W-Reading: Logic: Deductions with Assumptions

W-reading: Logic: Deductions With Assumptions

(Guttenplan, 126-138)

The (Harder) Rules of the Game: The Assumption Rules:

Let me begin with a general preface concerning how to use assumptions with the following rules: We often make assumptions in our everyday reasoning. For example in the course of an argument you may be told that something is being assumed 'for the sake of the argument'. The rule about to be discussed involves such an assumption, and it would help to get a bit clearer about assumptions before the rule is stated.

What is an assumption? In order to distinguish an assumption from a premise, we have to recognize that an assumption is used in an argument, but is itself 'dropped' from the argument before the final conclusion is reached. The fact that it is dropped from the argument after it has served its purpose is what distinguishes an assumption from a premise.

But what does it mean to say that an assumption is 'dropped'? It means that, even though an assumption is used on the way to reaching a conclusion, we do not, in the end, need to see the conclusion as dependent for its truth on the assumption. Or, in other words, the assumption does not serve as one of the premises for the conclusion. When an assumption is made and then later dropped, we will say that it has been discharged. This, and the general functioning of assumptions, can be illustrated by this first rule which makes use of them (the asterisk ‘*’ marks the assumed line):

Assumptions may be made at any stage of a deduction and you may assume anything you like. This sounds too good to be true, but that is only an appearance. Remember, the point of making an assumption is to help in the deduction. You do not want such assumptions to end up as further premises in the argument, and this means that you have to discharge them. Since, there are quite strict guidelines about how and when to discharge assumptions, you will find them useful only in cases where rules show you how to deal with them. In fact, I prefer to think of the assumptions being forced by the rules, as opposed to my wanting to assume something and then needing a justification. In other words, it is only when I see that I will be needing a rule that requires an assumption that I use one, and I plug in the rule to see what exactly I will need to assume.

Otherwise, what to assume can become quite a mystery.

DISJUNCTION ELIMINATION (۷ E**)

A ۷ B

….

...

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If we are able to derive the same sentence from both sides of a disjunction, then we know that this third sentence is true.

Notice that this rule requires two assumptions, one for each side of the original disjunction. This means that to keep score, we will need to cite five lines. You will need to write the line number of the original disjunction, the line where you assumed one of its sides, the sentence you were able to derive from it, the line where you assumed the other side, and the (same) sentence your were able to derive from it, followed by the rule (۷ E).

This is the most complicated rule of the ten, and you should go through the following informal justification of the rule slowly. What we want to show is that ۷ E is truth-preserving.

For the sake of argument, assume that A ۷ B is true because A is true. Since you have shown that C can be deduced from A, you know that C must be true. So, if it is A that makes A ۷ B true, A ۷ B must also make C true. Assume now that A ۷ B is true because B is true. Since you have also shown that C can be deduced from B, you know that C must be true. So, if what makes A ۷ B true is that B is true, A ۷ B must also make C true. But A ۷ B is a disjunction and, hence, is true only if one or both of its disjuncts are true. So, the two sides of our reasoning above show that in any of the circumstances in which A ۷ B is true, C must be true too.

Here is an example of the rule at work showing that the following sequent holds:

G ۷ H Premise
G → L Premise
H → L Premise
G Assumption*
L 2,4-→E
H Assumption*
L 3,6-→E
L 1,4,5,6,7-۷E**

CONDITIONAL INTRODUCTION (→ I*)

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A → B

To keep score, on the right you cite the line numbers of ‘A’ and ‘B’ followed by the rule (→ I).

To convince yourself that this pattern of reasoning preserves validity follow this argument: since the rules so far introduced are truth-preserving, the first stage of the rule establishes that it is not possible for A to be true, and B false. Since this is so, and given the meaning of ‘→’, there is no way in which A → B could be false. The only way a sentence of the form A → B could be false is when A is true and B is false. But we showed that to be impossible in this case by deducing B from A using truth-preserving rules.

In English, this rule basically says that if assuming a certain sentence allows us to deduce some other sentence, then we can assert that the assumption is a condition which produces that other sentence.

Here is a deduction which shows the following sequent to be true using the new rule:

P → Q Premise
Q → S Premise
P Assumption*
Q 1,3-→E
S 2,4-→E
P → S 3,5-→I*

When this rule is used properly, the line that depends on it will automatically discharge the assumption it required. So, in the above example, in our score-keeping, once we cite the assumed anetecedent line and the consequent line, followed by the rule (→I), this discharges the assumption. For this reason, I like to add a symbol, such as an asterisk to make it clearer when each assumption is discharged. This will prove very helpful in keeping track of lone deductions that may have multiple assumptions, because, if it is not clear that all assumptions have been discharged, your final line cannot be said to be truth-preserving.

NEGATION INTRODUCTION (~ I*)

…

B & ~B

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~ A

If, by making an assumption A, we can use the rules to deduce a sentence of the form B & ~ B (that is, any clear contradiction), then we are allowed to deduce ~A. In other words, to speak in English, if we assume some claim but then show that this claim leads to a contradiction, we can deduce that our original assumption was false (or its opposite must be true). For example:

1. K → R Premise

2. D & ~ R Premise

3. K Assumption*

4. R 1,3-→E

5. ~ R 2-&E

6. R & ~ R 4,5-&I

7. ~ K 3,6-~I*

To keep score, we cite the assumption line and the line that shows our derived contradiction, followed by the rule (~I). Once the rule is cited, its required assumption is discharged.

The process of reasoning described in the rule is sometimes called ‘reductio ad absurdum’ and sometimes simply ‘indirect proof’. The Latin phrase means ‘reduction to absurdity’ and it is an apt name because, in using the rule, you first make an assumption and then reduce it to absurdity (i.e. deduce a contradiction from it).

Putting the Rules to Work

Example #3:

Goal: J ۷ (K & C)

1. (F → J) & (M → C) premise

2. M premise

3. C → F premise

We are after the disjunction ‘J ۷ (K & C)’, but, in looking at the premises, we cannot find a disjunction at all. This means we will need to create one, using rule ۷I. So, if we can create either side of this disjunction, we can simply add the other side. Being as there is no ‘K’ to be found in the premises and ‘C’ looks tough to isolate, we should look at trying to derive ‘J’ as our sub-goal. There is a ‘J’ in the first premise, and we can eliminate the ‘&’ in the first premise quite easily. Since this ‘J’ is the antecedent in a conditional, we will need to see if we can find ‘F’ to eliminate this conditional. To get ‘F’, we will have to eliminate the conditional in premise 3. To do this we will need to derive ‘C’, and we can derive ‘C’ if we first isolate ‘M → C’ and then use the ‘M’ in premise 2 to eliminate ‘M → C’. This sounds very confusing and too hard to keep track ofin our memory, so let’s make some notes of all of this, noting our sub-goals at each step (working backwards):

Our goal: J ۷ (K & C)

Our last sub-goal: J

To get J, we need: F→J and F

To get these, we need: C and C → F

To get C, we need: M→C and M

So, we can now acquire these needs one-by-one from the premises. Here is the deduction:

Goal: J ۷ (K & C)

1. (F → J) & (M → C) Premise

2. M Premise

3. C → F Premise

4. M → C 1-&E

5. C 2,4-→E

6. F 3,5-→E

7. F → J 1-&E

8. J 6,7-→E

9. J ۷ (K & C) 8-۷I

Example #4:

Goal: D → ~X

1. D → H premise

2. X → ~H premise

To get to our conclusion, we will obviously need to create this new conditional, because it is not found in the premises. We also see that we don’t have a ‘~X’ in the premises either, so this will need to be created as well. These two realizations tell us that we will need to use the rules ‘→I’ and ‘~I’. So, we will just follow the rules. We’ll juts go in the order of the conditional in the conclusion. The ‘→I’ rule says to assume the antecedent then derive the consequent. To derive the consequent, we’ll need to, as the ‘~I’ rule says, first assume ‘X’. Then it says to derive a contradiction. It seems pretty clear that ‘H & ~H’ shouldn’t be too hard to create from the premises. So that will be our first sub-goal after we make our assumptions. Then, we will just obey our two assumption rules.

Goal: D → ~X

1. D → H Premise

2. X → ~H Premise

3. D Assumption* (for →I)

4. X Assumption# (for ~I)

5. H 1,3-→E

6. ~H 2,4-→E

7. H & ~H 5,6-&I

8. ~X 4,7-~I#

9. D → ~X 3,8-→I*