Lecture 1: Validity & Soundness 1
Goals Today Introduce one of our central topics: validity and soundness, and its connection to one of our primary course goals, namely: learning how to evaluate arguments Go over logistical and practical information about course, including: the various course requirements how to use the texts and other course resources suggestions for good study habits in the course 2
Two Dimensions of Evaluation So one of our primary goals in this course is to learn to evaluate arguments. In contradistinction from other courses which might help you learn how to evaluate arguments, here we use a set of formal tools which are applicable, more or less, to all subject matters. These tools speak to two different dimensions or facets of the evaluation of arguments, which in this tradition we call validity and soundness. Roughly, the idea is: validity is concerned with the relationship between the premises and the conclusion, whereas soundness is concerned with the truth of each. 3
Some Motivating Examples To see this difference, let s look at a couple of bad arguments. Argument 1: P1) If it rained all last night, then the sidewalks will be wet in the morning. P2) The sidewalks are wet this morning. C) Therefore, it rained all last night. Now, P1) seems in general true. Further, we can easily imagine situations in which both P2) and C) would be true. However, this is nonetheless a bad argument: for the premises could be true and yet the conclusion could be false. Imagine another night in which it didn t rain but the lawn sprinklers were on. 4
Some Motivating Examples What I want to draw attention to is that the way in which Argument 1 was bad is different from the way in which Argument 2 is bad. Argument 2. P1) The governor of California lives in the state capital. P2) The capital of California is Santa Cruz. C) Therefore, the governor of California lives in Santa Cruz. If we try to think about the reason that this argument is bad, we see that it pertains to P2): this premise is simply false! And it seems plausible that good arguments should have only true premises. 5
Two Features of Good Arguments So as these examples illustrate, good arguments should be such that: a) the premises of the argument are true b) the truth of the premises guarantees the truth of the conclusion. The first bad argument had a) but not b). The second bad argument had b) but not a). Good arguments should have both a) and b). Obviously, good arguments should have other features too: for instance, when possible, they should be short and easy to understand. Obviously, inductive arguments are important. But in this part of logic, we focus on a) and b). 6
Validity: A Definition Let s say that an argument is valid if the truth of the premises guarantees the truth of the conclusion. This definition of the word valid differs in a number of different ways from our ordinary usage, but it is standard in logic. E.g., on this definition, doesn t make sense to say that your feelings aren t valid. What s the sense of the word guarantee? Well, one explication it is this: whenever the premises are all true, then the conclusion is true. 7
Two more examples The definition: an argument is valid if the truth of the premises guarantees the truth of the conclusion; that is: whenever the premises are all true, then the conclusion is true. Argument 3: P1) Barry is a lawyer. P2) All lawyers can talk. C) Therefore, Barry can talk. Argument 4: P1) Aristotle is smart. C) Therefore, Aristotle is a philosopher. Looks like Argument 3 is valid (.... strange, but valid). However, Argument 4 is invalid: there s lots of smart people who aren t philosophers. 8
Abstract form of the argument So validity and invalidity seem to have to do with the abstract form of the argument. Consider again argument 5: Argument 3: P1) Barry is a lawyer. P2) All lawyers can talk. C) Therefore, Barry can talk. It kind of looks like this has the following form: Argument 3 : P1) b is L. P2) All L s are T s. C), Therefore, b is T. Consider other possible ways of filling in b, L, and T. All of these instances are such that if the premises are true, then conclusion is too. 9
Abstract form of the argument So validity and invalidity seem to have to do with the abstract form of the argument. Consider again argument 4: Argument 4: P1) Aristotle is smart. C) Therefore, Aristotle is a philosopher. It kind of looks like this has the following form: Argument 4 : P1) a is S. C) Therefore, a is P. This argument has some true instances (take S=P). But in general, not all of its instances are such that if the premises are true, then the conclusion is true (take S=smart, P=philosopher, and a=einstein). 10
Validity (Again) & A Presupposition An argument is valid if the truth of the premises guarantees the truth of the conclusion; that is: whenever the premises are all true, then the conclusion is true. So how do you tell whether an argument is valid? Well, you try to identify its abstract or underlying form or structure, and then you ask: for all possible ways of filling in the blanks in this structure, if the premises are true, then the conclusion is true. Big presupposition: natural language statements have some nearunique abstract form. As we will see, there s good reason to think that in general this presupposition is false. But it s a helpful idealization to make, and we ll learn more about its limitations as we go on. 11
How to study validity? We just said: So how do you tell whether an argument is valid? Well, you try to identify it s abstract or underlying form or structure, and then you ask: for all possible ways of filling in the blanks in this structure, if the premises are true, then the conclusion is true. So we will proceed in studying validity by studying these abstract or underlying forms and the formal relations which obtain between them. Part of the goal will be to develop a series of reliable mechanical tests for seeing whether an argument is valid. Again, the end goal in all this is to put us in a better position to evaluate arguments. 12
Soundness and Validity. An argument is valid if the truth of the premises guarantees the truth of the conclusion; that is: whenever the premises are all true, then the conclusion is true. Let s say that an argument is sound if it is valid, and if all the premises are true. From definition, it follows then that the conclusion of a sound argument is true. But there are valid but unsound arguments, i.e.: Argument 2. P1) The governor of California lives in the state capital. P2) The capital of California is Santa Cruz. C) Therefore, the governor of California lives in Santa Cruz. 13
The Four Possibilities Argument 1. P1) If it rained all last night, then the sidewalks will be wet in the morning. P2) The sidewalks are wet this morning. C) Therefore, it rained all last night. all premises true: Argument 2. P1) The governor of California lives in the state capital. P2) The capital of California is Santa Cruz. C) Therefore, the governor of California lives in Santa Cruz valid: argument 3* not all premises true: argument 2 Argument 3*: P1) The chief justice is a lawyer. P2) All lawyers can talk. C) Therefore, the chief justice can talk. invalid: Argument 4*: P1) Fido is smart. C) Therefore, Fido is a philosopher. 14 argument 1 argument 4*
How do we study truth? So on this way of putting it, validity is something different from the truth of the premises. So if another feature of good arguments is the truth of premises, then we should also study this. It s not as if logic has some unique access to the truth in the way in which other disciplines do not. Rather, the idea is that truth is part of the subject-matter of logic, in the way that plants are part of the subject-matter of biology. Logic studies truth by studying the way in which the truth of a sentence depends on the truth of its parts. This is the topic for next time. 15
Ω 16