12 2. Background

Exercise 2.1.3. Negate the following statements.

(i) 56894323 is a prime number.

(ii) If there is no coffee, I drink tea.

(iii) John watches but does not play.

(iv) I will buy the blue shirt or the green one.

Exercise 2.1.4. Write the following statements using standard log-

ical symbols.

(i) ϕ if ψ.

(ii) ϕ only if ψ.

(iii) ϕ unless ψ.

As an aside, let us have a brief introduction to truth tables. These

are nothing more than a way to organize information about logical

statements. The leftmost columns are generally headed by the indi-

vidual propositions, and under those headings occur all possible com-

binations of truth and falsehood. The remaining columns are headed

by more complicated formulas that are built from the propositions,

and the lower rows have T or F depending on the truth or falsehood

of the header formula when the propositions have the true/false val-

ues in the beginning of that row. Truth tables aren’t particularly

relevant to our use for this material, so I’ll leave you with an example

and move on.

ϕ ψ ¬ϕ ¬ψ ϕ & ψ ϕ ∨ ψ ϕ → ψ ϕ ↔ ψ

T T F F T T T T

T F F T F T F F

F T T F F T T F

F F T T F F T T

If we stop here, we have propositional (or sentential) logic. These

formulas usually look something like [A∨(B&C)] → C and their truth

or falsehood depends on the truth or falsehood of the assertions A,

B, and C. We will continue on to predicate logic, which replaces

these assertions with statements such as (x 0) & (x + 100 0),

which will be true or false depending on the value substituted for the

variable x. We will be able to turn those formulas into statements