Contents: Section 1: Truth Value Assignments and Truth Tables; Section 2: Possibilities and Full Tables; Section 3: Logical Concepts for SL Section 4 : Symbolizing Complex English Sentences
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In this chapter we construct truth tables which allow us to fully analyze the meaning of SL sentences. In turn, these tables will allows us to give very precise definitions of the logical concepts first presented in chapters one and two (e.g., the concept of validity).
By now you should have memorized the following truth table definitions for our connectives.
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 |
When we've mastered the analysis of SL sentences, we'll move back to the relationship to English: This is section 4 on Symbolization
The language SL consists of an infinite number of sentences. How can one tell? To begin with, there is the collection of atomic sentences from which to construct molecular sentences. You have 'A', 'B', 'C', and so forth (except 'V'). We can also use these as predicates, add a name, and write something like 'Da' or 'Ag', etc. But most of this chapter will be concerned with the simpler version of our lang ague without names. Then you may take these atomic sentences and make longer and longer molecular sentences. There is no end to it.
All 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.
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.
(If there are names and predicates in the language, this gets a little more complicated.)
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.
This pattern emerges:
Number of Atomic Components: | Number of Rows: |
---|---|
1 | 2 |
2 | 4 |
3 | 8 |
4 | 16 |
n | 2 ^{n} |
Every time an additional atomic sentence is added to
(Chapter Two) 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,
A sentence is self-contradictory (or as it's sometimes called logically false) if and only if it could not possibly be true.
A sentence of SL is self contradictory in SL if and only if it is true on no truth value assignment.
A sentence of SL is contingent (or as it's sometimes called 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.
Private schools implementing a voucher program will fail to provide equal educational opportunities across a community if they either skim off the best students and leave the poorer ones behind or they charge parents fees beyond what is paid for in private funds and so exclude children of poorer families.
Is this a conditional sentence? Or a disjunction? There are no parentheses to help us. But there are "grouping" words in the English. For instance, what goes between "either" and "or" will be a first disjunction and the word either works like a left hand parenthesis. Here's a reformulation.
Private schools implementing a voucher program will fail to provide equal educational opportunities across a community if [either (they skim off the best students and leave the problem students behind) or they exclude children of poorer families by charging parents fees beyond what is paid for in private funds].
This sentence is now more pretty clearly a conditional with consequent "Private schools implementing a voucher program will fail to provide equal educational opportunities". The antecedent is more complicated.
A hybrid formulation may help:
~P if [(S&L) or E]
(Here we've used P: "Private schools implementing a voucher program provide equal educational opportunities", S: "Private schools implementing a voucher program skim off the best students", L: "Private schools implementing a voucher program leave the problem students behind", E: "Private schools implementing a voucher program exclude...") But we recall that 'P if Q' is symbolized as 'Q>P', so '~P' is the consequent and we should symbolize the whole English sentence as:
[(S&L)vE]>~P
The following table summarizes some of the mechanisms English uses to group.
When one has a sentence of form: | Then: | Example (Hybrid): | Symbolization: |
---|---|---|---|
If P, then Q |
Antecedent is P | If A and B, then C. | (A&B)>C |
Either P or Q | First disjunct is P | Either both A and B, or C. | (A&B)vC |
Both P and Q | First conjunct is P | Both if A then B, and C. | (A>B)&C |
One other good indicator of grouping in sentences is a comma (or semicolon). These often mark a major structural division in the sentence corresponding to the main connective. So...
If Private schools implementing a voucher program succeed, then their students will benefit directly and traditional public schools will benefit from the competition.
Which might have the hybrid form...
If S, then D and C.
...and, because of the comma marking the main connective, would have symbolization as follows:
S>(D&C)
Now, make sure you have carefully read T4.2 and do lots of exercises. Experience is the key to symbolization.