# Monday 9/9

## Logic and Theoretical Analysis

### Synopsis

We began today discussing our upcoming (Wednesday, 9/11) first examination.

So far as content goes, I would expect approximately the following:

• Moral Dilemmas
• The Distinction between Inductive and Deductive Arguments
• Soundness, Validity, and Invalidity
• The Nature of Theory and the Standards of Clarity and Coherence

To be sure, this is a lot of material. Expect the exam to take approximately 45 minutes. Expect multiple choice, true/false, and at least one essay question. Bear in mind as you prepare that this exam is worth the least of all the exams, a mere 50 points. Thus it is possible, in theory, to do badly on this exam but well in the course.

Now, summarizing our study of logic, today I offered three crucial facts about valid arguments.

1. If an argument is valid and it happens to have all true premises, then it must have a true conclusion. (It is sound, that is to say.)
2. If an argument is valid and it happens to have a false conclusion, then at least one of its premises must be false.
3. If an argument is valid and it happens to have a true conclusion, then its premises might be true or they might be false. (No inference can be draw from the truth of the conclusion of a valid argument to the truth of the premises, since a valid argument with false premises can have a true conclusion.)

Even if you don't fully understand these three points, you should at least memorize them. They are critically important for all that is to follow.

For example, nowhere does understanding the facts of logic have greater impact than in grasping the role of theory in our understanding of the world. Today we took as our model scientific theory, considering a number of examples. Theories like Newtonian Mechanics, Quantum Mechanics, Relativity Theory, and the Theory of Evolution consist of a core set of propositions known variously as axioms (in mathematics) or natural laws (in the natural sciences) surrounded by indefinitely many propositions (also known as the theorems (mathematics again) or hypotheses (in the natural sciences) which are entailed by the axioms or natural laws. That is, there exists a valid (deductive) argument from the axioms or natural laws to the theorems or hypotheses of the theory.

Since an argument can be valid and have a true conclusion even though it has one or more false premises, confirming the hypotheses of a theory by conducting an experiment tells us absolutely nothing about the truth of the natural laws from which they derive. We cannot, that is to say, infer anything about the truth of the natural laws from the truth of the hypotheses. Science never proves anything true, in short.

However, a valid argument with a false conclusion must have at least one false premise. So if an experiment happens to disconfirm or show false a hypothesis, then we can infer that at least one of the so-called natural laws is in fact false. The upshot is that when a scientist conducts an experiment, she is trying to falsify her hypothesis. If repeated attempts at falsifying the hypotheses of a theory fail, then we say that the theory is well-confirmed. We never say the theory is true, for the (by now) obvious reason that we can have no justification for saying so.

Why are we worrying about the nature of scientific theory in a course on ethics?

The point of all our foundational work in logic last time is to establish a series of standards for the critical evaluation of moral normative theories. The standards are minimal in the sense that a theory which fails them is clearly false, while a theory which passes them might be true, but it could also be false. Thus, the Standards of Evaluation are used to exclude theories which do not have a chance of being true.

The analogy with science is important. Just as scientific theories consist of a core set of propositions (again, usually called laws in science or axioms in mathematics), which jointly entail descriptive propositions (often called testable hypotheses in science or theorems in mathematics), moral theories consist of a core set of propositions called principles which entail prescriptive propositions.

Now, as we've seen, validity is a curious relation. To put it in just slightly more technical terms, we say that a set of propositions S entails a proposition P if, and only if (iff) it is impossible for all the propositions s in S to be true and P false at the same time. Put another way, if S entails P and P is false, then at least one of the propositions s in S must also be false. Yet if P is true (and S entails P), you cannot conclude every s in S is true! All that is ruled out by validity (entailment, properly speaking) is that P be false when every s in S is true. P can be true and S entail P even though every s in S is false.

So this is, again, why scientific theories can never be proven true, they can only be proven false. For in a scientific theory, if the entailed testable hypothesis is shown to be false, we know that at least one of the entailing laws must be false. But if the entailed testable hypothesis in fact predicts what will happen, then we can only say that we have some confirmation of the theory. We cannot say the theory is proven true because one of the entailing 'laws' may still be false. At best our scientific theories will be well-confirmed, and scientists are always careful to speak in these terms. If one desires absolute certainty, science is certainly not the place to find it! I submit that that is a strength of science, however, and not a weakness.

Now, just as the scientist is correct to insist on clarity and coherence in her theories, ethicists, I argue, should insist on clarity and coherence in their theories. To wit,

With respect to Clarity,

1. Scientists don't tolerate vague or ambiguous terms in their theories, and neither should ethicists;
2. Scientists try to state their theories as precisely as possible, and so should ethicists.

With respect to Coherence,

1. Scientists cannot tolerate internal contradictions (because if S entails the proposition "P and not P", S has entailed a necessarily false proposition and so must have at least one member that is false), and neither, for the same reason, can ethicists;
2. Scientists are extremely cautious to avoid external contradictions, since theories which conflict with established, well-confirmed theories are very likely suspect. Ethicists are well-advised to be similarly cautious.

So far so good: Our problem now is that the experimental method of seeking to falsify a scientific theory is not available to the ethicist inasmuch as ethics is prescriptive or normative (it is about the way the world ought to be), whereas science enjoys the advantage of being purely descriptive (its lone ambition is to describe how the world is, not how it ought to be.)

That is, the analogy with science only goes so far. Science, you see, enjoys a significant advantage over ethics. If science seeks to understand or describe the way the world is, ethics seeks to prescribe the way the world ought to be. Scientists can always conduct experiments to determine whether what they think is the case is in fact the case. The world itself serves as a tribunal for scientific inquiry. Ethics has no such advantage since ethics seeks to understand how the world ought to be, not how it is.

To put it bluntly, scientists can conduct experiments, ethicists cannot. Our modest goal next time, after taking the first examination, will be to flesh out a counterpart in ethics to the experiments scientists are able to conduct.