Logic 101 · Series · Uncategorized

8 Basic Rules of Logic

In the last post I introduced the logical symbols that I will use when I write arguments out in logical form. If you aren’t familiar with logical operators and other symbols used in logic, then you aren’t going to follow the content of this post very well.

My intention is to give you eight valid rules of logic that form the basis of any other valid rules of logic. By a valid rule, I mean the logical form of an argument that produces a valid argument. For a deductive argument to be valid, the conclusion cannot be false when the premises are true.

Recall that the logical form of an argument is how an argument is organized and expressed in logical symbols. By “organized,” I mean how the propositions in the premises and conclusion are arranged so that the argument is valid. By “expressed” I mean how the argument is carried over into a logical language. So, a valid rule is one that guarantees that the conclusion follows from the premises because of the way the propositions in the argument are organized.

For each rule, I will give its technical name, a brief explanation, what it looks like in logical form, and an example of how it might work in an argument written in English. For each rule, I won’t number the premises of the argument. Instead, I will place each premise on different lines, and then indicate the conclusion that follows with the logical symbol for “therefore” (∴).

1. Modus Ponens

P → Q
P
∴ Q

Explanation: If you have a conditional statement whose antecedent (P) is true, the conclusion (Q) follows because the P is a sufficient condition for Q (the consequent).

Example:

  1. If it is raining outside my house, then my front lawn is wet.
  2. It is raining outside my house.
  3. Therefore, my front lawn is wet.

2. Modus Tollens

P → Q
~Q
∴ ~P

Explanation: If you have a conditional statement whose consequent (Q) is false, the conclusion (~P) follows because the Q is a necessary condition for P (the antecedent).

Example:

  1. If it is raining outside my house, then my front lawn is wet.
  2. My front lawn is not wet.
  3. Therefore, it is not raining outside my house.

By the way, did you notice that you’ve seen these two rules, and their examples, in our discussion of necessary and sufficient conditions?

3. Conjunction

P
Q
∴ P & Q

Explanation: When any two propositions appear as separate premises, you can create a conclusion that has those two propositions joined together by a conjunction.

Example:

  1. God is just.
  2. God is merciful.
  3. Therefore, God is just and God is merciful.

4. Addition

P
∴ P v Q

Explanation: For any proposition that stands by itself, you can create a conclusion that has that proposition joined to any another proposition by a disjunction.

Example:

  1. God exists.
  2. Therefore, God exists or the Earth has two moons.

5. Simplification

P & Q
∴ P
∴ Q

Explanation: For any number of propositions joined together by a conjunction, you can separate those propositions from the conjunction and create a conclusion that includes only a single one of them.

Example:

  1. The Sun is a star and the Earth is a planet.
  2. Therefore, the Sun is a star.
  3. Therefore, the Earth is a planet.

6. Absorption

P → Q
∴ P → (P & Q)

Explanation: For any conditional statement, you can create a conclusion that says that the original antecedent (P) implies P joined to the consequent (Q) by a conjunction. This rule works because it is always true that a propostion implies itself. In other words P → P is always true.

As Craig and Moreland point out in Philosophical Foundations for a Christian Worldview, this rule is mostly used for when you need P & Q in a further step in an argument (p. 36).

Example:

  1. If it is raining outside my house, then my front lawn is wet.
  2. Therefore, if it is raining outside my house, then it is raining outside my house and my front lawn is wet.

7. Hypothetical Syllogism

P → Q
Q → R
∴ P → R

Explanation: If an antecedent proposition (P) implies a consequent (Q), and Q implies another proposition (R), then you can conclude that P implies R.

Example:

  1. If you skip class, you will lose credit.
  2. If you lose credit, you will receive lower grades.
  3. Therefore, if you skip class, you will receive lower grades.

8. Disjunctive Syllogism

P v Q
~P
∴ Q

Explanation: If a disjunctive proposition has a disjunct (P) that is false (hence, ~P), then you can imply that the other disjunct (Q) is true.

Example:

  1. It is either sunny outside or it is overcast.
  2. It is not sunny outside.
  3. Therefore, it is overcast outside.

Wait, Aren’t There 9 Rules?

It’s common that books presenting these same rules I have will contain a ninth called the “Constructive Dilemma” rule. I haven’t included it here for two reasons. First, creating dilemmas is one way to object to (or deny; the ‘D’ in the R.E.A.D. Method) an argument. There are actually four common, valid rules for creating dilemmas, and I’d like to cover all of them at once. Second, the Constructive Dilemma rule (and the other three I will share) can be proven using the above rules and a couple of other simple rules I’ve omitted. I’ll share those other rules when I discuss conditional proofs.

The eight rules that I’ve shared in this post are the most basic and can be used to derive many other valid rules. In the next post I’ll discuss conditional proofs and some additional valid rules.

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