Logic 101 · Series

Conditional Proofs

In the previous two posts, I introduced logical symbols and eight basic rules of logic. This post will introduce four more valid rules and talk about what’s called a conditional proof.

Now that we know some rules of logic, it’s important that I introduce a couple of more conventions and show how to format arguments in logical form. Whenever we derive a conclusion in a logical argument, it’s a good idea to indicate how you reached that conclusion. To do this, you indicate to the right of the premise which rule you used, and which premises follow that rule. So, if we follow modus ponens, that would look something like this:

1. P→ Q
2. P
3. ∴ Q Modus Ponens: 1, 2

In this example, we could abbreviate modus ponens as “MP,” provided that we trust our readers will know what we’re referring to.

A conditional proof is just like any other argument, except for the fact that it begins with an assumption. In other words, a conditional proof assumes that something is true in order to draw a particular conclusion. And the conclusion that is drawn is based on what you end up with in each conditional proof. (This will make better sense in just a moment.)

A conditional proof can be used as a stand-alone argument or as arguments within other arguments. When they are used as arguments within other arguments, we indent each of the steps of the conditional proof to make it clear that it stands apart from the main argument. In the following example, let’s try to prove that P → R:

Goal: P → R

1. P Premise
2. P & Q Premise
3. Q → R Premise
4. P Assume for Conditional Proof
5. Q Conjunction Elimination: 2
6. ∴ P → Q From Conditional Proof: 4-5
7. ∴ P → R Hypothetical Syllogism: 3, 6

The conditional proof in steps 4-5 allows you to infer P → Q within the context of another argument. That’s why each step is indented. Of course, we can follow the same rule that steps 4-5 follow in a stand-alone argument if we want to. That might look something like this:

Goal: P → Q

1. P Assume for Conditional Proof
2. P & Q Premise
3. ∴ Q Conjunction Elimination: 2
4. ∴ P → Q From Conditional Proof: 1-3

This argument reaches the same conclusion as the first example, only it is an argument that stands by itself.

I’ve now used the same conditional proof rule twice without explaining how it works. Now it’s time to introduce you to three conditional proof rules and mention a fourth that deserves it’s own post at the end. Just like the eight valid rules I introduced before, I will give the name of the rule, how the rule functions in logical form, a brief explanation, and an example.

1. Conditional Introduction (CI)

P
[. . .]
Q
∴ P → Q

The [. . .] means there are any number of premises between P and Q.

Explanation: By assuming that P is true and showing that it leads us to Q, we are able to conclude that P is a sufficient condition for Q, and therefore P → Q.

Example: 

  1. Assume that water freezes at 32°F (at sea level).
  2. I am sea level.
  3. If I am at sea level, then my fountain outside will be frozen.
  4. Therefore, my fountain outside will be frozen.
  5. Therefore, if water freezes at 32°F (at sea level), then my fountain outside will be frozen.

2. Disjunction Elimination (DE)

P v Q

P
[. . .]
R

Q
[. . .]
R

∴ R

Explanation: If we have two propositions joined by a disjunction as a premise and want to derive a different proposition from that premise, we have to make two conditional proofs. In the first, we show that the first disjunct (P) leads to the conclusion we want (R). In the second, we show that the second disjunct (Q) also leads to R. We we have shown this, we are able to assert R as a conclusion.

Example:

  1. I will either go to class, or I will go to the gym.
    1. Assume that I go to class.
    2. If I go to class, then I will take notes.
    3. If I take notes, then I will get better grades.
    4. Therefore, if I go to class, then I will get better grades.
    5. Therefore, I will get better grades.

    6. Assume that I go to the gym.
    7. If I go to the gym, then I will be happier.
    8. If I will be happier, then I will get better grades.
    9. Therefore, if I go to the gym, I will get better grades.
    10. Therefore, I will get better grades.
  1. Therefore, I will get better grades.

Note: Don’t actually skip class for the gym. Not cool.

3. Biconditional Introduction (BI)

P
[. . .]
Q

Q
[. . .]
P

∴ P ↔ Q

Explanation: This is an application of the Conditional Introduction rule that allows us to conclude with a biconditional statement. There are two conditional proofs involved. In the first, we assume that some proposition (P) is true and that it leads to another (Q). In the second, we assume that Q is true and show how it leads to P. If both of these proofs are used, then we can assert P ↔ Q.

Example:

  1. Assume Jesus is God incarnate.
  2. If Jesus is God incarnate, then Jesus has all of God’s essential attributes.
  3. If Jesus has all of God’s essential attributes, then he is all-knowing (omniscient).
  4. Therefore, if Jesus is God incarnate, then he is omniscient.
  5. Therefore, Jesus is omniscient.

  6. Assume Jesus is omniscient.
  7. If Jesus is omniscient, then he shares an essential attribute with God.
  8. If Jesus shares an essential attribute with God, then he shares all of God’s essential attributes.
  9. If Jesus shares all of God’s essential attributes, then he is God incarnate.
  10. Therefore, if Jesus is omniscient, he is God incarnate.
  11. Therefore, Jesus is God incarnate.

  12. Therefore, Jesus is God incarnate if and only if he is omniscient.

Note: Assume for (8) that it is impossible for two omniscient beings to exist.

Summary

Conditional proofs are incredibly useful. The three rules that I have given you, in combination with the eight rules I gave in the previous post, allow us to prove that many other logical forms are valid rules.

Since that’s the case, I have to mention that I left out an important conditional proof rule: the Negation Introduction (NI) rule. This is also called a reductio ad absurdum argument or an indirect proof. Because this valid rule is so useful for denying a premise or conclusion, I will dedicate an entire post to it.

In addition, we can use all of these rules to prove that there are several valid rules for constructing dilemmas. Dilemmas are also useful for denying premises or conclusions in arguments. I will also dedicate an entire post to them.

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