5.3 Rules for Conjunctions and Disjunctions

This module presents the rules of conjunction introduction (&I) and conjunction elimination (&E), disjunction introduction (vI), distribution (DISTRIB), and disjunctive syllogism (DS). It also presents the fallacies of affirming a disjunct, false dilemma, and alternative advance.

Table of Contents

• 5.3 Rules for Conjunctions and Disjunctions
  • 5.3.1 Conjunction Rules
    • Conjunction Elimination (&E)
    • Valid and Invalid Uses of &E
    • Conjunction Introduction (&I)
    • Valid and Invalid Uses of &I
  • 5.3.2 Disjunction Introduction
    • Disclaimer
    • Disjunction Introduction (vI)
    • Valid and Invalid uses of vI
  • 5.3.3 Distribution
    • Truth Functional Equivalence
    • The Rule of Distribution (DISTRIB)
  • 5.3.4 Disjunctive Syllogism
    • Disjunctive Syllogism (DS)
    • Truth Table for Disjunctive Syllogism
    • Valid and Invalid Uses of Disjunctive Syllogism
  • 5.3.5 Fallacies with Disjunctions
    • Affirming the Disjunct
    • False Dilemma and Alternative Advance
  • Submodule 5.3 Quiz

5.3.1 Conjunction Rules

Conjunction ties two claims together.

Conjunction Elimination (&E)

There are two rules of inference for conjunctions. The first, conjunction elimination, says that from a conjunction we can derive either conjunct:

1. P & Q
2. P (1, &E)

or

1. P & Q
2. Q (1, &E)

It is easy to see why this rule is valid using a truth table. Our premise, the conjunction 1. P & Q only has the value T on the first row of the truth table, so we highlight this row. Our conclusion, 2. Q, is true on the highlighted row. So, since the conclusion is T on every row where the premise is T, the rule is valid.

P	Q	1. P & Q	2. Q
T	T	T	T
T	F	F	F
F	T	F	T
F	F	F	F

Valid and Invalid Uses of &E

The rule applies only when the & we are eliminating is the main connective. So, the rule applies here:

1. (P v Q) & R
2. P v Q (1, &E)

or here

1. (P v Q) & R
2. R (1, &E)

or here:

1. P & (Q & R)
2. Q & R (1, &E)

But the rule does not apply here:

1. (P & Q) v R
2. P !!INVALID!!

or here:

1. ~(P & Q)
2. Q !!INVALID!!

Conjunction Introduction (&I)

The second rule of inference for conjunctions, conjunction introduction, says that we can conjoin together any two sentences: from any two sentences on previous lines, we can derive their conjunction.

1. P
2. Q
3. P & Q (1, 2 &I)

Again, it is easy to see why this is valid using a truth table. On the only row where both P and Q are true, the first row, P & Q is also true.

P	Q	1. P	2. Q	3. P & Q
T	T	T	T	T
T	F	T	F	F
F	T	F	T	F
F	F	F	F	F

Valid and Invalid Uses of &I

Conjunction introduction can apply no matter what is on the previous line, provided that the & which is added is the main connective.

1. P & ~R
2. Q
3. (P & ~R) & Q (1, 2 &I)

or

1. P v R
2. Q v S
3. (P v R) & (Q v S)  (1, 2 &I)

Conjunction introduction cannot be used when the & which is added is not the main connective:

1. P
2. Q v R
3. (P & Q) v R !!INVALID!!

Also, contrary to what the name might suggest, “Conjunction Introduction” can’t be used to simply introduce any atomic sentence on lines. The sentence has to appear on a previous line.

1. P
2. Q
3. P & R !!INVALID!!

Lastly, it is important to avoid the fallacy of predicate conjunction. The rule &I allows us to conjoin whole sentences, but it does not allow us to conjoin parts of sentences. For instance:

Let

J = “John got married”

M = “Mary got married”

The following inference is valid:

1. J
2. M
3.  J & M (1, 2 &I)

However, it is important to interpret the sentence on line 3 correctly. It says, “John got married and Mary got married.” It does not say that “John and Mary got married”, however, in the sense that they married one another. They might have married other people!

---

5.3.2 Disjunction Introduction

Either you exist or the moon is edible.

Disclaimer

The rule explained on this page is not used in ordinary English, and it hits most students as strange, probably because it is a little strange. The reason it is studied in this textbook is because, until you understand this rule, you don’t yet understand what we’ve been saying by “disjunction” or “truth table” on previous pages. Keep at it, and go back and review if you need to. Once the rule finally makes sense to you, then you know you’ve really been following everything in this module so far.

Disjunction Introduction (vI)

The rule of disjunction introduction says that, if we know a sentence is true, then we know that the disjunction of that sentence with any other sentence whatsoever is also true. So, given a sentence on a prior line, we can infer the disjunction of that sentence with any other sentence, even if it did not occur previously. Notice that this is not like the rule of conjunction introduction, which only allows us to conjoin sentences which occured on the lines we derived them from.

1. P
2. P v Q (1, vI)

or

1. P
2. Q v P

The rule is easiest to prove with a truth table. On every row where 1. P is true, 2. Q v P is also true, so the rule is valid:

P	Q	1. P	2. Q v P
T	T	T	T
T	F	T	T
F	T	F	T
F	F	F	F

Valid and Invalid uses of vI

The rule can be used validly no matter how complex the sentence we add is, provided that the v we add is the main connective:

1. P
2. P v (((Q & ~B) v (H& Z)) & ~M)  (1 vI)

or

1. W & F
2. (W & F) v ~N (1 vI)

We can use the rule as often as we like:

1. P
2. P v Q (1 vI)
3. (P v Q) v R (2 vI)
4. ((P v Q) v R) v S (3 vI)

The rule is not used validly if the v we add is not the main connective, however:

1. ~P
2. ~(P v R)  !!INVALID!!

It is also important not to confuse the truth of a disjunction with the idea that each disjunct has a genuine chance of  being true. For instance, suppose that you know that you exist (Y). Then, you also know that it is true that either you exist or the moon is made of cheese (Y v M). But this doesn’t imply that there is a real chance that you don’t exist, or that the moon is actually made of cheese. It only says that, because you exist, and only one disjunct has to be true in order for the whole disjunction to be true, any disjunction in which Y is a disjunct is also going to be true.

Again, the rule may seem strange, but understanding why it is valid is a good test of your understanding of truth tables and rules.

---

5.3.3 Distribution

Distribution rearranges the placement of conjunctions and disjunctions

Truth Functional Equivalence

Two sentences are truth functionally equivalent when they have the same truth value assignments in every row on a truth table.  When two sentences are truth functionally equivalent, we draw the triple-bar ≡ sign between them. Here are two examples of truth functionally equivalent sentences, and truth tables which prove they are truth functionally equivalent.

P v (Q & R) ≡  (P v Q) & (P v R)

P	Q	R	Q & R	P v (Q & R)	P v Q	P v R	(P v Q) & (P v R)
T	T	T	T	T	T	T	T
T	T	F	F	T	T	T	T
T	F	T	F	T	T	T	T
T	F	F	F	T	T	T	T
F	T	T	T	T	T	T	T
F	T	F	F	F	T	F	F
F	F	T	F	F	F	T	F
F	F	F	F	F	F	F	F

P & (Q v R) ≡  (P & Q) v (P & R)

P	Q	R	Q v R	P & (Q v R)	P & Q	P & R	(P & Q) v (P & R)
T	T	T	T	T	T	T	T
T	T	F	T	T	T	F	T
T	F	T	T	T	F	T	T
T	F	F	F	F	F	F	F
F	T	T	T	F	F	F	F
F	T	F	T	F	F	F	F
F	F	T	T	F	F	F	F
F	F	F	F	F	F	F	F

The Rule of Distribution (DISTRIB)

The rule of distribution comes from the two relationships of truth-functional equivalence shown above. It tells us that the four following inferences are all valid. The first two are disjunctive distribution:

1.  P v (Q & R)
2.  (P v Q) & (P v R) (1, DISTRIB)

1.  (P v Q) & (P v R)
2. P v (Q & R) (1, DISTRIB)

The second two are conjunctive distribution:

1. P & (Q v R)
2. (P & Q) v (P & R) (1, DISTRIB)

1. (P & Q) v (P & R)
2. P & (Q v R) (1, DISTRIB)

Because, in each case above, what is on line 2 is truth-functionally equivalent to what appears on line 1 above it, the inference is always valid; if both sentences are true on all of the same rows, then 2 will always be true in all the rows where 1 is true.

The important thing is not to get conjunctive distribution mixed up with disjunctive distribution. For instance, these two inferences are invalid:

1.  (P & Q) v (P & R)
2. P v (Q & R) !!INVALID!!

1.  P v (Q & R)
2.  (P & Q) v (P & R) !!INVALID!!

---

5.3.4 Disjunctive Syllogism

If you have two choices, and you don’t take one, you must take the other.

Disjunctive Syllogism (DS)

Most of the rules we have introduced so far have not been terribly interesting. We now turn to a rule that is very powerful for modeling real-world arguments, called disjunctive syllogism, abbreviated DS. Disjunctive syllogism is one of the most common forms of argument; you may have heard it referred to before as reasoning through a “process of elimination”. A disjunctive syllogism contains two premises: one a disjunction, and another the negation of one of the disjuncts; it then concludes the remaining disjunct must be true.  An example of a disjunctive syllogism in ordinary English is this argument:

1. Either we should accept some loss of privacy or we should accept the risk of terror attacks.
2. We should not accept the risk of terror attacks.
3. We should accept some loss of privacy. (1, 2 DS)

Symbolized, the rule looks like this:

1. P v Q
2. ~Q
3. P (1, 2 DS)

The order of the premises, or which disjunct is negated, does not make a difference. So, all of the following are valid:

1. P v Q
2. ~P
3. Q (1, 2 DS)

1. ~P
2. P v Q
3. Q (1, 2 DS)

1. ~Q
2. P v Q
3. P (1, 2 DS)

Truth Table for Disjunctive Syllogism

We can prove the validity of disjunctive syllogism using a truth table:

P	Q	1. P v Q	2. ~Q	3. P
T	T	T	F	T
T	F	T	T	T
F	T	T	F	F
F	F	F	T	F

The premise 1. P v Q is true in the first three rows; however, of those first three rows, 2. ~Q is only true in the second row, so we highlight only the second row, since that is the only row in which both premises are true. The conclusion 3.  is true in the highlighted row. So, the rule of inference is valid.

Valid and Invalid Uses of Disjunctive Syllogism

Remember that P or Q could themselves be positive or negative sentences; the negation of a negation is a double negation. The following argument is a valid use of DS:

1. ~P v ~Q
2. ~~P
3. ~Q (1, 2 DS)

The disjuncts can be very complex; the argument is still valid, so long as v is the main connective and one of the disjuncts is negated:

1. ~(A & ~B) v (C & ~D)
2. ~~(A & ~B)
3. C & ~D (1, 2 DS)

It is not valid to use disjunctive syllogism when the disjunction which has one side negated is not the main connective, however:

1. P v ~(Q v R)
2. ~R
3. P v ~Q !!INVALID!!

---

5.3.5 Fallacies with Disjunctions

You could have chicken, or you could have waffles… or you could have both.

Affirming the Disjunct

A fallacious argument that looks like disjunctive syllogism, but which is actually an invalid argument, is called the fallacy of affirming the disjunct:

1. P v Q
2. P
3. ~Q(!!INVALID!!)

For example:

1. Either the job is fun or it pays well.
2. The job is fun.
3. The job doesn’t pay well. (!!INVALID)

Another example:

1. Either you study Spanish or you study French.
2. You study Spanish
3. You don’t study French (!!INVALID!!)

This argument would be valid if premise 1 were the “exclusive or”, meaning “either P or Q but not both”. Remember that our disjunctive symbol v represents the “inclusive or”, however, which is true if either or both disjuncts are true — it means the same thing as “P or Q or both P and Q”. So, it could be true that the job is either fun or it pays well because it is both fun and also pays well.

False Dilemma and Alternative Advance

Lastly, there are two valid uses of disjunctive syllogism which, while valid, are fallacies. They are fallacies because the disjunctive premise is probably not true, so they are unsound.

The first is the fallacy of false dilemma. A false dilemma occurs when someone uses disjunctive syllogism, but the disjunctive premise doesn’t exhaust all of the options. For example, suppose that there are many places that Kim and Leslie could go out to eat. Leslie argues:

1. Either we want to eat Mexican food or we want to eat Italian food.
2. We don’t want to eat Italian food.
3. We want to eat Mexican food. (1, 2 DS)

The argument is a valid use of disjunctive syllogism, but the first premise restricts the options too tightly. There are other options which the first premise is ignoring.

The second is the fallacy of alternative advance, which is a combination of a false dilemma and a distinction without a difference. Alternative advance is a disjunctive syllogism in which both disjuncts in the disjunctive premise entail something that isn’t argued for. For instance, suppose someone is trying to get you to sign up for a subscription, and you are skeptical of the offer. They argue:

1. Either you want to pay full price, or you want to save 50% by signing up now.
2. You don’t want to pay full price.
3. You want to save 50% by signing up now (1, 2 DS)

The argument is valid, but the argument presupposes that you’re going to sign up one way or another, either for full price or for 50% off, something that you aren’t persuaded of. The salesperson ignores that you have the option to spend no money at all, and if you reject the offer, they might even object: why in the world would you want to pay full price?

---

Submodule 5.3 Quiz

Licenses and Attributions

Key Sources:

• Watson, Jeffrey (2019). Introduction to Logic. Licensed under: (CC BY-SA).

Next Page: 5.4 Practice with Conjunctions and Disjunctions