# 2.1.10: 2.1.J Classifying propositions

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Certain types of proposition will play a special role in our further work with logic. In particular, we define tautologies, contradictions, and contingencies as follows:

Definition 2.4.

A compound proposition is said to be a **tautology** if and only if it is true for all possible combinations of truth values of the propositional variables which it contains. A compound proposition is said to be a **contradiction** if and only if it is false for all possible combinations of truth values of the propositional variables which it contains. A compound proposition is said to be a **contingency** if and only if it is neither a tautology nor a contradiction.

For example, the proposition ((*p *∨ *q*) ∧ ¬*q*) → *p *is a tautology. This can be checked 2with a truth table:

The fact that all entries in the last column are true tells us that this expression is a tautology. Note that for any compound proposition *P*, *P *is a tautology if and only if ¬*P*is a contradiction. (Here and in the future, I use uppercase letters to represent compound propositions. *P *stands for any formula made up of simple propositions, propositional variables, and logical operators.)

Logical equivalence can be defined in terms of tautology:

Definition2.5.

Two compound propositions, *P *and *Q*, are said to be **logically equivalent** if and only if the proposition *P *↔ *Q *is a tautology.

The assertion that *P *is logically equivalent to *Q *will be expressed symbolically as “*P *≡ *Q*”. For example, (*p *→ *q*) ≡ (¬*p *∨ *q*), and *p *⊕ *q *≡ (*p *∨ *q*) ∧ ¬(*p *∧ *q*).

What if *P *→ *Q *and *P *is false? From a false premise we can derive any conclusion (check the truth table of →). So if *k *stands for “I’m the King of the Netherlands”, then *k *→ *Q *is true for any compound proposition *Q*. You can substitute anything for *Q*, and the implication *k *→ *Q *will hold. For example, it a logically valid deduction that: If I’m the King of the Netherlands, then unicorns exist. Taking this further, from a contradiction we can derive any conclusion. This is called the **Principle of Explosion**.

**Exercises**

1. Give the three truth tables that define the logical operators ∧, ∨, and ¬.

2. Some of the following compound propositions are tautologies, some are contradictions, and some are neither (i.e.,, so are contingencies). In each case, use a truth table to decide to which of these categories the proposition belongs:

a) (*p *∧ (*p *→ *q*)) → *q *b) ((*p *→ *q*) ∧ (*q *→ *r*)) → (*p *→ *r*)

c) *p *∧ ¬*p *d) (*p *∨ *q*) → (*p *∧ *q*)

e) *p *∨ ¬*p* f) (*p *∧ *q*) → (*p *∨ *q*)

3. Use truth tables to show that each of the following propositions is logically equivalent to *p *↔ *q*.

a) (*p *→ *q*) ∧ (*q *→ *p*) b) ¬*p *↔ ¬*q*

c) (*p *→ *q*) ∧ (¬*p *→ ¬*q*) d) ¬(*p *⊕ *q*)

4. Is → an associative operation? This is, is (*p *→ *q*) → *r *logically equivalent to *p *→ (*q *→ *r*)? Is↔ associative?

- Let
*p*represent the proposition “You leave” and let*q*represent the proposition “I leave”. Ex- press the following sentences as compound propositions using*p*and*q*, and show that they are logically equivalent:a) Either you leave or I do. (Or both!) b) If you don’t leave, I will.

- Suppose that
*m*represents the proposition “The Earth moves”,*c*represents “The Earth is the centre of the universe”, and*g*represents “Galileo was falsely accused”. Translate each of the following compound propositions into English:

a) ¬*g *∧ *c *b) *m *→ ¬*c*

c) *m *↔ ¬*c *d) (*m *→ *g*) ∧ (*c *→ ¬*g*)

7. Give the converse and the contrapositive of each of the following English sentences:

a) If you are good, Sinterklaas brings you toys.

b) If the package weighs more than one kilo, then you need extra postage.

c) If I have a choice, I don’t eat courgette.

8. In an ordinary deck of fifty-two playing cards, for how many cards is it truea) that “This card is a ten and this card is a heart”?

b) that “This card is a ten or this card is a heart”?

c) that “If this card is a ten, then this card is a heart”?

d) that “This card is a ten if and only if this card is a heart”?

9.Define a logical operator ↓ so that *p *↓ *q *is logically equivalent to ¬(*p *∨ *q*). (This operator is usually referred to as ‘nor’, short for ‘not or’.) Show that each of the propositions ¬*p*, *p *∧ *q*,*p *∨ *q*, *p *→ *q*, *p *↔ *q*, and *p *⊕ *q *can be rewritten as a logically equivalent proposition that uses↓ as its only operator.

10. For our proof that {¬, ∨} is functionally complete, we need to show that all formulas in pro- positional logic can be expressed in an equivalent form using only {¬, ∧, ∨, →, ↔}.

a) How many unique truth tables exist for formulas containing two atoms?

b) Create a function for each of the possible truth tables that uses only the 5 operators listed

above.

c) Give an (informal) argument why this means all formulas in propositional logic can be

expressed using only these five operators.