by Corry Shores
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[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]
Summary of
David W. Agler
Symbolic Logic: Syntax, Semantics, and Proof
Ch.3: Truth Tables
3.6 Truth Table Analysis of Arguments
Brief Summary:
There are a couple ways we can use truth tables to test for the validity of arguments. (a) We look for a row where all the premises are true and the conclusion false. If there is such a row, it is invalid. And it is valid otherwise. (b) We convert the argument into a set of propositions with the premises left intact but the conclusion is negated. If that set of propositions is inconsistent, that is, if there is no row where they are all true, then the original argument is valid. However, if that set is consistent, that is, if there is at least one row where they are all true, then the original argument is invalid.
Summary
3.6 Truth Table Analysis of Arguments
Previously we used truth tables to evaluate singular propositions for the logical properties of tautology, contradiction, and contingency (see section 3.3) and sets of propositions for the logical properties of equivalence and consistency (see section 3.4). Now we examine whole arguments to determine whether or not they are valid (88).
3.6.1 Validity
[Recall from section 1.3 how we defined validity: “An argument is deductively valid if and only if, it is necessarily the case that if the premises are true, then the conclusion is true. That is, an argument is deductively valid if and only if it is logically impossible for its premises/assumptions to be true and its conclusion to be false” (16). And an argument we said was invalid “if and only if the argument is not valid” (16).] “An argument is valid if and only if it is impossible for the premises to be true and the conclusion to be false” (88). In chapter 1 [section 1.3.2], we used the negative test for validity, where we tried to imagine a situation where the premises could be true while the conclusion is false. But this is not the most efficient or reliable method, because it depends on our psychological capacities, which become overtaxed when evaluating lengthy arguments in this way. However,
Using a truth table, an argument is valid in PL if and only if there is no row of the truth table where the premises are true and the conclusion is false. If there is a row where the premises are true and the conclusion is false, then the argument is invalid.
(88)
Agler then introduces the turnstile symbol (⊢). He says that later we will give it a more specific meaning, but for now we will use it just to mean the presence of an argument:
the propositions to the left of the turnstile are the premises, while the proposition to the right of the turnstile is the conclusion. For example, the turnstile in
P∧R ⊢ R
indicates the presence of an argument where ‘P∧R’ is the premise and ‘R’ is the conclusion. Likewise, the turnstile in
P∧R, Z→Z, ¬ (P∧Q) ⊢ R
indicates the presence of an argument where ‘P∧R,’ ‘Z→Z,’ and ‘¬(P∧Q)’ are premises, and ‘R’ is the conclusion. Lastly, the turnstile in
⊢ R| indicates the presence of an argument that has no premises but has ‘R’ as a conclusion (88-89).
Agler then defines validity and invalidity using these concepts and the turnstile symbol. [When testing with a truth-table, we are looking for a row where all the premisses are true but the conclusion false. If we find such a row, the argument is invalid. If we do not, then it is valid]:
Validity: An argument ‘P, Q, ..., Y ⊢ Z’ is valid in PL if and only if it is impossible for the premises to be true and the conclusion false. A truth table shows that an argument is valid if and only if there is no row of the truth table where the premises are true and the conclusion is false.
Invalidity: An argument ‘P, Q, ..., Y ⊢ Z’ is invalid in PL if and only if it is possible for the premises to be true and the conclusion false. A truth table shows that an argument is invalid if and only if there is a row of the truth table where the premises are true and the conclusion is false.
(89)
Agler then illustrates with the truth table for this argument:
P→Q, ¬Q ⊢ ¬P
(89)
We see that there is no row where both premises are true and the conclusion false. Thus the argument is valid (89).
Agler then given another example, this time for:
P → Q, Q ⊢ P
This time we do see a row with true premises and a false conclusion, namely, row three. Thus this argument is invalid (89).
3.6.2 Validity and Inconsistency
[Recall from section 3.4 the definition for the inconsistency of sets of propositions, “Inconsistency: A set of propositions ‘{P, Q, R, ... , Z}’ is logically inconsistent if and only if there is no valuation where ‘P,’ ‘Q,’ ‘R,’ ... , ‘Z’ are jointly true. A truth table shows that a set of propositions is inconsistent when there is no row on the truth table where ‘P,’ ‘Q,’ ‘R,’ ... , ‘Z’ are all true” (84).] Agler notes another way that we can evaluate an argument for validity. We negate the conclusion, and we see if the premises along with the negated conclusion are inconsistent. If they are inconsistent, then the original argument is valid (89). [If there is no row where the premises and negated conclusion are all true, that mean for the original argument, there is no row where the premises are true and the conclusion is false.] Agler illustrates by evaluating the argument:
P → Q, P ⊢ Q
As we can see from the truth table, there is no row where the premises are true but the conclusion is false. Thus it is valid.
So what we are wondering in this case is if it is possible for there to be the following truth assignments:
v(P→Q) = T, v(P) = T, and v(Q) = F
(90)
But, if v(Q) = F, then v(¬Q) = T. That means we can determine the validity of the argument by seeing if the following truth assignments are possible:
v(P→Q) = T, v(P) = T, and v(¬Q) = T
(90)
[Notice this time that all the propositions are true.] We can therefore ask this question to determine an argument’s validity: “Are the premises and the negation of the conclusion logically consistent (i.e., all true under the same truth-value assignment)?” If they are inconsistent, then the original argument is valid. And if they are logically consistent, then the original argument is invalid.
So let us evaluate the same argument:
P → Q, P ⊢ Q
Only this time using the consistency test. For this, we need to change it from an argument to a set of propositions, with the conclusion negated. Hence we get:
{P→Q, P, ¬Q}
And we will use the truth table to see if this set of propositions are logically consistent with one another.
As we can see from the table, there is no row where all the propositions are true. Thus the set of propositions {P→Q, P, ¬Q} is inconsistent, which means that the argument “P → Q, P ⊢ Q” is valid.
Agler then compares the definitions of validity and inconsistency. [The idea is that we can define validity in terms of inconsistency, by saying that a valid argument is one where its propositions and negated conclusion are inconsistent.]
To consider this more generally, compare the definitions for validity and inconsistency. An argument is valid if and only if it is impossible for the premises ‘P,’ ‘Q,’ ..., ‘Y’ to be true and the conclusion ‘Z’ to be false. This is just another way of saying that an argument is valid if and only if it is impossible for the propositions ‘{P, Q, ..., Y, ¬Z}’ all to be true. Notice, however, that if it is impossible for the propositions ‘{P, Q, ..., Y, ¬Z}’ to all be true, then the propositions ‘{P, Q, ..., Y, ¬Z}’ are inconsistent. Thus, validity can be defined in terms of inconsistency.
(91)
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
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