# 1.1.10: Classifying propositions

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

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 ¬Pis 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?

1. 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.

2. 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.

1.1.10: Classifying propositions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.