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
3.4 Truth Tables Analysis of Sets of Propositions
Brief Summary:
We can use decision procedures with truth tables to determine whether or not a set of propositions are equivalent or consistent. If the truth table for a set of propositions shows them each to have identical truth values for any truth assignments, then they are logically equivalent. And they are not equivalent otherwise. Another way to conduct this test is to combine the propositions into one larger proposition using the biconditional operator. If the new proposition is a tautology, then the original two propositions are logically equivalent. A set of propositions are consistent if there is at least one value assignment that makes them all true. So on the truth table, we look for at least one line where all the propositions in question have the value true, and that tells us they are consistent. The propositions are inconsistent if no truth value assignment makes all the propositions jointly true. This can also be tested by combining a pair of propositions with a conjunction. If the new proposition is a contradiction, then the original propositions are inconsistent.
Summary
3.4 Truth Tables Analysis of Sets of Propositions
Previously we used truth tables to analyze singular propositions [to see if they were tautologies, contradictions, or contingencies]. Now we will use truth tables to evaluate sets of propositions to test them for logical equivalence and consistency (81).
3.4.1 Equivalence
Consider the following two sentences:
John is a married man.
John is not an unmarried man.
(81)
No matter what truth assignments we make, these two sentences will always have the same truth value, and thus they are logically equivalent.
Equivalence: A pair of propositions ‘P’ and‘Q’ is logically equivalent if and only if ‘P’ and ‘Q’ have identical truth values under every valuation. In a truth table for an equivalence, there is no row on the truth table where one of the pair ‘P’ has a different truth value than the other ‘Q.’
(81)
So first consider a truth table that evaluates both propositions:
P→Q
Q∨¬P
We see that under the same truth assignments, they have the same evaluations, and thus they are logically equivalent:
The truth table above shows that ‘P→Q’ is logically equivalent to ‘Q∨¬P’ because whenever v(P→Q) = T, then v(Q∨¬P) = T, and whenever v(P→Q) = F, then v(Q∨¬P) = F.
(Agler 81)
[So this methodology involves us using our eyes and comparing two columns. The next method will involve an extra step, but it will display one column of values which will make the equivalence obvious.] Another way to determine whether or not a pair of propositions is logically equivalent is by joining the two with the biconditional and then seeing if that biconditional relation produces all true values, that is, seeing if it is a tautology. [Recall from section 3.3 our decision procedures for singular propositions. A proposition is a tautology when all the truth values are true, regardless of the truth assignments. Here was one example for P→(Q→P).
This biconditional technique involves taking the two different propositions and making them one larger proposition and then applying the decision procedure for singular propositions to see if it is tautological.] The the biconditional is true whenever both sides have the same truth value, and false otherwise. Thus a biconditional is a tautology if both sides have the same truth value, regardless of the truth assignments. So let us take our two propositions from before:
P→Q
Q∨¬P
And we will make it into a singular biconditional:
(P→Q) ↔ (Q∨¬P)
Now when we make the truth table for it, we see that the biconditional is true for all assignments, thus the biconditional proposition is a tautology; and hence furthermore, both of the original propositions are logically equivalent.
And of course, if the biconditional relation between the two original propositions is not a tautology, then they are not logically equivalent (82).
3.4.2 Consistency
Consider the following three propositions.
John is a bachelor.
Frank is a bachelor.
Vic is a bachelor.
(83a)
It is possible for them all to be, that is to say, there are value assignments where all three are jointly true, and thus these sentences are logically consistent.
Consistency: A set of propositions ‘{P, Q, R, ... , Z}’ is logically consistent if and only if there is at least one valuation where ‘P,’ ‘Q,’ ‘R,’ ... , ‘Z’ are true. A truth table shows that a set of propositions is consistent when there is at least one row in the truth table where ‘P,’ ‘Q,’ ‘R,’ ... ,‘Z’ are all true.
(83)
For example, let us consider the following three propositions:
P→Q
Q∨P
P↔Q
And let us make a truth table for them:
As we can see, for line 1, all three main operators have the value true, and thus all three are logically consistent. For, there is at least one value assignment, namely v(P)=T and v(Q) =T, that makes all three propositions true (83).
Agler then notes that empirical evidence might say that certain logically consistent propositions are not together true, but that does not make them any less logically consistent. It only makes them factually inconsistent (83). For, “at the level of logical or semantic analysis, they could both be true” (83).
Now, whenever a set of propositions cannot all be true together, that is, whenever there is no value assignment that can make them all true, then they are inconsistent.
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 illustrates by giving the truth table for the following two propositions:
(P∨Q)
¬(Q∨P)
As we can see, there is no value assignment where both propositions are jointly true. Thus they are inconsistent.
Another way of showing this is by joining the propositions by a conjunction and seeing if they form a contradiction, that is to say, seeing if they are false under all evaluations. As we can see, they are, which indicates that the original two propositions are inconsistent (84).
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
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