Logic and Problems
Chapter 8b exercises
Translate the following arguments and construct proofs to show they are valid. Check your translation against the posted translation answers before attempting the proof. If your translation is off, the proof may be impossible to complete. If your translation was wrong, when you turn in the assignment, post your original attempt of the translation and then the translation used in the proof.
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The death penalty should be abolished. Although proponents of the death penalty make many arguments, even they concede that the death penalty is not necessary for a well-ordered society. If the death penalty is not necessary for a well-ordered society, then it is an unnecessary killing of humans. If the death penalty is an unnecessary killing of humans, the death penalty is wrong. And if it is wrong, then it should be abolished. (A: The death penalty should be abolished. U: The unnecessary killing of humans is wrong. D: The death penalty is an unnecessary killing of humans. N: The death penalty is necessary for a well-ordered society. W: The death penalty is wrong.)
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If workers should be paid, then either they should be paid according to their needs, or they should be paid for services rendered. If workers should be paid according to their needs, then singles mothers should be paid more than their coworkers, and so should workers who have large families. If workers should be paid for services rendered, then workers should receive equal pay for equal work. Workers should be paid, but it is not the case that workers having large families should be paid more than their coworkers. Hence, workers should be paid for services rendered. (P: Workers should be paid. N: Workers should be paid according to their needs; S: Workers should be paid for services rendered; M: Single mothers should be paid more than their coworkers; F: Workers who have large families should be paid more than their coworkers; E: Workers should receive equal pay for equal work.)
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Common sense tells us that if humans burn more fuel, then humans make the earth warmer. If humans make the earth warmer, then humans cause climate change. Since the industrial revolution, we have massively increased the amount of fuel we burn. Therefore, humans cause climate change. (V: The vast majority of scientists who study the climate believe in anthropogenic climate change. M: We burn more fuel. W: The earth is made warmer. H: Humans cause climate change.)
8.5 Constructing Proofs
To recap the past eight and a half chapters, logic began as an attempt to solve the problem of sophistry, i.e., the problem of arguments appearing to be good when they are not. As attempts to solve this problem have evolved, they have progressed from intuitive and imaginative approaches, such as the counterexample method, to highly formal methods, such as truth tables. Each one of these methods for evaluating arguments suffered from a defect that the next system attempted to remedy.
The counterexample method relied on someone being imaginative enough to think up a counterexample and could only show invalidity. Venn diagrams were introduced to avoid these problems by allowing us to map premises onto diagrams and see if the premises required the conclusion to be true. While Venn diagrams resolved the problems with the counterexample method, they become very unwieldy with complex arguments (that would require more than 3 circles) and they cannot easily represent all statements (How do you put a disjunction onto a Venn diagram?). Truth tables were introduced to address both of these problems. As truth tables for complex arguments can also be unwieldy, there was a shift to abbreviated truth tables. These work brilliantly for evaluating validity. They use a completely formal and iterative approach to evaluating arguments. They can show validity and invalidity for all types of arguments. And they are fairly simple—never requiring more than the construction of one line of a truth table. While abbreviated truth tables can flawlessly evaluate arguments for validity, they do not allow us to see the chain of reasoning between the premises and conclusion. For valid arguments, this is problematic because it is the chain of reasoning that can be convincing (and the entire goal of presenting an argument is to convince/persuade). When an argument is invalid, truth tables don’t allow us to see where the mistake was made. So, we can know that the argument is invalid, but not diagnose the problem. This may not be troubling when evaluating someone else’s argument, but when you are trying to construct an argument, truth tables don’t tell you where your mistakes are, just that you made some.
Thus, we advance to proofs. Proofs are constructed within a system constructed of two basic elements, rules for formulating statements and rules for deriving statements. The first element is a familiar fact that is put in an unfamiliar way. For instance, in English, you know that, ‘the platypus is an extraterrestrial’ is a statement and that ‘plastic Martian orange sleep’ is not. In mathematics, you can recognize ‘(8xy) – 3z = 17i’ as a proper statement while ‘x92/dog - = + ))’ is not. Similarly, in statement logic, statements are built from atomic statements and the logical connectives introduced previously (and, or, not, if…then, if and only if). Everything you can assert (claim to be true) in English can be expressed as one of these statements.
The other part of the system, the rules of derivation (also called implication rules), is slightly more complex. To begin with, proof construction builds on the rules for formulating statements. Once you have a proper statement or set of statements (well-formed formulas, as logicians like to call them) you can then use the implication rules to derive other statements. For instance, the rule Disjunctive Syllogism tells you that if ‘P v Q’ and ‘~P’ are true, then you can derive ‘Q.’ But proofs are more than just rules allowing you to derive one statement from others. There are rules for how to construct proofs. Essentially, proofs are similar to arguments put into standard form—each statement is numbered and statements derived from the premises are annotated to state which lines the new statement is derived from and which rule allows for the derivation. Here’s two simple examples:
P & Q
2. P 1, Simplification (S)
1. P v Q
2. ~Q
3. P 1, 2 Disjunctive Syllogism (DS)
In both instances, we have a proof of ‘P’ from a premise set.
Now, what makes these proofs? Besides the fact that they fit the formal definition for being proofs, these arguments count as proofs because every step they take is governed by a rule that is truth preserving (that is to say, valid). All the rules can be shown to be truth preserving using the truth table method for evaluating validity. Additionally, all of the rules are fairly intuitive and most people can “see” that disjunctive syllogism is a good argument without constructing truth tables.
Let’s consider an example of a more complex proof.
1. R → B
2. R v J
3. ~B
4. J → L \ L v W
Now, before we begin to construct the proof from the premises to the conclusion, (L v W), here are a couple of strategies for constructing proofs. It is common for students to look at the four lines above and just not know what to do. These strategies are meant to help you know what steps to take. One strategy to adopt is to break down the premises into their component parts and then to use those parts to build up the conclusion. Let’s employ that strategy by using the rules we have been given to break down the premises into atomic statements as much as we can. In order to do this, we need to employ another strategy, which is to look for repetition. When we see the same letter repeated in two lines, there may be a rule that we can apply to those two lines. When we look at the premise set above, we notice that ‘R’ occurs in premise one and premise two. Now, we must ask, is there a rule that applies to premises one and two? Premise one is a conditional statement and premise two is a disjunction. Is there a rule that operates on two such statements? Constructive Dilemma (CD) operates on a disjunction and two conditional statements, but we don’t have two conditional statements, so that rule cannot be applied. There is no other rule that involves both the symbols ‘→’and ‘v.’ So, we must look elsewhere to get started. Premises one and three both have a ‘B.’ Is there a rule that applies to a conditional statement and the negation of the consequent (the part after the arrow)? Yes! Modus Tollens (MT) is such a rule. Remember, MT is the following rule:
1. P → Q
2. ~Q
So, 3. ~P
Now, let’s apply that rule to our proof:
1. R → B
2. R v J
3. ~B
4. J → L \ L v W
5. ~R 1,3 MT
We have started our proof, but what next. Well, a good strategy is to see if you can use new information. We just added line 5, ‘~R.’ Is there another line with an R? Well, there is line 1, but since we used line 1 to get line 5, it is unlikely (but not impossible) that we can use it with line 5. There is also an ‘R’ in line 2. Now, is there a rule that applies to lines 2 and 5, a disjunction and the negation of one of the disjuncts? (Or, as I like to say, is there a syllogism that involves a disjunction? Maybe Disjunctive Syllogism?)
Recall that Disjunctive Syllogism (DS) looks like this:
1. P or Q 1. P v Q
2. ~P 2. ~Q
So, 3. Q So, 3. P
If we apply DS to lines 4 and 7, our proof looks like this:
1. R → B
2. R v J
3. ~B
4. J → L \ L v W
5. ~R 1,3 MT
6. J 2,5 DS
Now if we repeat the process, we look to use our new line ‘J.’ Line 4 also has a J, so we look for a rule that can be applied to lines 4 and 6. Is there a rule that applies to a conditional statement and the first part of a conditional statement (the antecedent)? Yes. Modus Ponens (MP) fits the bill and it looks like this:
1. P → Q
2. P
So, 3. Q
Continuing our proof with the use of MP, we now have:
1. R → B
2. R v J
3. ~B
4. J → L \ L v W
5. ~R 1,3 MT
6. J 2,5 DS
7. L 4,6 MP
With line 7, we have arrived at ‘L.’ Other than line 4, which we used to get line 7, there isn’t another ‘L’ on a line in the proof. However, ‘L’ is in the conclusion, ‘L v W.’ But there isn’t a ‘W’ anywhere in the proof! How did that get into the conclusion??? It’s as if they just added it in. (Lightbulb goes off over student’s head as they remember the rule of Addition.) There is the rule of Addition (Add) that allows us do just that. It looks like this:
1. P
2. P v Q
Although this rule seems odd, it is based on a very simple idea. A disjunction is simply the statement that at least one of two claims is true. As long as one of the disjuncts is true, then the disjunction is true no matter what the other disjunct says. So, for any true statement ‘P,’ it follows that ‘P’ or any other statement (call that statement Q) is true. When we take this next step, we complete our proof as follows:
1. R → B
2. R v J
3. ~B
4. J → L \ L v W
5. ~R 1,3 MT
6. J 2,5 DS
7. L 4,6 MP
8. L v W 7, Add
As we have arrived at a line of the proof that contains that conclusion, our proof is complete.
Here is a summary of the strategies we used to construct this proof.
Strategies for Constructing Proofs
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Break down compound statements into atomic statements.
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Look for statement letters that repeat on different lines.
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See if any rules apply to cases where a statement appears more than once.
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After adding a new line, attempt to use the new information.
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Use the atomic statements to build the conclusion (unless the conclusion is an atomic statement).