108.04
Logical PHI and Critical Reasoning
Spring 2008
The Basics of Arguments III
1. NECESSARY AND SUFFICIENT CONDITIONS A necessary condition is a condition which must be present in order for a given result to follow. For example, there are a number of necessary conditions that govern whether or not our car will start moving when we press the accelerator. There must be gas in the car, the car must be turned on, the engine must be working, etc. A necessary condition is necessary to bringing about a result, but does not guarantee that result. A sufficient condition is one which, whenever it is present, always produces a given result. If my Saturn SL2 lacks an engine, this is a sufficient condition for knowing that it will not work properly. A sufficient condition is just that it is sufficient, it is all that is needed, to bring about a certain result. These terms can be applied to some of the statement forms we've looked at so far. For example, in the statement "P Q", we know that if P is true then Q must also be true. Therefore, P is sufficient to guarantee Q; in other words, P is a sufficient condition for Q. On the other hand, Q is a necessary condition for P: Q must be true, in order for P to be true. Q is necessary for P.
2. EQUIVALENT FORMS In evaluating arguments symbolically, we must use the following rules establishing equivalent (and therefore substitutable) argument forms. An equivalent form is another way of stating the same thing. Many of these will look like common sense, but it is necessary to know them in their standard forms. a. Double Negation (DN) --P P This is certainly common sense: a thing is the opposite of its opposite; P is not the opposite of P; P is not not-P. b. Commutation (Com) (P Q) (Q P) This allows us to switch the two terms of a conjunction: "P and Q" is the same as "Q and P". It also applies to disjunctions, "either/or" statements. We can switch their terms because the order of terms in "and" and "either/or" statements doesn't matter. (On the other hand, this is not true in the case of implication: "If Q then P" is not the same as "If P then Q.") For example, Albert has a pet lizard and a pet monkey. ...is equivalent to... Albert has a pet monkey and a pet lizard. or (P v Q) (Q v P) Either James will win the game or Marie will. ...is equivalent to... Either Marie will win the game or James will. c. Contraposition (Contra) (P Q) (-Q -P) This might remind you of Modus Tollens it is the same basic function. That is, if (P Q), then if -Q then -P (-Q -P). For example, If I go to Joan's birthday party tomorrow, then I have to buy a gift today. ...is equivalent to... If I don't buy a gift today, then I won't go to Joan's birthday party tomorrow.
Hurley, Logic, and Munson and Black, The Elements of Reasoning were used as sources for the above information.
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PHI 108.04
Logical and Critical Reasoning
Spring 2008
d. Definition of Implication (Imp) (P v Q) (-P Q) Saying this statement aloud helps to make clear that it is true: If it is true that "P or Q", then "not P" means that "Q" must be true. In an either/or statement, if one of the elements is not true, then the other one has to be. Either cats are better pets, or dogs are. ...is equivalent to... If cats are not better pets, then dogs are. or (P v Q) (-Q P) Either cats are better pets, or dogs are. ...is equivalent to... If dogs are not better pets, then cats are. e. Exportation (Exp) [(P Q) R+ *P (Q R)+ This rule is less obvious at first sight and needs a bit of thinking-through. If P and Q are sufficient conditions for R, then if P is present, all we need is Q to guarantee R (If P, then if Q then R.) This rule is easier to understand when we replace the variables with statements. If we win the game and we have the money, then we'll get ice cream. ...is equivalent to... If we win the game, then if we have money, we'll get ice cream. f. De Morgan's Rules (DM) -(P Q) (-P v -Q) Although it looks complex, this is another relatively intuitive rule. If P and Q are not both true, then either P is not true, or Q is not true. A person cannot both win the competition and make friends. ...is equivalent to... Either a person cannot win the competition, or they cannot make friends. and -(P v Q) (-P -Q) On the other hand, if it is not the case that P or Q, then both P and Q must be false. In other words, if neither P nor Q, then not P and not Q. Neither today nor tomorrow works for me. ...is equivalent to... Today doesn't work for me and tomorrow doesn't work for me.
Establishing equivalent forms is necessary for carrying out step-by-step logical proofs of complicated arguments. In order to "prove" something we have to demonstrate it based on rules. Before we can use common sense to prove something, we have to turn it into a rule that is, we have to state it as a rule for our system. The following is an example of a proof using both the argument and equivalence forms. EXAMPLE Premises: (-P v Q) -(R -S), R, and P Conclusion: S 1. (-P v Q) -(R -S) 2. R 3. -P 4. -P v Q 5. -(R -S) 6. -R v S 7. S
given given given 3 Add 1, 4 MP 5 DM 2, 6 DS
Hurley, Logic, and Munson and Black, The Elements of Reasoning were used as sources for the above information.
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PHI 108.04 3. VENN DIAGRAMS
Logical and Critical Reasoning
Spring 2008
***Note: Due to technological limitations, you will have to color in the diagrams in this section yourself. I've indicated in each case which sections should be shaded.*** Venn Diagrams are visual representations of statements categorical or categorical syllogisms (three part arguments with "all", "some", or "none" statements). Not only do these diagrams make it easier for us to see how the syllogism works (similar to the use of variables), but they can also be very helpful tools in determining whether or not a particular argument is valid. 1. First, decide which elements of the argument should be replaced with variables and put the argument into variable-format. 2. For each of the variables, draw a circle. Each circle represents a category or set the set of all cats, the set of all dogs, the set of all things with fur, etc. This means that everything belonging to that set is in the circle, and everything not belonging to the set is outside the circle. In your diagram, every circle should overlap every other circle, because every category or set could potentially overlap with all others. Two circles should look like this..............and three circles should look like this:
Draw your circles this way regardless of whether or not the groups overlap. Even if your argument states that "No cats are dogs", your "cat" and "dog" circles should still overlap. 3. Now, considering the two-circle diagram, the two circles have been divided into three segments. A B
On the left, a chunk of Circle A by itself; on the right, a chunk of Circle B by itself; and between them a chunk that is part of both Circle A and Circle B that is to say, where A and B overlap. An object in the left hand chunk belongs to A but not to B, an object in the middle belongs to both A and B, and an object in the right hand chunk belongs to B but not to A. A group of three circles is more complicated. Consider the following. A 1) 2) 3) 4) 5) 6) 7) All things that are A, but not B or C. All things that are both A and B, but not C. All things that are B, but not A or C. All things that are A, B, and C. All things that are both A and C, but not B. All things that are both B and C, but not A. All things that are C, but not A or B. 1 2 4 5 7 C 6 3 B
Hurley, Logic, and Munson and Black, The Elements of Reasoning were used as sources for the above information.
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PHI 108.04
Logical and Critical Reasoning
Spring 2008
4. With this diagram format (using the two-circle diagram as an example), we can visually represent any relation between the category represented by circle A and the category represented by circle B. a. To indicate that there is nothing in the set represented by a segment, we shade it. b. To indicate that there is something in the set represented by the segment, we place an "x" in this segment. No A are B. or All A are not B. A B Only the middle is shaded. A x B There is an x in the middle.
Some A are B.
All A are B.
A
B
The left (A-only) segment is shaded.
Some A are not B.
A x
B There is an x in the left segment.
c. In a three-circle diagram, the "some" statements can get tricky. For example, take the following statement: Some A are not C. If we were to place the x in segment 1, this would be ignoring the fact that the As that are not Cs could still be Bs. On the other hand, if we place the x in segment 2, we run into a similar problem, because we can't know that the As that are not Cs are definitely also Bs. A 1 2 4 5 7 6 C 3 B
In this case, we would place the x on the line between 1 and 2. This indicates that the As that are not in C could either be only As, or they could also be Cs.
x
One final note: The "x" indicating "some" should always be placed on the circumference of a single circle, between two segments never on an intersection of two or more lines. Using these rules, we should be able to successfully and fully represent arguments in visual form, and we should be able to determine the validity of the conclusion to a categorical statement or syllogism.
Hurley, Logic, and Munson and Black, The Elements of Reasoning were used as sources for the above information.
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PHI 108.04
Logical and Critical Reasoning
Spring 2008
The analysis of a complete syllogism would look like the following examples: 1. All teachers are baseball fans. No baseball fans are swimmers. Therefore, no teachers are swimmers. First Premise: All T are B. T B Segments 1 and 5 shaded. S Second Premise: No B are S. T B Segments 4 and 6 shaded. S At this point, we can determine whether the conclusion is valid by checking to see if the diagram fits the conclusion. For example, does the diagram tell us that "No T are S"? Because the entire area of overlap between circle T and circle S (segments 4 and 5)is shaded, we can answer this question affirmatively: Yes, our diagram shows that there is nothing that is a T and an S. Since the diagram supports the conclusion, the argument is valid. 2. All beeps are chirps. Some chirps are honks. Therefore, some honks are beeps. First Premise: All B are C. B C Segments 1 and 5 shaded. H Second Premise: Some C are H. B C An x on the line between 4 and 6. H Does the diagram fit the conclusion that "Some honks are beeps"? The diagram tells us that there are no beeps that are not also chirps (that is, there are no beeps that are honks and not chirps). It also tells us that there are some chirps that are honks, but it can't tell us whether these chirps are also beeps or not. Therefore, the conclusion is not fully supported by the diagram and is therefore not valid. (see numbering above, p. 2 or 3) T = teachers B = baseball fans S = swimmers
Hurley, Logic, and Munson and Black, The Elements of Reasoning were used as sources for the above information.
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