§1. Negation, Conjunction, and Alternation

Negation

A “statement” can be affirmed or denied.

Denying a statement is the same as affirming another statement.

For example, if you denied the previous statement you would be affirming the following:

Denying a statement is NOT the same as affirming another statement.

Sometimes it is more complicated to figure out what the “negation” of a statement is, but with a little effort the negation of a statement can be figured out and worded clearly.

In logic, the symbols “-” and “~” are signs of negation. So

~(I am in Cafe Doma)

could be rewritten as

It is not the case that I am in Cafe Doma.

The parenthesis work just like they do in math: to indicate that something should be dealt with as a whole.

Sometimes in logic “it is not the case” is shortened to just “not”. And statements are often represented with letters, especially starting with “p”.

So “~p” should be read in your mind as “it is not the case that p”, or just “not p”.

Note that sometimes a sign of negation is written above a letter rather than in front of it.

Conjunction

The “conjunction” of multiple statements forms a single statement. This is represented by writing each statement next to the others. For example,

(I am in Cafe Doma)(Lindsey is next to me)(It is 40 degrees Fahrenheit)

could be written as

I am in Cafe Doma, Lindsey is next to me, and it is 40 degrees Fahrenheit.

If each statement in the above conjunction were represented by a letter, it would be written as

pqr

After explaining all this Quine says,

The meanings of negation and conjunction are summed up in these laws. The negation of a true statement is false; the negation of a false statement is true; a conjunction of statements all of which are true is true; and a conjunction of statements not all of which are true is false.

- p. 2

Note that, just like mathematical addition and multiplication, the “associative” property holds true of conjunctions. This means that internal grouping is irrelevant. For example

2 + (2 + 2) = 6,
(2 + 2) + 2 = 6,
2 + 2 + 2 = 6, and
(2 + 2 + 2) = 6

all express the same statements.

Similarly, note that, just like addition and multiplication, conjunctions are “commutative”. This means that internal order is irrelevant. For example

1 + 3, and
3 + 1

express the same statements.

But, unlike multiplication and addition, logical conjunctions of statements are “idempotent”. All this means is that conjoining a statement to itself is unnecessary. For example

(I am in Cafe Doma)(I am in Cafe Doma)

or, in everyday language

I am in Cafe Doma, and I am in Cafe Doma

is simply redundant. But adding a number to itself is meaningful: 2 + 2 = 4.

Revolutionary, no doubt.

So “negation” is like “not”, and “conjunction” is like “and”. Next, Quine looks at “or”, and calls it “alternation”.

Alternation

In everyday language “or” can be ambiguous. In logic, there is a distinction be an exclusive "or", and a nonexclusive "or".

An exclusive "or" is like saying

Either p or q, but not both.

Whereas a nonexclusive "or" is like saying

Either p or q or both.

The exclusive "or" was represented in Latin with the word "aut", and the nonexclusive "or" by the word "vel".

"Alternation" references only the nonexclusive "or". It is sometimes simply written as "v"; reminiscent of it's Latin term.

Quine proposes to the reader that the nonexclusive use of the term be understood when he writes the ambiguous English "or".

So an alternation is true if at least one of its member statements is true.

In order to achieve an exclusive "or", one can write it out

p or q but not both

or

p~q v ~pq

This setup is also often written

p≡~q

Alternation is associative, cummutative, and indempotent like conjunction.

Combining negation, conjunction, and alternation provides for the ability to quickly write complex statements, but you have to be careful to think clearly in order to sort out the truth values of statements that combine conjunction and alternation.

On page 7 Quine lays out something helpful:

1. ~(pq): It is not the case that both p and q.
2. ~p~q: It is not the case that p and it is not the case that q.
3. ~(p v q): It is not the case that either p or q.
4. ~p v ~q: It is not the case that p or it is not the case that q.

(I added the numbers). He notes that 1 and 4 are the same statement, and 2 and 3 are the same statement.