∧ (∼ b ∨ q) Conjunction, 3,4
6. (∼ a∧ ∼ b) ∨ q Distributive Law, 5
7. ∼ (a ∨ b) ∨ q De Morgan, 6
8. (a ∨ b) → q Switcheroo

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NAME:
Prove that the following arguments are valid using the Rules of Inference. Some hints are given
below. Use them ONLY IF NEEDED.
3. s → r
1. (p ∨ r) → (s ∧ t) (p ∨ q) → (∼ r)
p (∼ s) → (∼ q → r)
∴t p
∴q

.
.
2. a → (b ∧ c)
∼b
∴∼ a

.
Hints (again, use only if needed!):
In Item 1, the following will be used (in this order): Addition, Modus Ponens, Simplification
In Item 2, from ∼ b, use Addition to obtain ∼ b∨ ∼ c. Then use De Morgan’s to “factor” out
the negation.
In Item 3, you will use the premises in this order: fourth premise, then Addition. Then second
premise (apply Modus Ponens). Next, use the first premise (apply Modus Tollens). By this point
you should obtain ∼ s. Now use the third premise to get (∼ q → r). To finally reach q, use Modus
Tollens again.

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 L.5. Rules of Inference 47

L.5 Exercises
In each of Exercises 1–34, supply the missing statement or reason, as the case may be. (To
make life simpler, we shall allow you to write ~(~p) as just p whenever it occurs. This saves an
extra step in practice.)

Statement Reason Statement Reason

1. 1. p’~q Premise 2. 1. ~p’q Premise
2. p Premise 2. ~p Premise
3. - - - - 1,2 Modus Ponens 3. - - - - 1,2 Modus Ponens

3. 1. (~pæq)’~(q…r) Premise 4. 1. (~p…q)’(q…~r) Premise

2. ~pæq Premise 2. ~p…q Premise
3. - - - - 1,2 Modus Ponens 3. - - - - 1,2 Modus Ponens

5. 1. (~pæq)’~(q…r) Premise 6. 1. (~p…q)’(q…~r) Premise

2. q…r Premise 2.~(q…~r) Premise
3. - - - - 1,2 Modus Tollens 3. - - - - 1,2 Modus Tollens

7. 1. ~(~pæq) Premise 8. 1. ~(p…~q) Premise

2. - - - - 1, DeMorgan 2. - - - - 1, DeMorgan

9. 1. (p…r)’~q Premise 10. 1. (~p…q)’(q…~r) Premise

2. ~q’r Premise 2. (q…~r)’s Premise
3. - - - - 1,2 Transitive Law 3. - - - - 1,2 Transitive Law

11. 1. (p…r)’~q Premise 12. 1. (~p…q)’(q…~r) Premise

2. ~q’r Premise 2. (q…~r)’s Premise
3. ~r Premise 3. ~s Premise
4. - - - - 1,2 Transitive Law 4. - - - - 1,2 Transitive Law
5. - - - - 3,4 Modus Tollens 5. - - - - 3,4 Modus Tollens

13. 1. (p’q)ær Premise 14. 1. ~(p…q)æs Premise

2. ~r Premise 2. ~s Premise
3. - - - - 1,2 Disjunctive 3. - - - - 1,2 Disjunctive
Syllogism Syllogism

15. 1. p’(r…q) Premise 16. 1. (pæq)’r Premise

2. ~r Premise 2. ~r Premise
3. - - - - 2, Addition of ~q 3. - - - - 1,2 Modus Tollens
4. - - - - 3, DeMorgan 4. - - - - 3, DeMorgan
5. - - - - 1,4 Modus Tollens 5. - - - - Simplification1

1 Use simplification in the following form:

A…B
____
2.4 Invalid Arguments or Fallacies
In Example 4 on page 4, you already encountered an argument which is not valid. In fact, Example
4 showed three methods that you can use to prove that an argument is not valid.
In this section we will learn more about invalid arguments or fallacies.

Definition.
An argument
(p1 ∧ p2 ∧ . . . ∧ pn ) → q,
which is not valid is called a fallacy.

In a fallacy, it is possible for the premises p1 , p2 , . . . , pn to be true, while the conclusion q is false.
Equivalently, for this case, the conditional

(p1 ∧ p2 ∧ . . . ∧ pn ) → q

is not a tautology.

A common fallacy
One of the common fallacies is called the Fallacy of the Converse. Symbolically, it is the
argument
p→q
q
∴p

An example of the Fallacy of the Converse is this argument:

If you drink softdrinks daily, then you will develop diabetes.
You developed diabetes.
Therefore you drink softdrinks daily.
This is an invalid argument. Perhaps you can develop diabetes some other way (possibly
by eating a lot of cake). The point is, the premise only says that drinking softdrinks will lead to
diabetes. It does NOT say that drinking softdrinks is the only reason for you to develop diabetes.

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 People who do not like math are good-looking
There is an often-shared joke in the Internet which says that if you do not like math, then
you are good-looking.
You like math. Does that mean you are not good-looking?

In the argument above, you cannot conclude that you are not good-looking just because you
like math (this is the Fallacy of the Inverse). The premise only states a conclusion for people
who do not like math; it does not give any statement regarding people who like math. So don’t
worry! Even if you like math, you are still good-looking.
The Fallacy of the Converse and Inverse are just two of several fallacies. The following table
lists some common fallacies in logic. You can prove that they are fallacies by applying any of the
three methods demonstrated in Example 4.

Table of Fallacies
Let p, q, and r be propositions.
Propositional Form Standard Form
p→q
Fallacy of the Converse ((p → q) ∧ q) → p q
∴p
p→q
Fallacy of the Inverse ((p → q) ∧ (∼ p)) → (∼ q) ∼p
∴∼ q
p∨q
Affirming the Disjunct ((p ∨ q) ∧ p) → (∼ q) p
∴∼ q
p→q
Fallacy of the Consequent (p → q) → (q → p)
∴q→p
∼ (p ∧ q)
Denying a Conjunct (∼ (p ∧ q) ∧ (∼ p)) → q ∼p
∴q
p→q
Improper Transposition (p → q) → ((∼ p) → (∼ q))
∴ (∼ p) → (∼ q)

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 It is very difficult to memorize all the fallacies!
As with the Rules of Inferences, try to make sense of the fallacies; do not just memorize
them.
For example, in Affirming the Disjunct, if p ∨ q is true, then we know (from the definition
of ∨) that if p is true, then p ∨ q is automatically true. In fact, q can be either true or false,
and p ∨ q would still be true. Thus, there is no reason to conclude that q must be true.

EXAMPLE 10. Determine whether the given is a valid argument or a fallacy.

Either Alvin sings or dances with Nina.
Alvin sang with Nina.
Therefore, Alvin did not dance with Nina.
Solution. In this example, it is not given whether the arguments are valid or not. Knowing the
Rules of Inferences and the Table of Fallacies can be a big help.
The given argument is of the form
p∨q
p
∴∼ q

Method 1: Determine if ((p ∨ q) ∧ p) →∼ q is a tautology.

p q ∼q p∨q (p ∨ q) ∧ p [(p ∨ q) ∧ p] →∼ q
T T F T T F
T F T T T T
F T F T F T
F F T F F T
Since the argument is not a tautology, then it is a fallacy.
Method 2: Determine if it is possible for all the premises to be true, but the conclusion is
false.
Suppose the conclusion is false. Then ∼ q is false or q is true. This automatically makes the first
premise (p ∨ q) true. The second premise p must also be true. Therefore, if p is true and q is true,
then all premises are true and the conclusion is false, as summarized below.
Premise Conclusion Premise
p q ∼q p∨q
T T F T
The table shows that it is possible for all premises to be true, but the conclusion is false.
Therefore, the argument is a fallacy.

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 Method 3: If you recognize this argument is the fallacy of Affirming the Disjunct, then you
can immediately know that the argument is a fallacy. Simply justify your reasoning by writing
“Affirming the Disjunct.”

Try these:
Determine whether the arguments below are valid or are fallacies.
Present two methods of justification (choose any of the three methods).

Either Alvin sings or dances with Nina.

1. Alvin did not dance with Nina.
Therefore, Alvin sang with Nina.

It is not true that Alvin sings and dances with Nina.

2. Alvin did not sing with Nina.
Therefore, Alvin danced with Nina.

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Decide whether each of the following arguments are valid. Justify your answer using any of the
methods discussed in the module.