The Logic Café
|
| P | Q | P&Q | PvQ | P>Q | P=Q | ~P | |
| row one: | T | T | T | T | T | T | F |
| row two: | T | F | F | T | F | F | F |
| row three: | F | T | F | T | T | F | T |
| row four: | F | F | F | F | T | T | T |
The language SL consists of an infinite number of sentences. How can one tell? To begin with, there is an unending collection of atomic sentences from which to construct molecular sentences! You have 'A', 'B', 'C', and so forth (except 'V'), but also each of these uppercase letters with a numerical subscript also counts: 'A1', 'B17', 'L229', and so on without end.
All these SL sentences are without meaning until we give them an "interpretation". We might do so by associating them with English sentences as we did in chapter two. This is usually the most valuable means of interpreting. So, 'J' may stand for "Lyndon Johnson was US president in 1965". (Or, one might pick 'P1965'.) Either way, we have assigned an atomic sentence a meaning making it true.
An interpretation must do at least this: it assigns a truth value (true or false) to atomic sentences of SL. In this chapter, we will be interested in one particular kind of interpretation: a truth value assignment.
A truth value assignment is an association of a single truth value with every atomic sentence.
So, a truth value assignment provides an infinite number of truth values, one for each atomic sentence. You can imagine starting:
(*) 'A' has value true, 'B' has value false, 'C' has value ...
But of course you can't finish: there is an unending list of atomic sentences including those with subscripts.
Fortunately, practical purposes require only a partial assignment of truth values. For example, we may be asked whether or not 'Av~B' is true. To decide requires an interpretation: we need to know a bit about what 'A' and 'B' mean. The easiest way to do this is to have something like (*) in hand: Enough of a truth value assignment to determine a truth value for the molecular sentence in question.
Because our connectives are truth functional, we can tell the truth value of 'Av~B' just from the partial truth value assignment (*). (It's true, right?)
A partial truth value assignment associates a truth value (true or false) with some of the atomic sentences of SL.
You will have become familiar with partial truth value assignments from the tutorials even though we did not often use the name "partial": A row of a truth table represents a partial truth value assignment. We use a table to work out the truth value of 'Av~B' like so:
| A | B | A | v | ~ | B | |
| T | F | T | T | T | F |
Here we represent the partial truth value assignment of (*): "T" is under 'A' and "F" under 'B' to give the association.
Recall from just above that (*) is an example truth value assignment; it makes 'A' true and 'B' false. Now, of course we don't have any reason to think that (*) is in any way the unique correct truth value assignment: We have no idea what 'A' and 'B' mean! Either could both be true or false. In fact there are four possible ways to assign truth values to 'A' and 'B'. You know them well by now: they are given by the rows of a truth table. For example, the table ...
| A | B | A | v | ~ | B | ||
| row one: | T | T | T | T | F | T | |
| row two: | T | F | T | T | T | F | |
| row three: | F | T | F | F | F | T | |
| row four: | F | F | F | T | T | F |
... has four rows, one for each way that truth values can be assigned to 'A' and 'B'.
If we just consider just the four partial truth value assignments, we get:
| A | B |
| T | T |
| T | F |
| F | T |
| F | F |
For 'Av~B' we need (for obvious reasons) to be concerned only with possible truth value assignments to 'A' and 'B'. These partial truth value assignments encapsulate all that is needed in any possible situation which could make 'Av~B' true. So, as far as SL is concerned,
The partial truth value assignments to the atomic components of a sentence (or set of sentences) represent all possibilities for that sentence (or set of sentences).
This is quite important for the logical concepts taken up in the next section.
If we add a third atomic sentence, then the number of rows doubles.
| A | B | C |
| T | T | T |
| T | T | F |
| T | F | T |
| T | F | F |
| F | T | T |
| F | T | F |
| F | F | T |
| F | F | F |
Notice how on this and all tables, the atomic sentence on the far right
is assigned 'T', 'F', 'T', 'F', and so fourth (read down the right column
under 'C'). The next from the right is assigned 'T','T','F','F' etc. doubling
the repetition. Then, comes 'T','T','T','T','F','F','F','F' doubling once
again. For a fourth sentence, the partial truth value assignments are
as follows:
| A | B | C | D |
| T | T | T | T |
| T | T | T | F |
| T | T | F | T |
| T | T | F | F |
| T | F | T | T |
| T | F | T | F |
| T | F | F | T |
| T | F | F | F |
| F | T | T | T |
| F | T | T | F |
| F | T | F | T |
| F | T | F | F |
| F | F | T | T |
| F | F | T | F |
| F | F | F | T |
| F | F | F | F |
This pattern emerges:
| Number of Atomic Components: | Number of Rows: |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| n | 2n |
A fuller account is given in the tutorials.
(Chapter One) A sentence is logically true if and only if it could not possibly be false.
In SL, as we have just seen, we have a better handle on the possibilities. They are the truth value assignments. Thus, in the context of SL, we can think about logical truth of a sentence as meaning "no truth value assignment makes it false".
A sentence of SL in logically true in SL if and only if it is false on no truth value assignment.
Then notice that 'Av~B' is NOT logically true in SL: it can be false, i.e., in row three:
| A | B | A | v | ~ | B | ||
| row one: | T | T | T | T | F | T | |
| row two: | T | F | T | T | T | F | |
| row three: | F | T | F | F | F | T | |
| row four: | F | F | F | T | T | F |
To move from the old, chapter one definitions in terms of "possibility" to the SL version, we need to keep this idea in mind:
Something is possible if it happens in some possibility. For SL the possibilities are the truth value assignments.
So, "could possibly be false" comes to "is false in a truth value assignment". We use this to provide special SL version for all our logical concepts involving possibility.
For another example,
(Chapter One) A sentence is logically false if and only if it could not possibly be true.
comes toA sentence of SL is logically false in SL if and only if it is true on no truth value assignment.
Furthermore,A sentence of SL is logically indeterminate in SL if and only if it is true on some truth value assignment and false on some truth value assignment.
More importantly, our definitions of validity and of logical equivalence can now be given precise form without reference to "possibilities".
(Chapter One) An argument is valid
just in case it is not possible that its conclusion be false and its premises
all be true.
An argument is invalid
if and only if it is not valid.
We can now restate this for SL sentences as:
An argument is valid in SL just
in case there is no truth value assignment on which its conclusion is
false and its premises are all true.
An argument is invalid
in SL if and only if it is not valid in SL, i.e., if and only if
there is a truth value assignment on which its premises are true and its
conclusion is false.
Similarly,
The two members of a pair of SL sentences are logically equivalent in SL if and only if there is no truth value assignment on which one of the pair is true while the other is false.
And...
A set of SL sentences is logically consistent
in SL if and only if there is some truth value assignment on which
all members of the set are true.
A set of SL sentences is logically inconsistent
in SL if and only it it is not logically consistent, i.e., there
is no truth value assignment on which all members of the set are true.
Each of the definitions above corresponds to a truth table test. Just remember that the rows correspond to the (partial) truth value assignments. For instance, to test for the validity of the argument
~A>C
C
~A
We could look to see if there is a (partial) truth value assignment making the premises true and conclusion false. If there is, then it's invalid in SL by the definition just given.
| A | C | ~ | A | > | C | , | C | / | ~ | A | |
| T | T | F | T | T | T | T | F | T | |||
| T | F | F | T | T | F | F | F | T | |||
| F | T | T | F | T | T | T | T | F | |||
| F | F | T | F | F | F | F | T | F |
Again, the columns under main connectives are in bold. So, you notice that on the very first row, the premises are both true and the conclusion false. Thus the argument is not valid in SL, it's invalid.
Finally, you are allowed to do short-cut truth tables when they are sufficient to prove your point. That is, if doing only some of the rows is enough to determine that an argument has or lacks a certain property you need to know about, then you need only do those rows.
For example, to show that our argument
~A>C
C
~A
is NOT valid, we need only point to the first row of the above table. Or, if you want, you may simply show this one row in a short-cut table:
| A | C | ~ | A | > | C | , | C | / | ~ | A | |
| T | T | F | T | T | T | T | F | T |
and ignore the other rows. This is enough, because the definition of "invalid" merely requires a single truth value assignment making all premises true and conclusion false.
To reiterate,
Short-cut Tables: One may give an incomplete table and give only some of a table's rows to support a claim if but only if these rows are enough to determine that the claim is true.
If in doubt, just do the whole table.
Back to chapter three
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