Logic, An Introductory Lecture

Michael F. Goodman
Department of Philosophy
Humboldt State University

Concepts explained below:

Beginnings

Logic is the study of reasoning with a view to distinguishing correct from incorrect reasoning. Reasoning involves making inferences and hence logicians have historically been interested in distinguishing correct from incorrect inference. The concept of inference in logic has to do with drawing a conclusion based upon evidence. For example, if you see muddy footprints on your carpet, you make the inference that someone didn't wipe her/his feet. The inference goes from "There are muddy footprints on the carpet" to "Someone didn't wipe her/his feet". To make this sort of inference is to have constructed an argument. (This "study of reasoning" is not the study of the psychological process of reasoning. That is, it is not the study of what goes on in the psyche of the person who reasons.)

The word "argument", as it is used in logic, does not refer to people heatedly disagreeing over some issue, as when neighbors have an argument over their shared back fence about who should pay the cost of the repair for the damaged fence posts. However, when two people have a disagreement of this sort, usually each will be presenting an argument (in the logician's sense) to the other. Example:

Neighbor Dave: "I shouldn't have to pay because I'm not the one who set the posts in the first place."

Neighbor Steve: "I didn't set them either and I shouldn't have to pay because I wasn't even in town when the storm hit."

As you can see, both Neighbor Dave and Neighbor Steve are making the same point, that is, that "I shouldn't have to pay for the repair of the fence posts." The point they are making is what we call the conclusion; that is what they are arguing for. In logic, to "argue for" something is to give reasons in support of something. In this case, the neighbors are giving reasons in support of the alleged fact that neither should have to pay for the repair of the fence posts. Dave's reason is that he wasn't the one who installed the fence posts. Steve has two reasons. First, he didn't install the fence posts and, second, he wasn't around when the storm hit (presumably the storm is responsible for the fence posts needing repair (probably high winds and rain)).

The reasons given are what logicians call premises. They are intended as supporting evidence for the conclusion. One of the most important things we can ask is: Do the premises present good supporting evidence for the conclusion? The common way to refer to an argument with premises which support well the conclusion is "good argument". It's the sort of argument people believe in, one they accept. Note that one argument may be acceptable to one person but not acceptable to another. This is difficult for some people to understand, so let's give an example.

The second premise in each of these arguments presents us with a problem, because they seemingly contradict one another. It appears that no one could accept both of them at the same time. Here is where the assessment of argumentation begins for some people.

Below are a number of passages. Some contain arguments, some do not. See whether you can determine the premises and the conclusions in the passages you identify as containing arguments.

A biology teacher says:
"No one-celled animal has a brain."
A student in the class responds:
"How do we know this?"
Teacher answers:
"Extensive studies have been done over the past 80 years on all known one-celled animals and all have proved to be brainless."

Dave bought a Martin D-28 guitar from Elderly Instruments in Lansing. The folks at Elderly treated Dave very well in terms of sales and service. Dave couldn't be happier.

Without the aid of telescopes or of elaborate mathematical arguments that have no apparent relation to astronomy, no effective evidence for a moving planetary earth can be produced. (T.S. Kuhn, The Copernican Revolution)

Anyone who can think a thought of the form Rab can think a thought of the form Rba. (R. Cummins, "Systematicity", Journal of Philosophy, XCIII, 12.)

If any team can win the World Series 2 years in a row, then it will be clear that their wins are not just good luck. Some teams not only can but will win the series two years in a row. This is excellent evidence that some teams winning the Series two years in a row is not just good luck.

Because at least four strategies may be used to indicate the attributes a word connotes, there are at least four kinds of intensional definitions: synonymous definition, etymological definition, operational definition, and definition by genus and difference. (P.J. Hurley, A Concise Introduction to Logic)

Validity & Invalidity

All arguments are either valid or invalid. If an argument is such that it is impossible for the premises to be true at the same time as the conclusion is false, then the argument is valid. And, any argument that is not valid is invalid. There is no in-between. Consider the following argument:

Question: Is it possible for each of the premises in the argument above to be true and the conclusion false (that is, at the same time)? If no, then the argument is valid; if yes, then the argument is invalid. The argument above is valid. Here is an invalid one:

Question: Is it possible for the premises to be true while the conclusion is false? If no, the argument is valid; if yes, the argument is invalid. The argument is invalid, since Ceasar may be your Aunt Penny's cat, and since no cats are dogs, the conclusion would be false, even though each of the premises is true.

Perhaps the crucial question to ask, when one is trying to determine validity (that is, whether some argument is valid or invalid) is this: If these premises were true, would it be possible for the conclusion to be false? If the answer is yes, then the argument is invalid; if no, valid.

When trying to determine validity, it is very important that one not be concerned with whether the premises and conclusion are in fact true or false. Actual truth value is of no consequence in determining validity. Here's an example:

Since there is most certainly not a monster under my bed, we see that the second premise is false. And if someone tried to convince me to accept this conclusion based on these premises, I'd call the second premise into question. But, in trying to see whether the argument is valid, I only ask: "What if the premises were true? Would the conclusion follow?" My answer has to be "yes". I couldn't consistently agree that the premises are true (if I did agree with them) and then deny that the conclusion was also true.

Acceptability

The discussion above leads right into talk about acceptability. Arguments are either acceptable or unacceptable. This is also true for sentences (premises). Any time you find an argument to have one or more false premises, then the argument would be unacceptable to you. An argument may be valid and it may be the case that if the premises were true you would accept the conclusion. However, since accepting an argument is tantamount to thinking (knowing or believing) that the premises are true and that the premises lend a good deal of support for the conclusion, you wouldn't find yourself accepting any argument with a false premise (or one in which the premises don't support the conclusion, or support it weakly). Consider the following example:

Acceptable or unacceptable? First, the argument isn't valid. That is, if the premises were true, still, the conclusion might be false. Since the first premise refers to "most" rather than "all" movie stars, it might be the case that Al Pacino is simply one of the exceptions. The fact that the argument isn't valid, however, does not of itself make it unacceptable.

Second, we need to ask whether the premises are true. Certainly the second one is true. It may be the case that we can't discover, very easily, whether the first is true or not. We could read the Hollywood magazines, I guess. Would that help? It's hard to know just how one would come to reasonably decide whether the first premise is true. Let's say, for a moment, that you have good evidence that the first is true (say you're intimately connected with the movie industry, and so on). Your judgment would then be that the premise is true and hence acceptable. (It must be noticed that accepting this premise as true is not like accepting other sentences as true. Imagine that you are going to the store to buy some apples to make a pie. You look out the door and see that it is raining; you see the wet streets, you see the drops falling, you hear the drops make a sound as they hit the leaves of your rhododendren plant near the door. In this sort of example you have the evidence of your senses that the sentence "It is raining" is true. We might want to say that there are grades, or degrees, of knowledge regarding the truth of the sentences we can accept as true. Epistemic justification (having to do with how our beliefs are justified) can be very complicated, philosophically, and, the essential point having been made, this is not the right place to go into great detail.)

OK, then. So let's say the premises are both true. Do those premises lend much support for the conclusion? If the word "most" in the first premise means something like 98%, then from the statistical point of view, there is good evidence that the conclusion is true. If 'most' means about 51%, the support for the conclusion, statistically, is much less.

There is a very simple way of talking about whether premises are acceptable or not: If you think some premise is true, then the premise is acceptable to you. Now just because we believe premise X to be true does not make it true; we might have a false belief. We will come to believe that X is true based on good evidence, all the while being openminded that there may well be evidence that we haven't come across which could speak against X. That's what "changing our mind" is all about. We might come to believe that the number of planets is 8 rather than 9 (say we agree, for example, with some scientists who believe that Pluto is not a true planet). But, it will take evidence that we understand and believe. And, if someone presents us with an argument which has premises the truth value of which we are unsure, then of course we will neither accept nor reject the argument. We will withhold our judgment on it until such time as we do come to believe one way or the other about the premises. In this way, both arguments and premises can be acceptable, unacceptable or neither.

Inductive Arguments

An inductive argument is an argument in which the conclusion follows from the premises with some degree of probability below 1.0. Let's say it has been determined that there is a 67.3% chance that some event will occur. Since there is 32.7% chance that it won't occur, we see that the premises leading up to the conclusion (67.3%) do not provide conclusive support for the conclusion. That is, there is still a chance that all the premises are true and the conclusion is false. We see, then, that the argument is not valid. It turns out that all inductive arguments are invalid (which doesn't make them automatically unacceptable.)

There is still the question, however, as to whether the argument itself is acceptable. This is very difficult to judge in some cases. What percentage probability would be considered high? If it comes to the weather, then what percentage would have to be forcasted for rain that would influence you to take your umbrella? 67.3%? 63.2%? 79.8%? Or, would it not be a percentage that you would count on at all? Perhaps you'd go outside, look at the sky, smell the air, feel the wind direction and force, feel the temperature, and decide to take or not take your umbrella on the basis of past experience. You'd kind of be acting as your own meteorologist.

I think it also makes a difference what the subject of the argument is. The weather is one good example. So is gambling, say, on horses, or baseball teams or at the roulette table. So is judging the probability that your business partner, who is chronically late, will be on time for the important presentation you have to make at 2pm. If you think the chance that your partner will be late is very high, then you will probably want to prepare for that (assuming that the presentation is really very, very important). If your experience is that your partner is late 80% of the time, then you may well judge that there is an 80% chance that your partner will be late for the presentation. If you judge 80% to be high probability, you will act accordingly. In logic we say that the higher the probability, the better the argument, and therefore, the more acceptable the argument.

This last comment is thought provoking, perhaps. Question: Does the last sentence in the paragraph directly above imply that there are degrees of acceptability? I think it does and I think that there are degrees of acceptability. Contrast this, now, with the concept of validity. There are no degrees of validity; there is no such thing as an argument that is more valid than another argument, at least not the way we use the term "valid" in logic. Which is to say: if you prove argument S valid, then that is the end of it on the validity score; there isn't anything more to say.

There is much more to say on the concept of inductive arguments, however. Inductive arguments come in many forms, but they all have the same "definition", that is, that the premises support the conclusion with some degree of probability greater than zero (0.0) but less than one (1.0). The following is an inductive argument:

This argument commits a fallacy known as Appeal to Authority; just because Einstein believed something, that doesn't make it true. However, it is important that Einstein was an expert in the field which is the subject of his belief. We would want to say that it is more likely that Einstein was right about space than wrong (though he could have been wrong). The phrase "it is more likely" indicates a probability greater than 50% but less than 100%. Hence, the argument is inductive. Consider another example.

This is called an analogical argument. It refers to similarities common to a number of things and then draws a conclusion referring to a further similarity. Note how the conclusion contains the term "probably". This is a vital clue telling us that we've got here is an inductive argument. Consider another example: This is another analogical argument, with "probably" again in the conclusion. This is a good example of an inductive argument that does not proceed from the particular to the general (as so many people believe all inductive arguments do). This argument proceeds from the particular to the particular, but whose conclusion is only probable.

An argument with many of the same premises as the one above draws a different conclusion, a "universal" conclusion (not all arguments with universal conclusions are either inductive or invalid).

This sort of argument is called an induction by simple enumeration. This is very much like an analogical argument, the difference being the universal (more general) conclusion.

With these examples of inductive arguments in mind, we will be able to identify many of the arguments used in daily life, in science, and sometimes even in philosophical discourse. The one most significant idea in the assessment of any argument, inductive or not, is that we must judge whether the premises are true and whether the premises support the conclusion to such an extent that we believe, or would be willing to act on, the conclusion, given the premises.

Deduction

You will note that I haven't spoken of deductive arguments here so far. This is because I have never seen a definition of a deductive argument that is not deeply flawed. Some definitions leave us with the conclusion that all deductive arguments are valid arguments. In that case, what work does the word "deductive" do for us? We might just as well keep to "valid". Other definitions attempt to make a case for there being invalid deductive arguments. However, they yield arguments which are either inductive or arguments whose conclusions don't follow with any degree of probablility at all. For example, consider the following argument: Certainly this is an invalid argument. But, does the conclusion follow with any degree of probability (above 0.0) from the premises? I don't think so. The most we could say of this conclusion, given the premises, is that the premises do not rule out that all mammals are dogs. Even if the subject and predicate terms ("mammals" and "dogs", respectively) in the conclusion were reversed (so that you had "All dogs are mammals" for the conclusion), and even though this is a true sentence, the premises themselves (upon which we must judge whether the conclusion is true) do not give us anything but the "possibility" that the conclusion is true. Since there is only "possibility" here, we cannot proceed to saying there is therefore "probability" also.

Suffice it to say that I will stick with the concepts of "validity", "invalidity", "acceptability" and "unacceptability" when assessing arguments in this class. I believe these concepts do all that is needed in this regard and that all arguments can be assessed adequately using them. But, since I want to be openminded about these sorts of things, we'll need to be on the lookout for arguments that do not fit the sorts of patterns I've been talking about throughout this lecture.

Copyright 2007 by Michael F. Goodman


Not sure what Popeye has to do with Logic, but his under-the-breath comments seem logical and funny.

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