21 Validity and Soundness37
Recall that a sentence is a meaningful expression that can be true or false. The sentence ~~~D is true if and only if the sentence ~~D is false, and so on through the structure of the sentence until we arrive at the atomic components:
~~~D is true if and only if the atomic sentence D is false.
A “well-formed formula” (wff) like (Q&R) must be surrounded by parentheses, because we might apply the definition again to use this as part of a more complicated sentence. If we negate (Q&R), we get ~(Q&R). If we just had Q&R without the parentheses and put a negation in front of it, we would have ~Q&R. It is most natural to read this as meaning the same thing as (~Q&R), something very different than ~(Q&R). The sentence ~(Q&R) means that it is not the case that both Q and R are true; Q might be false or R might be false, but the sentence does not tell us which. The sentence (~Q&R) means specifically that Q is false and that R is true. As such, parentheses are crucial to the meaning of the sentence.
So, strictly speaking, Q&R without parentheses is not a sentence of SL (Sentential Logic). When using SL, however, we will often be able to relax the precise definition so as to make things easier for ourselves. We will do this in several ways.
First, we understand that Q&R means the same thing as (Q&R). As a matter of convention, we can leave off parentheses that occur around the entire sentence.
Second, it can sometimes be confusing to look at long sentences with many, nested pairs of parentheses. We adopt the convention of using square brackets `[' and `]' in place of parenthesis. There is no logical difference between (P vQ) and [P v Q], for example. The unwieldy sentence
(((H & I) v (I & H))&(J v K))
could be written in this way:
(H & I) v (I & H)
&(J v K)
Third, we will sometimes want to translate the conjunction of three or more sentences. For the sentence `Alice, Bob, and Candice all went to the party', suppose we let A mean `Alice went', B mean `Bob went', and C mean `Candice went.' The definition only allows us to form a conjunction out of two sentences, so we can translate it as (A&B)&C or as A&(B &C). There is no reason to distinguish between these, since the two translations are logically equivalent. There is no logical difference between the first, in which (A&B) is conjoined with C, and the second, in which A is conjoined with (B &C). So we might as well just write A&B &C. As a matter of convention, we can leave out parentheses when we conjoin three or more sentences.
Fourth, a similar situation arises with multiple disjunctions. `Either Alice, Bob, or Candice went to the party' can be translated as (AvB)vC or as Av(BvC). Since these two translations are logically equivalent, we may write A v B v C. These latter two conventions only apply to multiple conjunctions or multiple disjunctions. If a series of connectives includes both disjunctions and conjunctions, then the parentheses are essential; as with (A&B) v C and A&(B v C). The parentheses are also required if there is a series of conditionals or biconditionals (which will be covered in the next chapter); as with (A ⊃ B) ⊃ C and A ↔ (B ↔ C).
We have adopted these four rules as notational conventions, not as changes to the definition of a sentence. Strictly speaking, AvB v C is still not a sentence. Instead, it is a kind of shorthand. We write it for the sake of convenience, but we really mean the sentence (A v (B v C)).
The connective that you look to first in decomposing a sentence is called the main logical operator of that sentence. For example: The main logical operator of ~(E v (F & G)) is negation, ~. The main logical operator of (~E v (F & G)) is disjunction, v.
This chapter introduces a way of evaluating sentences and arguments of SL. Although it can be laborious, the truth table method is a purely mechanical procedure that requires no intuition or special insight.
Any non-atomic sentence of SL is composed of atomic sentences with sentential connectives. The truth-value of the compound sentence depends only on the truth-value of the atomic sentences that comprise it. In order to know the truth-value of (D & E), for instance, you only need to know the truth-value of D and the truth-value of E. Connectives that work in this way are called truth-functional.
We write the variables we are using at the top and then we generate a table where we have all of the possible combinations of True and False for the variables. For example, when we just have A, we need to know what the value of the negation is when A is true or when A is false. What this means is that the statement A has the value of being either true or false. Here’s how this works: If I say that statement A is “Superman wears a cape” and this statement is true, then the negating this statement would make a new False statement “Superman does not wear a cape.” However, if I say that A is “The Incredibles wear capes” and that this statement is False, then negating it would make a new True statement, “The Incredibles do not wear capes.” So what this table means is that when A is True, its negation is False, and when A is False, its negation is True.
However, we hardly ever have arguments with only 1 statement. So when we have 2 statements, we need to have all of the possibilities of Truth for them. What this means (and you can see it in the table below) is that we have to make a table that shows us what the truth value of things are when A and B are both True, when A is True and B is False, when A is False and B is True, and when A and B are both False. Then we can add the connectives and see when they are true, given the assumed truth of A and B. For example, the conjunct (&) is only True when they are both True and the disjunct (v) is true when at least one of them is (and only False when they are both False).
A B A &B AvB
T T T T
T F F T
F T F T
F F F F
Now you should be able to some analyses on some basic sentences. For the following, assume that A, B, C are True and that X, Y, Z are False. Assuming these truth values, are the following sentences True or False?
1) ~X v Y
2) (A v Z) & B
3) ~A v (Z & ~X)
4)(A v Z) v ~(~(B & ~Z) & ~(C v ~Y))
Answers: 1) is T because ~X is True and it only takes one side of a disjunct to be True to make it all True; 2) is True; 3) is False; and 4) is True (hint: All you need to know is that A is True and it becomes really simple)
Is the following statement true?
The sky is blue or the moon is hot pink and dogs are not animals.
To figure this out, we need to symbolize it. I propose the following:
B = The sky is blue. P = The moon is hot pink. D = Dogs are animals.
B v P & ~D
But is it clear what this means? We need to use parentheses! But it is unclear where the parentheses go, and depending on where we put them it can be True (in the first case) or False (in the second).
B v (P & ~D)
(B v P) & ~D
Complete truth tables
The truth-value of sentences that contain only one connective is given by the characteristic truth table for that connective. To put them all in one place, the truth tables for the connectives of SL are repeated in the table.
The characteristic truth table for conjunction, for example, gives the truth conditions for any sentence of the form (A &B). Even if the conjuncts A and B are long, complicated sentences, the conjunction is true if and only if both A and B are true. Consider the sentence (H &I) vH. We consider all the possible combinations of true and false for H and I, which gives us four rows.
We then copy the truth-values for the sentence letters and write them underneath the letters in the sentence.
H I (H & I) v H
T T T T T
T F T F T
F T F T F
F F F F F
Now consider the subsentence H &I. This is a conjunction A &B with H as A and with I as B. H and I are both true on the first row. Since a conjunction is true when both conjuncts are true, we write a T underneath the conjunction symbol. We continue for the other three rows and get this:
H I (H & I) v H
T T T T T T
T F T F F T
F T F F T F
F F F F F F
The entire sentence is a disjunction AvB with (H &I) as A and with H as B. On the second row, for example, (H &I) is false and H is true. Since a disjunction is true when the either disjunct is True, we write a T in the second row underneath the disjunction symbol. We continue for the other three rows and get this:
H I (H & I) v H
T T T T T T T
T F T F F T T
F T F F T F F
F F F F F F F
The column of Ts underneath the conditional tells us that the sentence (H &I) v I is true whenever H is true, and the truth of I doesn’t determine the truth of the sentence. It is crucial that we have considered all of the possible combinations. If we only had a two-line truth table, we could not be sure that the sentence was not false for some other combination of truth-values.
Most of the columns underneath the sentence are only there for bookkeeping purposes. When you become more adept with truth tables, you will probably no longer need to copy over the columns for each of the sentence letters. In any case, the truth-value of the sentence on each row is just the column underneath the main logical operator of the sentence; in this case, the column underneath the conditional.
A complete truth table has a row for all the possible combinations of T and F for all of the sentence letters. The size of the complete truth table depends on the number of different sentence letters in the table. A sentence that contains only one sentence letter requires only two rows, as in the characteristic truth table for negation. This is true even if the same letter is repeated many times, as in the sentence [(C v C) & C]&~(C & C). The complete truth table requires only two lines because there are only two possibilities: C can be true or it can be false. A single sentence letter can never be marked both T and F on the same row. The truth table for this sentence looks like this:
C [( C vC )&C ] & ~ ( C &C )
T T T T T T F F T T T
F F T F F F F F F T F
Looking at the column underneath the main connective, we see that the sentence is false on both rows of the table; i.e., it is false regardless of whether C is true or false.
A sentence that contains two sentence letters requires four lines for a complete truth table, as in the characteristic truth tables and the table for (H &I) v I.
A sentence that contains three sentence letters requires eight lines. For example:
M N P M & (N v P)
T T T T T T T T
T T F T T T T F
T F T T T F T T
T F F T F F F F
F T T F F T T T
F T F F F T T F
F F T F F F T T
F F F F F F F F
From this table, we know that the sentence M &(N vP) might be true or false, depending on the truth-values of M, N, and P.
A complete truth table for a sentence that contains four different sentence letters requires 16 lines. Five letters, 32 lines. Six letters, 64 lines. And so on. To be perfectly general: If a complete truth table has n different sentence letters, then it must have 2n rows.
In order to fill in the columns of a complete truth table, begin with the rightmost sentence letter and alternate Ts and Fs. In the next column to the left, write two Ts, write two Fs, and repeat. For the third sentence letter, write four Ts followed by four Fs. This yields an eight line truth table like the one above.
For a 16 line truth table, the next column of sentence letters should have eight Ts followed by eight Fs. For a 32 line table, the next column would have 16 Ts followed by 16 Fs. And so on.
Tautologies, contradictions, and contingent sentences
An English sentence is a tautology if it must be true as a matter of logic. With a complete truth table, we consider all of the ways that the world might be. If the sentence is true on every line of a complete truth table, then it is true as a matter of logic, regardless of what the world is like.
So a sentence is a tautology in SL if the column under its main connective is T on every row of a complete truth table. Conversely, a sentence is a contradiction in SL if the column under its main connective is F on every row of a complete truth table.
A sentence is contingent in SL if it is neither a tautology nor a contradiction; i.e. if it is T on at least one row and F on at least one row.
Two sentences are logically equivalent in English if they have the same truth value as a matter logic. Once again, truth tables allow us to define an analogous concept for SL: Two sentences are logically equivalent in SL if they have the same truth-value on every row of a complete truth table. Consider the sentences ~(A v B) and ~A&~B. Are they logically equivalent? To find out, we construct a truth table.
A B ~ (A v B) ~ A & ~ B
T T F T T T F T F F T
T F F T T F F T F T F
F T F F T T T F F F T
F F T F F F T F T T F
Look at the columns for the main connectives; negation for the first sentence, conjunction for the second. On the first three rows, both are F. On the final row, both are T. Since they match on every row, the two sentences are logically equivalent.
An argument in English is valid if it is logically impossible for the premises to be true and for the conclusion to be false at the same time. In other words, an if the premises are true, then the conclusion must also be true. An argument is valid in SL if there is no row of a complete truth table on which the premises are all T and the conclusion is F; an argument is invalid in SL if there is such a row. Consider this argument:
~L v (J & L)
Is it valid? To find out, we construct a truth table
J L ~ L v (J & L) L J
T T F T T T T T T T OK!
T F T F T T F F F T
F T F T F F F T T F
F F T F T F F F F F
Yes, the argument is valid. The only row on which both the premises are T is the first row, and on that row the conclusion is also T.
What about this one?
~L v(J v L)
Is it valid? To find out, we construct a truth table.
J L ~ L v (J v L) L J
T T F T T T T T T T OK!
T F T F T T T F F T
F T F T T F T T T F Invalid!
F F T F T F F F F F
No, the argument is not valid. There are two rows where both premises are True. In one of them, the conclusion is also True. However, when J is False and L is True (row 3), the premises are True and the conclusion is False, making it invalid.
Soundness is the easiest concept to understand, provided you understand validity. Logic is all about structure of arguments and determining validity since that’s all we can do: ensure that the reasoning we are using is proper and actually takes us to the conclusions we want. Valid arguments can be wholly uninteresting, but at least their reasoning is solid. For example, this is valid:
If you eat a sandwich, then you will turn purple.
You ate a sandwich.
Therefore, you will turn purple.
It’s valid…but who cares? What we really care about in real-life is soundness. An argument is sound when it is valid and all of the premises are actually true. This means you have just learned something new and real that we care about. For example,
Either you will understand this or you will re-read it.
If you re-read this, then you will understand it.
Therefore, you will understand this.
This is valid, and it is actually true, since I assume you will re-read it if you didn’t catch it the first time. Here is its symbolization so you can check it for validity yourself:
U v R
R > U