NATURAL SCIENCE 120, R.S. STEWART, LOGIC EXERCISES

The following is provided in the hope that it will assist you in understanding the rudimentary logic which is involved in (the basics of) scientific methodology. Recall that, according to the hypothetico-deductive method, one begins by guessing at the "correct" answer to the problem under investigation: that is, one begins by formulating an hypothesis. In order to have any confidence in one's hypothesis, however, we must test it. In science, tests take a wide variety of guises: the point to be remembered here, however, is that no matter how different in detail such tests might be, they all have -- or ought to have -- a similar logical structure. It is because of this fact that a full understanding of scientific methodology requires some knowledge of deductive logic, for all scientific tests occur within the context, in the first instance, of deductive reasoning. You will recall the following as the definition of deductive validity:

A deductive inference is valid if and only if it is impossible to have a case where all the premises are true and the conclusion is false.

Invalidity, then, is a deductive inference where it is possible to have a case of all true premises and a false conclusion. Besides knowing these two definitions, you also need to know the following logical terms.

Simple statement: any statement which asserts one and only one thing. All simple statements are either true (T) or false (F). E.g., "It is raining outside."

Compound statement: any statement containing two or more simple statements where the truth of the compound statement is determined entirely by the truth values of the simple statements contained within it, and the value of the logical connective., E.g. "If it is raining outside, then the streets are wet."

Conditional or hypothetical statement: a particular type of compound statement which is (typically) expressed in an "if ... then" form. E.g., "If it is raining outside, then the streets are wet." In a conditional statement, the two simple statements have names. The part which comes before the "then" -- which is printed in bold above -- is called the antecedent: the simple statement which comes after the "then" -- which is underlined -- is called the consequent. In symbolic logic, conditional statements are represented by an "arrow" ( --- > ). Since symbolic logic attempts to be very precise, the --- > is given an exact definition which can be expressed in a truth table which gives a truth value for every different permutation of the conditional. The truth table for --->, that is, the definition of --- >, is as follows: (This, and all the other tables below, will look strange on the page because I could not format it properly, but we'll strighten things out in class.)

Where "p" and "q" stand for any two simple statements:

p q p ----> q

T T T

T F F

F T T

F F T

Negation: the denial of any simple statement such that if the statement is true, the denial of that statement is false and vice- versa. Negation in symbolic logic is represented by a tilde (~): The ~ has the following truth table, or definition:

Where "p" stands for any simple statement:

p ~ p

T F

F T

Besides being employed for defining various sorts of logical concepts, truth tables can be used as a simple and mechanical way to test for validity. Suppose someone argued in the following manner:

If Socrates is a man, then he is mortal. Moreover, Socrates is indeed a man. I conclude, therefore, that Socrates is mortal.

Upon hearing, this, you know intuitively that although this person seems to be a pretentious dolt, he/she is also arguing in a deductively valid way. By using truth tables we can explicitly confirm our intuition. We do so in the following manner:

Let "S" stand for, "Socrates is a man."

Let "M" stand for, "Socrates is mortal."

S M S --- >M S M

T T T T T

T F F T F

F T T F T

F F T F F

All we have done here is: (1) set out all the possible truth value permutations of all our simple statements: here we have two simple statements, so there are four possible permutations. (If we had three simple statements, we would have 8, if we had 4 simple statements, we would have 16 permutations, and so on. However, all you have to concern yourselves with here are two simple statements and four permutations.) (2) Set out, and given truth values for each of the premises. Here we have two premises to our argument: (i) S --- > M, and (ii) S. We figure out the truth values for these by: a) using the truth values for our simple statements, and b) using our definitions for various logical symbols. (3) Set out and give truth values for the conclusion.

Once this procedure is completed, testing for validity is very simple. All we must do is remember our definition of validity -- where it is impossible to have all true premises and a false conclusion -- and then look at our truth table. If there are no cases where all the premises are true and the conclusion is false, then our inference is valid; if we do have a case of all true premises and a false conclusion -- and remember, we need only one such case -- then we know our inference is invalid. Since, in the case above, there are no cases of all true premises and a false conclusion, the inference in the argument is valid. (This is an argument form that is seen often: because of this, it has been given a name, modus ponens or affirming the antecedent.)

Here are two other commonly seen argument forms. Indeed, these are the two forms you will have to concern yourselves with since they are the two you will be dealing with with respect to scientific methodology. As you will also see, the first case is valid, while the second is invalid.

If Socrates is a man, then Socrates is mortal. Socrates is not mortal; therefore Socrates is not a man.

Let S = Socrates is a man.

Let M = Socrates is mortal.

S M S ---> M ~M ~S

T T T F F

T F F T F

F T T F T

F F T T T

Here we have only one case where all the premises are true (the fourth row), but in this case the conclusion is true as well. Since, then, we have no case where all the premises are true and the conclusion is false, the inference here is valid. This argument form is called modus tollens or denying the consequent. (Do not get thrown off by this example. Intuitively, you can see that this as a "bad" argument -- after all, it concludes that Socrates is not a man. Your intuition here is true enough, as far as it goes, but you must remember that validity is only one ingredient in a "good" or, as logicians call it, a "sound" deductive argument. Besides being valid, a sound deductive argument must also have all true premises. Presumably, the second premise here -- Socrates is not a mortal -- is false. Thus, the argument, while valid, is unsound.

If Socrates is a man, then Socrates is mortal. Of course, Socrates is mortal; therefore he is man.

Let S = Socrates is a man.

Let M = Socrates is mortal.

S M S ---> M M S

T T T T T

T F F F T

F T T T F

F F T F F

We see here that in row 3 all the premises are true while the conclusion is false. As a result, we know that the inference here is invalid. This argument form is called either modus morons or affirming the consequent. Again, there are a number of things here that could confuse you, but don't let them. (1) Row one has a case of all true premises and a true conclusion. Some of you will think that -- and unfortunately some of you will say this an a test -- this makes the inference here valid. It does not. The reason you might think it does is that you are working from an incorrect definition of validity; something like, a deductive inference is valid when you have a case of all true premises and a true conclusion. This is wrong: compare this to the correct definition of validity. (2) Some of you may get confused by row two where we have a case of all false premises and a true conclusion. You may argue on this basis that the inference here is invalid. If you said this, you would be right, but for the wrong reasons. Invalidity is defined as the possibility of all true premises and a false conclusion. This is not the same thing as saying, "a case of all false premises and a true conclusion." (3) Finally, you may get confused by thinking that, since you know the premises are true and the conclusion is true, that the inference must be valid. Here, you would, in a sense, be making both of the above two mistakes. What you have really said here is that there is a case here of all true premises and a true conclusion (Case (1) described above.) This is true enough, but that does not constitute validity. Moreover, though the conclusion is true, this argument -- as in case (2) above -- gives bad reasons (or more precisely, uses a bad or invalid inference) f o r what is, we can all agree, a true conclusion.

Logic seems to be one of those things that one learns by doing: given this, I thought we would begin class today by breaking up into three groups and doing some simple exercises. Group one will work an the two arguments given in Section A below, group two will work on 8, and group three will work on C. When I say "work on" I mean, (1) translate the argument into symbolic form, (2) put the symbolized argument in a truth table, (3) say whether the inference is valid or invalid, including a justification for your determination. One member of each group will graciously accept the group's nomination to act as the person who shall work this out on the board.

Section A

(1) If the moon is made of green cheese, then the cat in the hat can jump over the moon. The cat in the hat cannot jump over the moon. Therefore the moon is not made of green cheese.

(2) If childbed fever is caused by putrid material which is transferred from the hands of medical students who have just completed autopsies and helped in delivery without sterilizing their hands, then proper sterilization of the medical student's hands should decrease the rate of childbed fever. (As tests have shown) proper sterilization of the medical student's hands does decrease the rate of childbed fever. Therefore, Childbed fever is caused by putrid material which is transferred from the hands of medical students who have just completed autopsies and helped in delivery without sterilizing their hands.

Section B

(3) If the moon is made of green cheese, then the cat in the hat can jump over the moon. The moon is made of green cheese. Consequently, the cat in the hat can jump over the moon.

(4) If overcrowding is the cause of childbed fever, then making Ward II just as overcrowded as Ward I will increase the rate of childbed fever in Ward 11. However, (as tests show) making Ward II just as overcrowded as Ward I did not increase the rate of childbed fever in Ward II. Therefore, overcrowding is not the cause of childbed fever.

Section C

(5) If the moon is made of green cheese, then the cat in the hat can jump over the moon. The moon is not made of green cheese. Therefore, the cat in the hat can not jump over the moon.

(6) If childbed fever is caused by women delivering while lying on their backs, them having women lie an their sides during labour ought to reduce the rate of childbed fever. However, (as tests show) having women lie on their sides during labour does not reduce the rate of childbed fever. Therefore, childbed fever is not caused by women delivering while laying on their backs.