Rules of Equivalence

Rules of Equivalence or Replacement

I. DeMorgan’s Rule

 

Statements that say the same thing, or are equivalent to one another are very important to a system of logical deduction. As you know, for instance, if we have a true conjunction, we can infer that either of its parts is true. Well, if we have a “neither…nor” statement, we happen to have a conjunction, since saying “It will neither rain nor snow” is the same as saying “It won’t rain and it won’t snow either.” So we can simplify a “neither…nor” statement, if we first confirm that it is equivalent to a conjunction of negatives. This is one of two versions of a rule known as DeMorgan’s Rule.

 

DM       ~(p v q) :: (~p ∙ ~q)

 

DM also works with a disjunction of negatives. Saying “Either this is false or that is false” is the same as saying “These are not both true.” In the following formula, the left side says “Not both,” and the right says “Either not this or not that.”

 

DM ~(p ∙ q) :: (~p v ~q)

 

It is quite important that you not mistake these formula as expressing what is known as the “distributive property” in mathematics, concerning distribution of negative signs (for negative numbers). We are not dealing in sums or numbers here, but with meanings. You can assure yourself with a truth table that “Neither…nor” does not mean “Either not this or not that.” You can also assure yourself of it by imaging a situation in which “neither…nor” really matters to you! You won’t be fooled by someone who misinterprets it. For example, let’s say you’re planning a party and someone asks if you are going to invite Scott and Rhonda: you say “Are you crazy? I’m not going to invite either of them!” (which means “neither of them”). If your friend now asks: “So which one aren’t you going to invite?” (because he thinks that “neither” — ~(S v R)– means (~S v ~R), you’re going to reply something like “Do you have a brain?” It is only in a classroom that you would fail to know that “neither” is not the same as “either not the one or not the other.”
Like all statement forms about which there are rules of equivalence, any of these can be replaced by the other. In terms of how you see them written on the page, you could say that they “work” in both directions, from the formula on the left to the one on the right, or from the one on the right to the one on the left.

II. Distribution

 

We’ve already used the word “Distribution” in Logic twice: once was when we contrasted collective predication with distributive predication, to understand the fallacy called Composition. We all appreciate that “America is a rich country” does not entail that “All Americans are rich.” From a true predication of the property of being rich to the whole made up of all Americans, it does not follow that you can “distribute” that property to all of the individuals who are its parts. The other time we encountered Distribution was in determining the validity of Categorical Syllogisms: we saw two rules that invoked the notion of a reference being made to every member of the class named by a term.

In identifying equivalent expressions, Distribution” comes back again. Here is the formula (there are two versions):

DIST     (p v (q ∙ r)) :: ((p v q) ∙ (p v r))

DIST    (p ∙ (q v r)) :: ((p ∙ q) v (p ∙ r))

What’s going on ? It looks like when you have a disjunction with a subordinate conjunction, or a conjunction with a subordinate disjunction, one of the statements can be distributed onto the parts of the other, by making the main operator subordinate and using it twice, while making the subordinate operator the new main.

That’s not all that catchy or memorable. (Neither… nor)

Try this: If I know that either p is true or else that both q and r are true, then that’s the same as knowing that both p or q, and p or r, are true.

Either Russia withdraws from Crimea or else Germany and the US impose sanctions.

Either Russia withdraws from Crimea or Germany imposes sanctions, and either Russia withdraws from Crimea or the US imposes sanctions.

The other version of distribution strikes me as more intuitive (but not less valid):

Russia annexed Crimea, and either the US or Germany will impose sanctions.

Either Russia annexed Crimea and the US will impose sanctions, or else Russia annexed Crimea and Germany will impose sanctions.

But the moral for the task of writing proofs is to realize that DIST offers the possibility of rewriting a dot statement as a wedge, or a wedge as a dot. Not all the time, but if there is a subordinate clause with the other operator. It’s pretty mechanical and unintuitive, but we’ll find it useful sometimes. Such a change might allow a SM (simplification) where a DS wouldn’t work.

III. Transposition

 

The next one is very intuitive: Transposition. In categorical logic, there is a very similar move, known as Contraposition. It says that All Popes are Catholics is equivalent to All nonCatholics are nonPopes. Here we are saying that with two statements instead of just within one, because we’ve seen that universal predications can be represented very effectively as Conditional statements:

TRAN      (p ⊃ q) :: (~q ⊃ ~p)

If he’s the Pope, he’s Catholic ≡ If he’s a non-Catholic, he’s a nonPope.

So you can switch the order of antecedent and consequent as long as you negate each. You cannot use Commutation on a conditional, however.

IV. Material Implication

 

When we first talked about “or” we observed that besides its strong and weak senses, it means “unless,” which is intuitively, the same as “if not…” So here we have a rule that lets us change wedges to horseshoes and horseshoes to wedges. That could be useful for doing MT, MP, DS, etc. This is known as Material Implication (IMP). It’s helpful to realize that what it amounts to is that when you change a “⊃” to a “v” or a “v” to a “⊃”, you change the left hand statement by a tilde (i.e., either add one or remove one).

 

IMP       (p v q) :: (~p ⊃ q)

 

V. Material Equivalence

 

We’ve talked about the triple bar as having two ways to be understood, and the two versions of the EQ rule address them. One is to see it is equivalent to a biconditional (i.e., a conjunction of conditionals), and in this case, it asserts that each thing is necessary to the other and also sufficient for the other.

EQ         (p ≡ q) ::   ((p ⊃ q) ∙ (q ⊃ p))

The other version says that the two statements have the same truth-value: they are either both true or else both false:

EQ      (p ≡ q) ::    ((p ∙ q) v (~p ∙ ~q))

This is what we said a truth table for the triple bar really means (the triple bar gets a T when both statements are T and when they are both F).

VI. Exportation

You might recall this exercise from earlier:

If you are the Vice-President, then if your aide is found guilty of obstruction of justice, then if you don’t distance yourself from him, you will find yourself under a cloud on the cover of TIME magazine.

V ⊃ (G ⊃ (~D ⊃ U))

This rule establishes for us that when a consequent of a conditional is a conditional, its antecedent can be expressed as a conjunct of the antecedent to which it is subordinate.

So, we rewrote the VP statement this way: If you are the Vice President, then if your aide is found guilty of obstruction of justice and you don’t distance yourself from him, you will find yourself under a cloud on the cover of TIME. V ⊃ ((G ∙ ~D) ⊃ U).

And that can be rewritten by the same principle, because it has a conditional for a consequent. So we take the antecedent G ∙ ~D and conjoin it to V:

((V ∙ (G ∙ ~D)) ⊃ U,

which yields: If you are the Vice President and your aide is found guilty of obstruction of justice and you don’t distance yourself from him, then you will find yourself under a cloud on the cover of TIME.

And of course, AS (association) and CM (commutation) can be applied to this to regroup and reorder the three statements that make up this antecedent. This is known as Exportation, and like all of these, it works “both ways.”

EXP (p ⊃ (q ⊃ r)) :: ((p ∙ q) ⊃ r)

 

VII. Tautology

 

The last rule is enormously trivial and will only be used when something enormously trivial is needed. One version of the rule is not even needed, because you can simplify a redundant conjunction, and you can add anything to whatever you have. But sometimes introducing a redundant statement is the only way to get some other rule to apply (e.g., if you need to set up a CD, you need a disjunction of antecedents, so you might need to generate that by addition). My bet is that if you ever use this, it will be to go from

p v p    to    p

or it will be to introduce a redundant conjunction:

going from p     to    p ∙ p

It is called the rule of Tautology.

TAUT     (p v p) :: p         (p . p) :: p

 

VIII. Please Note …

 

A final, very important observation, is that these rules apply anywhere. Anywhere you want, wherever you see the appropriate main operator of the rule -whether it is at the main operator of the line in the proof, or within some subordinate clause within the line. So for instance, see if you can tell where to apply DM to this one, and what the result will be:

A ⊃ {G ≡ [(~ C ∙ ~ S) v (P ∙ N)]}

 

 

It applies to the conjunction of negatives (~C ∙ ~S), which can be rewritten as “neither…nor.” It’s also the case that anywhere you see a horseshoe, you could change it to a wedge and negate the left hand side (IMP). If we do both of those things, we’ll get this:

 

~A v {G ≡ [~(C v S) v (P ∙ N)]} DM, IMP

 

If you do a truth table for each of these, you’ll find they are equivalent.

Doing proofs using all of these rules takes a good deal of practice. I’ll provide a set of exercises that build from easier to harder, and will expect you to be working on them daily for at least an hour. You can write each other or me with questions, you can come and see me with questions. Try them, and when you have a problem, either move on or ask for some help. Don’t waste time unproductively, that’s the most important thing. Learn all the rules of inference. Be able to write them. Copy the rules of replacement over and over until you begin to get used to them. Then try to apply them. And ask whatever questions arise.

Attached to this page you’ll find four powerpoints on Rules of Equivalence.

Seeing rules of replacement

9 Rules of Replacement 1

9-Rules-of-Replacement-2 3.2014

9 some examples with EQ

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