Logic 6: Equivalence & Rules of Replacement

In the last chapter, we saw that several expressions, such as ~ (A ^ B) and (~A v ~B), are equivalent.  We can use the biconditional connective to see whether or not two expressions are equivalent.  If we connect two expressions with a biconditional symbol, evaluate this new biconditional expression with the truth table method, and find that it is true in all possible cases, then we know that the two expressions are equivalent.  Likewise, if we find that the new biconditional expression is not true in all possible cases, then we know that the two expressions are not equivalent.

In formal logic, there are several basic equivalences known as the rules of replacement that are used to make substitutions in proofs, including Commutativity, DeMorgan’s theorem, the Material Conditional, and Modus Tollens.  We are now going to prove each of these using truth tables.

Commutativity is when an expression is the equivalent of itself when reversed.  We can show with truth tables that AND and OR are commutative.  This means that saying, “I have an apple and a banana” is the same thing as saying, “I have a banana and an apple”, and saying, “I have an apple or a banana” is the same thing as saying, “I have a banana or an apple”.

We can also see by constructing a truth table that IF-THEN is not commutative.  This means that saying, “If I have an apple then I have a banana” is not the same as saying “If I have a banana then I have an apple”.

We could also use a truth table to show that biconditionals are themselves biconditional.  This means that saying, “I have an apple if and only if I have a banana” is the same thing as saying, “I have a banana if and only if I have an apple”.

We saw one half of DeMorgan’s Theorem in the last chapter.  The two halves of DeMorgan’s Theorem are:

1) ~ (A ^ B) is equivalent to ~A v ~B

2) ~ (A v B) is equivalent to ~A ^ ~B

This means that saying, “I do not have both an apple and a banana” is the same as saying “I do not have an apple or I do not have a banana”, and that saying, “I do not have either an apple or a banana” is the same thing as saying, “I do not have an apple and I do not have a banana”.  We can use truth tables to show this:

The Material Conditional shows us the relationship between OR and IF-THEN.

 There are three parts, which are:

1) (A > B) <> (~A v B)

2) (A v B) <> (~A > B)

3) (A v B) <> (~B > A)

The first part means that saying, “If I have an apple then I have a banana” is the same thing as saying, “Either II don’t have an apple or I have a banana”.  The second and third parts means that saying, “Either I have an apple or I have a banana” is the same thing as saying, “If I do not have an apple then I have a banana”, and is also the same thing as saying, “If I do not have a banana then I have an apple”.  This can be confusing, as it seems to say that OR is used exclusively when we know that it is used inclusively, but when we construct the truth tables we can see that an inclusive OR does mean that if one side is false then the other side must be true, both ways.

Modus Tollens, Latin for the “way of denial”, is a dual reversal of the conditional connective.  While the name comes from medieval European logicians, who gave it the Latin name, it was known by the ancient Greek stoics and can be found in the negative basic proof of the ancient Indian Nyaya school of Gotama.

We have seen that conditionals are not commutative.  A > B is not the same as B > A.  This means that if we know that the statement, “If I have an apple, then I have a banana” is true, we do not know whether the statement, “If I have a banana then I have an apple” is true.  We can see this by looking at the first and third cases of the conditional connective truth table.

In both of these cases, A > B is true and B is true, but in the first case A is true, and in the third case A is false.  This means that we do not know whether or not A is true, even if we know both B and A > B are true.  We can, however, know something about A if we know that B is false and that A > B is true.  If we know that A > B is true, and I do not have a banana, we can only be dealing with the fourth and final case.

This means that if we know A > B is true, but B is false, then A must also be false, and I do not have an apple.  This is because if A > B is true, we know that we cannot be dealing with the second of the four cases, when A > B and B are false but A is true.  Therefore, if we know the statement, “If I have an apple then I have a banana” (A > B) is true, we also know that the statement, “If I do not have a banana then I do not have an apple” (~B > ~A) is also true.  This is why medieval logicians called this Modus Tollens, the way of denial.  We can show that A > B and ~B > ~A are equivalent with a truth table:

In the Nyaya Sutra of Gotama, we can see Modus Tollens in the difference between the positive and negative basic forms of proof.  In an example of the positive basic form of proof, we read:

Whatever is created is impermanent, like a cup.

Because sound is created, sound is impermanent.

In an example of the negative basic form of proof, we read:

Whatever is permanent is not created, like the soul.

Sound is created, therefore sound is impermanent.

Notice that both come to the same conclusion, but in the second negative example, ‘created’ and ‘permanent’ are negated and reversed, just like in Modus Tollens.  In ancient India, the Nyaya, an orthodox Hindu school, argued that the soul or self was permanent and uncreated, against Jains and Buddhists, who argued that the self was impermanent and thus created at some point in time.

Exercise 5.1: On a separate piece of paper, without looking at the truth tables above, construct a truth table that proves biconditional expressions for Commutativity, DeMorgan’s Theorem, the Material Conditional, and Modus Tollens.

In addition to the rules of replacement, there are two standard types of replacement useful for proofs, the Hypothetical syllogism and the Dilemma, known as derived rules.  For these two, we will need to construct truth tables that involve three statements, symbolized by A, B and C.  This will mean we will need eight cases of truth values rather than four.

The Hypothetical Syllogism is nothing more than the Barbara form of Aristotle.  When we say, “All As are Bs”, this is the same as saying, “If it is an A, then it is a B”, or A > B.  This means that the expression, “If, if A then B and if B then C, then if A then C” is equivalent to Barbara, and so can be symbolized as:

Notice that, unlike the rules of replacement, the hypothetical is a conditional expression, not a biconditional expression.  This means that if we know ((A > B) ^ (B > C)) then we also know (A > C), but it is not commutative such that if we know (A > C) then we also know ((A > B) ^ (B > C)).  The following truth table shows us that the biconditional expression is true in SEVEN of the possible eight cases, but not in all eight of them:

Notice that in the sixth of the eight cases, ((A > B) ^ (B > C)) is false, but (A > C) is true.  However, if we change the expression to a conditional, it is true in all eight cases.

Why is the Hypothetical Syllogism conditional and not biconditional?  Consider our example of Barbara in which which all puppies are evil.  We can imagine that we live in a hypothetical universe in which all puppies are green and all green things are evil.  We could conclude in this universe that all puppies are evil.  Now consider a second hypothetical universe where all puppies are known to be evil.  We cannot conclude in this second universe that all puppies are green, or that all green things are evil.  In the second universe, it could be that all puppies are brown and made of peanut butter, such that no puppies are green even if all green things are indeed evil.  It could also be that not all green things are evil, even if all puppies are evil.  The sixth of our eight possible cases shows us that if we know that (A > B) and (A > C) are true, (B > C) can still be false.  We can see this with an illustration:

The Dilemma, also known as, “Damned if you do, damned if you don’t”, shows us that if we know that (A > C), (B > C) and (A v B) are true, then we also know that C must be true.  If we know that both A and B lead to C, and that A or B must be true, then either way C will be true.  Like the Hypothetical Syllogism, the Dilemma is conditional, not a biconditional.

Let us say that I like to eat both apples and bananas with coconut, but I do not like eating apples and bananas together.  Whenever I go to the market, I make sure to buy either an apple or a banana (A v B).  When I buy an apple I also buy a coconut (A > C), and when I buy a banana I also buy a coconut (B > C).  If my friend wants me to buy either an apple or a banana, but is afraid of coconuts, she is faced with the dilemma that, either way, whichever fruit I choose to buy, I am certainly going to buy a coconut (C).  Here is the truth table that shows how a dilemma functions:

Notice that first we must determine the truth values for (A > C) and (B > C), then determine the truth values for ((A > C) ^ (B > C)), then determine the truth values for (((A > C) ^ (B > C)) ^ (A v B)), and only then, using those truth values and the truth values for C, can we determine the final truth values for the entire expression.  Here are the three steps illustrated:

Exercise 5.2: On a separate piece of paper, construct a truth table for the conditional expressions of the Hypothetical Syllogism and the Dilemma.  You should practice such that, after copying the expression above the horizontal line, you can construct the table without looking at the truth tables above.

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