Arguments in Propositional Logic

---

Arguments in Propositional Logic

---

1. What is an Argument?

An argument in propositional logic is a sequence of statements (called premises) followed by a conclusion. It is used to determine whether the conclusion logically follows from the premises.

Form:

$$\text{Premise}_1, \text{Premise}_2, \dots, \text{Premise}_n \vdash \text{Conclusion}$$

The symbol $\vdash$ means "logically implies."

---

2. Valid and Invalid Arguments

• An argument is valid if, whenever all premises are true, the conclusion is also true.
• An argument is invalid if it is possible for all premises to be true but the conclusion is false.

Validity depends only on logical structure, not actual content.

---

3. Example of a Valid Argument

Premises:

1. $p \rightarrow q$
2. $p$

Conclusion:
$q$

Form:

$$p \rightarrow q,\ p \vdash q$$

This is known as Modus Ponens, and it is a valid form of argument.

---

4. Common Valid Argument Forms

a) Modus Ponens (Law of Detachment)

If $p \rightarrow q$, and $p$ is true, then $q$ is true.

$$p \rightarrow q,\ p \vdash q$$

b) Modus Tollens

If $p \rightarrow q$, and $\neg q$, then $\neg p$.

$$p \rightarrow q,\ \neg q \vdash \neg p$$

c) Hypothetical Syllogism

If $p \rightarrow q$ and $q \rightarrow r$, then $p \rightarrow r$.

$$p \rightarrow q,\ q \rightarrow r \vdash p \rightarrow r$$

d) Disjunctive Syllogism

If $p \lor q$, and $\neg p$, then $q$.

$$p \lor q,\ \neg p \vdash q$$

e) Addition

If $p$, then $p \lor q$

$$p \vdash p \lor q$$

f) Simplification

If $p \land q$, then $p$

$$p \land q \vdash p$$

---

5. Invalid Argument Forms (Fallacies)

a) Affirming the Consequent (Invalid)

$$p \rightarrow q,\ q \vdash p \quad \text{(Invalid)}$$

Example:
If it is raining, then the ground is wet.
The ground is wet.
∴ It is raining. → Not necessarily true (maybe someone watered the garden)

---

b) Denying the Antecedent (Invalid)

$$p \rightarrow q,\ \neg p \vdash \neg q \quad \text{(Invalid)}$$

Example:
If it is raining, then the ground is wet.
It is not raining.
∴ The ground is not wet. → Not necessarily true

---

6. Testing Argument Validity with Truth Tables

Construct a truth table including all premises and the conclusion. An argument is valid if every time all premises are true, the conclusion is also true.

Example:

Premises: $p \rightarrow q$, $p$
Conclusion: $q$

p	q	$p \rightarrow q$	Conclusion $q$
T	T	T	T
T	F	F	F
F	T	T	T
F	F	T	F

Check the row(s) where all premises are true. If in those rows the conclusion is also true, the argument is valid.

Here, only the first row has all premises true, and the conclusion is also true → Valid argument.

---

7. Use of Rules of Inference

Arguments can also be validated using a sequence of logical steps (called rules of inference) to derive the conclusion from the premises.

---