﻿ "And"

# Conjunction

Let P and Q be statements, each of which is either true or false (but neither is both; there are weird statements that are both---"this statement is false"). "P and Q" is a compound statement, which is true only when both P and Q are true (and false in all other cases). Just to make it VERY official, here is a table which lists every possible combination of truth and falseness for P and Q and states whether "P and Q" is true for each possible combination. I apologize for stating the obvious! In the tables, below, T stands for true, and F stands for false.

Truth Table For conjunction
P Q P and Q
T T T
T F F
F T F
F F F

Consider one of the rows of this table, say the third. That row represents this situation: P is false and Q is true; in this situation "P and Q" is false.

What is the truth of "Q and P"? What difference does it make to "commute" the two statements P and Q by putting Q ahead of P? Having both P and Q true has nothing to do with which is stated first (please forgive this review of what may seem obvious). It makes no difference, which of P and Q is stated first in a conjunction. "P and Q" MEANS EXACTLY THE SAME AS "Q and P"; the two compound statements are true in exactly the same situations. Again, to make it very official, we display this sameness of truth values in the table below.

Conjunction Is Commutative
P Q P and Q Q and P
T T T T
T F F F
F T F F
F F F F

What is the truth of "P and P"? If P is true, both conjuncts of this conjunction are true; by our understanding of conjunction, the compound statement is true. If P is false, both conjuncts of this conjunction statement are false, and thus the compound statement is false. Thus, "P and P" MEANS THE SAME THING AS "P". It may be redundant, even boring, but it's true. Surprisingly, it is useful as well. Sometimes one wants to remove redundancy to make shorter but equivalent statements. Sometimes one wants to introduce redundancy as part of a transformation into something that IS interesting. The following table is intended to make this observation quite official. Note that this table has only two rows because there are only two situations to consider: P is true or P is false.

Redundancy with conjunction
P P P and P
T T T
F F F
Consider, finally, a conjunction within a conjunction: "(P and Q) and R". Here P, Q, and R are statements (each one is either true or false, but not both). The parentheses indicate that "P and Q" is compounded with a third statement R (think of "P and Q" being evaluated first for truthfulness, and then R is taken into account). Inevitably, someone will create the statement "P and (Q and R)" [I just did]. By our understanding, of "and", "(P and Q) and R" should mean the same as "P and (Q and R)"; in fact both are true exactly when all three of P, Q, and R are true (and false in all other situations). There are 8 situations to consider, T or F for each of P, Q, and R in all possible ways. These are listed in the table below, which lists the truth values of "(P and Q) and R".

Conjunction Is Associative, part 1
P Q R P and Q (P and Q) and R
T T T T T
T T F T F
T F T F F
T F F F F
F T T F F
F T F F F
F F T F F
F F F F F
Let us review one row of this table, say the fifth. There, P is false, Q is true and R is true. Thus "P and Q" is false; from that, it follows that "(P and Q) and R" is false because R is being conjoined with a statement which is false. Note that "(P and Q) and R" is true only in the first situation, when all of P, Q, and R are true.

We now repeat this table for "P and (Q and R)":

Conjunction Is Associative, part 2
P Q R Q and R P and (Q and R)
T T T T T
T T F F F
T F T F F
T F F F F
F T T T F
F T F F F
F F T F F
F F F F F

Again, let us review the fifth row of this table. Because Q and R are true there, "Q and R" is true; however, P is false; this makes "P and (Q and R)" be false because "Q and R" is being conjoined with a false statement. Again, note that "P and (Q and R)" is true only in the first situation when all three of P, Q and R are true.

Some Questions To Test Understanding.
• How many different compound statements can be made from "(P and Q) and R" by permuting the letters P, Q, and R? How many more can be made by permuting the letters P, Q, and R and moving the parentheses to enclose the last two statements of the compound expression? Do the compound statements created by permutation and/or moving parentheses all mean the same thing? Try to list all the possibilities in an order in which each one is obtained from the previous one by one of two actions: commuting two statements in an "and" or by moving parenthesis.
• Consider four statements "and"-ed: "(P and Q) and (R and S)". List all the possible equivalent statements obtained by moving parentheses and/or permuting the letters P, Q, R and S. How many are there? Do they all mean the same thing?
• Suppose that 27 statements are "and"-ed in some order, with parentheses placed so that each "and" connects only two statements (possibly compound themselves). When is this compound statement true?