A **logical inference** is a type of reasoning process that enables us to draw conclusions or make judgements based on the information or premises available to us.

Logical inferences can be used to fill in missing information and make sense of the relationships between different statements or propositions. By using logical inference, we can move from what we already know to what can be logically deduced or inferred from that knowledge.

Logical inference involves applying logical rules and principles – **inference laws** – to reach new knowledge or beliefs that are logically supported by what we already know. These laws provide a set of rules that allow us to make logical deductions and draw conclusions based on given premises.

In this post, we will look at 8 essential inference laws (also referred to as *implicational rules):*

- Modus Ponens
- Modus Tollens
- Disjunctive Syllogism
- Simplification
- Conjunction
- Hypothetical Syllogism
- Addition
- Constructive Dilemma

These are all **logically valid**. In other words, the conclusions of these inferences follow (logically) from the premises.

By understanding and applying these laws, you can enhance your logical reasoning skills and evaluate arguments more effectively.

## 1. Modus Ponens

**Modus Ponens** is a simple yet powerful inference law that allows us to draw a conclusion from a conditional statement and its antecedent. It follows the pattern:

- If P, then Q.
- P.
- Therefore, Q.

For example:

- If it is raining, then the ground is wet.
- It is raining.
- Therefore, the ground is wet.

Modus Ponens enables us to conclude that the consequent is true based on the truth of the conditional statement and its antecedent.

Modus Ponens looks deceptively similar to the fallacy of the converse. But don’t get confused – the fallacy of the converse is an *invalid* argument format, unlike Modus Ponens.

## 2. Modus Tollens

**Modus Tollens** is another valuable inference law that enables us to draw a conclusion by negating the consequent of a conditional statement and its denial. It follows the pattern:

- If P, then Q.
- Not Q.
- Therefore, not P.

For example:

- If the battery is dead, then the device won’t turn on.
- The device is turned on.
- Therefore, the battery is not dead.

Modus Tollens enables us to conclude that antecedent is false based on the negation of the conditional statement’s consequent.

It looks deceptively similar to the fallacy of the inverse. But don’t get confused – the fallacy of the inverse is an *invalid* argument format, unlike Modus Tollens.

## 3. Disjunctive Syllogism

**Disjunctive Syllogism** is employed when we have a statement that presents two alternatives (A or B) and one of them is negated. It follows the pattern:

- A or B.
- Not A.
- Therefore, B.

For example:

- Either I will go to the park or stay at home.
- I will not go to the park.
- Therefore, I will stay at home.

Disjunctive Syllogism enables us to deduce the truth of the alternative that remains when one of the options is negated.

## 4. Simplification

**Simplification** is an inference law used to simplify compound statements (statements with multiple components) by focusing on individual components. It follows the pattern:

- P and Q.
- Therefore, P.

For example:

- The car is red and it has four doors.
- Therefore, the car is red.

Simplification enables us to extract one component from a compound statement while disregarding the other.

## 5. Conjunction

**Conjunction** is the inverse of simplification. It enables us to combine individual statements into a compound statement. It follows the pattern:

- P.
- Q.
- Therefore, P and Q.

For example:

- The sun is shining.
- It is a beautiful day.
- Therefore, the sun is shining, and it is a beautiful day.

Conjunction enables us to join two separate statements into a single compound statement.

## 6. Hypothetical Syllogism

**Hypothetical Syllogism** allows us to make logical deductions by chaining together two conditional statements. It follows the pattern:

- If P, then Q.
- If Q, then R.
- Therefore, if P, then R.

For example:

- If it is sunny, then the flowers will bloom.
- If the flowers bloom, then the garden will look beautiful.
- Therefore, if it is sunny, then the garden will look beautiful.

Hypothetical Syllogism enables us to establish a relationship between the antecedent of the first conditional statement and the consequent of the second conditional statement.

## 7. Addition

**Addition** is a simple inference law that enables us to add a statement to an existing set of premises. It follows the pattern:

- P.
- Therefore, P or Q.

For example:

- It is raining.
- Therefore, it is raining or the sun is shining.

Addition enables us to include additional possibilities or options based on a given statement. It is sometimes referred to as *disjunction introduction*.

You may have come across addition in the context of the definition of knowledge and Gettier cases. Addition is used in the *second* Gettier case when Smith infers that “either Jones owns a Ford or Brown is in Barcelona” from “Jones owns a Ford”.

## 8. Dilemma

**Constructive dilemma** is an inference law used when we have a complex conditional statement involving two options and their corresponding consequences. It follows the pattern:

- If P, then Q.
- If R, then S.
- Either P or R.
- Therefore, either Q or S.

For example:

- If we study hard, we will pass the exam.
- If we don’t study hard, we will fail the exam.
- Either we study hard or we don’t.
- Therefore, either we will pass the exam or we will fail the exam.

Dilemma allows us to deduce the possible outcomes based on the given conditional statements and the choices presented.

## Summary: Inference Laws

The 8 inference laws above are the basic logical inferences of statement logic. They enable us to make *logically valid* moves from:

**Statements we know are true.**

To:

**Conclusions that must also be true.**

These inference laws are powerful because they *logically guarantee* your conclusion (when used correctly). In other words, it’s logically impossible for your premises to be true and your conclusion be false when you use these inference laws.

Understanding valid and invalid logical inferences is a great way to improve your philosophical reasoning skills, which can help you hit those top grade boundaries in the A level philosophy exam (particularly for the 25 mark AO2 questions).

Beyond the exam, understanding logical inferences enables you to engage in more rigorous critical thinking, evaluate other people’s reasoning more precisely, and strengthens your ability to construct your own arguments. This can be incredibly useful in all areas of life.