Chapter 21, Occurrence Tables
The Special Usage of Three Words: not, and & or
Given a situation A, we can talk about the negative
situation not A. Given a situation A and another
situation B, we may talk about two further situations
-
A and B (conjunction), and
-
A or B (inclusive or).
The meanings of the terms or phrases are explained next.
NOT A and NOT (NOT A)
Given a single situation A, we can speak of another situation NOT A. The
situation NOT A is said to occur when the situation A does not occur.
Further, the situation NOT A is said not to occur when the situation A
occurs. This is summarized in the following table.
|
row
|
A
|
NOT (A)
|
|
1
|
occurs
|
occurs not
|
|
2
|
occurs not
|
occurs
|
Language note: a situation A is said to be true when it
occurs and not true (false) when it does not occur.
The following table
|
row
|
A
|
NOT A
|
NOT (NOT A)
|
|
1
|
occurs
|
occurs not
|
occurs
|
|
2
|
occurs not
|
occurs
|
occurs not
|
shows that the situation NOT (NOT A) occurs when A occurs and that the
situation NOT (NOT A) does not occur when A does not occur. This suggests
that the situation A is equivalent to the situation NOT (NOT A).
The Special Usage of Three Words (Continued)
The word AND
The situation A and B is said to occur if both
situations A and B occur. Otherwise, it is said not to
occur. See the table below.
|
row
|
situation A
|
situation B
|
A and B
|
|
1
|
occurs
|
occurs
|
occurs
|
|
2
|
occurs
|
occurs not
|
occurs not
|
|
3
|
occurs not
|
occurs
|
occurs not
|
|
4
|
occurs not
|
occurs not
|
occurs not
|
The situation A and B occurs provided
rows 2, 3 and 4 in the above never occur.
In each row, a possible combination of the occurrence or nonoccurrence of
the situations A and B is shown in the middle two columns. In the last
column, we put a note to say whether or not, the situation A and
B occurs or occurs not.
* Language Note. The phrase A and B is
also labelled (called) the conjunction of the situations A and B. The
situation A and B is said to be true when and only when
both the situations A and B occur (= are true).
The Special Usage of Three Words
The At-Least-One-Usage of the word OR
In everyday speech when you use the word or in a phrase
like John or Andrew will go to the store, the usual
expectation is that only one will go, not both. But there is another use
of the word or favored in logic. The word
or is employed in the at least one
sense (as is done in logic and mathematics). With this sense or
usage, the previous phrase is understood in the inclusive sense:
John or Andrew, or both, will go to the store. We now
proceed and we will use the word or in the at
least one sense.
The situation (A or B ) is said to occur if at least one
of the two situations A and B occurs.
Otherwise, it is said not to occur. This is summarized in the following
table.
|
row
|
situation A
|
situation B
|
A or B
|
|
1
|
occurs
|
occurs
|
occurs
|
|
2
|
occurs
|
occurs not
|
occurs
|
|
3
|
occurs not
|
occurs
|
occurs
|
|
4
|
occurs not
|
occurs not
|
occurs not
|
The situation A or B can be said to occur
provided the situation in row 4 does not occur.
Remember the at least one usage differs from the exactly
one usage of A or B which means either A or
B occurs, but not both. In contrast, in the at least
one usage, A or B means either
A or B occurs, or both.
We have to be careful with the word or. Its meaning
depends on the speaker and possibly the listener. That is, confusion and
ambiguity results when two people in question use the same words but give
them different meanings. To eliminate this ambiguity in everyday speech,
write and say one of the following:
-
A or B, or both,
-
A and/or B
-
A or B, but not both.
When listening, you will have to ask what is meant. Legal texts use the
phrase A and/or B to signal that at least one of the two
cases A and B can occur.
2. Occurrence Table for One-Way Implications
Any rule which can be stated in the form if a first situation A
occurs, then a second situation B occurs, in brief, if A
then B or A implies B, is called a one-way
implication.
A one-way implication which is never disobeyed is said to hold and to be
(always) true. For a one-way implication rule if A then
B, we recall the following:
- The rule is obeyed when both situations occur.
- The rule is not disobeyed when the first situation A does not occur
but the second B occurs.
- The rule is not disobeyed when the first situation A does not occur
and also the second situation B does not occur.
- The rule is disobeyed if the first situation A occurs but the second
situation B does not.
The last two items 3 and 4 can be summarized by saying that disobeying a
one-way implication rule is impossible when the first situation A does
not occur. When not disobeyed, the rule is said to be obeyed by
default. The following table, an occurrence table for the
one-way implication rule if A then B, summarizes what
has been said.
|
row
|
situation A
|
situation B
|
if A then B
|
|
1
|
occurs
|
occurs
|
obeyed
|
|
2
|
occurs
|
occurs not
|
disobeyed
|
|
3
|
occurs not
|
occurs
|
not
disobeyed
|
|
4
|
occurs not
|
occurs not
|
not
disobeyed
|
In each row, a possible combination of the occurrence or non-occurrence
of the situations A and B is shown in the middle two columns. In the last
column, we put a note to say whether or not the if-then rule is obeyed,
disobeyed, or not disobeyed.
Row 2 represents the situation in which A occurs but B does not. Observe
that in this situation, the rule is disobeyed. In the situations
represented by the other three rows, the rule is not disobeyed. A one-way
implication rule if A then B is said
-
to be always true,
-
to always hold
when it is never disobeyed. The one-way implication if A then
B is always true when the situation described in row 2 in the
above table never occurs.
Remark. If situation A never occurs,
the implication rule if A then B is never disobeyed amd
it is said to be vacuously true.
3 Occurrence Table for Two-Way Implication Rules
A rule which can be stated, or restated, in the form
The first situation A occurs when and only when the
second situation B occurs
or in the form
The first situation A occurs if and only if the second
situation B occurs
is called a two-way implication rule. For each two-way implication rule
note that:
-
The rule is obeyed when both situations occur.
-
The rule is disobeyed when the first situation A
occurs without the second situation B occurring.
-
The rule is disobeyed when the second situation B occurs without the
first situation A.
- In brief, the two situations in a two-way implication rule must both
occur or both must not occur, for the rule to be not
disobeyed.
The next table summarizes the above remarks for any two-way implication
rule A if and only if B.
|
row
|
situation A
|
situation B
|
A if and only if B
|
|
1
|
occurs
|
occurs
|
obeyed
|
|
2
|
occurs
|
occurs not
|
disobeyed
|
|
3
|
occurs not
|
occurs
|
disobeyed
|
|
4
|
occurs not
|
occurs not
|
not disobeyed
|
As said before, a two-way implication rule is said to be always true when
it is never disobeyed. This requires that the situations in rows 2 and 3
of the above table do not occur. That is, the above two-way implication
rule A iff B is true (never disobeyed) provided neither
of the situations A and B occurs without the other.
Converses for One-Way Implications
A Definition
The converse to the implication rule if A then
B is the rule if B then A. Note that
interchanging the first and second situation A and
B yields the converse to a rule. From this definition or
perspective, we see that the converse of the converse is the original
rule. Check this.
When we know a rule if A then B is never disobeyed, we
have no guarantee that the converse rule if B then A is
never disobeyed. The reason for this is as follows. The rule if A
then B is true if the situation A never occurs without the
situation B. The converse rule if B then A is true if
the situation B cannot occur without the situation A.
Reminder. Now we can easily answer the following
question: What can we say for sure about the event A when (i) the rule
if A then B is never disobeyed, and (ii) the event B
occurs? Your answer should be not much, or nothing, without further
information.
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