Logic

Instructions

This is a four-video/four-test course

Watch each video and then take the subsequent test.

A transcript of each video is provided.

The test for each video is immediately below the transcript.

Transcript of Logic 1

Lisa:  John, what is the discipline of logic about?

John: Logic studies relations of dependence among statements. In this context, a statement is anything that is true or false. And a dependence-relation holds between two statements when the truth of one of them depends on the truth of the other

Lisa: Please give me an example.

John: John cannot have exactly two cars unless John has an even number of cars. Therefore, the truth of the statement “John has exactly two cars” depends on that of the statement “John has an even number of cars,” meaning that the first statement cannot be true if the second is false.

Lisa: Ok. I think I understand. So if P is a statement and Q is some other statement, it may be happen that there is no way that there is no way for P to be true unless Q is also true. You just gave one example, and another example would be if P is the statement “Smith can read” and Q is the statement “Smith is an animate being.”

John: Excellent example, Lisa. And when the truth of P depends on that of Q, we say that entails Q. We also say that Q is a logical consequence of P.

Lisa:  So “John has an even number of cars” is a logical consequence of “John has exactly two cars”, and “John has exactly two cars” therefore entails “John has an even number of cars.”

John: Correct. The concept of negation is important to logic. The negation of “snow is white” is “it is false that snow is white” or simply “snow is not white.” In general, the negation of a P is not P.

Lisa: I follow.

John: A related concept is that of incompatibility. Two statements are incompatible if they cannot both be true.

Lisa: So “Smith is in London” and “Smith is in Australia” are incompatible.

John: Correct. And for P to entail Q is for P to be incompatible with the negation of Q.

Lisa: I see. “Smith is in London” is incompatible with “Smith is in Australia” because “Smith is in London” entails “Smith is not in Australia.”

John: That’s correct. There are a few extra details, but that is really the gist of the discipline of logic.

Lisa: Ok. But what does the discipline of logic do exactly?

John: Great question. I will answer it in the next video.

Transcript of Logic 2

Lisa: John, last time I asked you what the discipline of logic aspires to do, but you did not have time to give me an answer. Do you have time now?

John: Yes I do. First of all, the word “logic” has two meanings. Sometimes it refers to this or that specific system of logic. And sometimes it refers to the activity of constructing such systems.

Lisa: What is a system of logic?

John: A system of logic is a recursively defined class whose members are statements. If k is a class of objects, k is recursively defined if its members are identified by a true statement of the form alpha belongs to k and whenever x belongs to k, so does phi of x, where phi is some function.

Lisa: I think I follow. if k is the class of natural numbers, then 0 belongs to k and x+1 belongs to k whenever x belongs to k. and that is a recursive definition of the class of natural numbers.

John: Exactly. Of course, k is a class of numbers, not of statements. But there is a corresponding statement-set. Here it is. Let k star be the smallest class such that the statement “0 is a natural number belongs to k” and such that, for any statement x, k star contains the statement “x+1 belongs to k” whenever it contains the statement “x belongs to k.”

Lisa: And that statement is a logic.

John: Correct. Any recursively defined statement set is a logic.

Lisa: But this still doesn’t explain what the discipline of logic does.

John: Logic attempts to generate recursive definitions of statement-sets that are scientifically important.

Lisa: This rings a bell. Please continue.

John: If a class of statements is recursively defined, that means that there is a mechanical way of determining whether or not a given statement belongs to that class. So if somebody discovered some recursion that generated the class of true statements of psychology, then we could literally compute the truths of psychology the same way we can compute truths of arithmetic.

Lisa: So logic is supposed to generate recursive definitions of the truths composing scientific disciplines or at the very least large fragments of those disciplines.

John: Exactly.

Lisa: Does it ever do this?

John: Sometimes, but not often. In recent years, logic has become more concerned with generating theorems concerning its own limitations.

Lisa: Please explain.

John: It turns out that it can be mathematically proved that certain classes of truths cannot be recursively defined. A mathematical theorem to the effect that some class of statements cannot be recursively defined is known as an incompleteness theorem. And a mathematical theorem to the effect that some class of truths can be recursively defined is known as a completeness theorem.

Lisa: And you are saying that in recent years, logicians have focused more on incompleteness theorems than on completeness theorems.

John: Correct. The problem is that an incompleteness theorem tells us what logic cannot do, and that is all it tells us. So logic is useful only to the extent that it generates completeness theorems.

Lisa: And does that happen?

John: Yes it does. There is a completeness theorem corresponding to any given system of logic, since a system of logic is a recursively defined statement class. But specific systems of logic tend to be of interest only in the context of extremely specific investigations. Anybody who designs a working machine or computer program or an app or a school curriculum either implicitly or explicitly generates a logic.

Lisa: That makes sense, since a computer program is a set of recursions, and so is a school curriculum or, as we might also say, a school program.

John: Exactly. But of course those systems of logic are much too context specific to be of general scientific interest. And the systems of logic that are sufficiently generic to be of general interest are weak and lacking in depth.

Lisa: Your point being that what is referred to in universities as quote on quote logic consists of these generic systems of logic.

John: Correct.

Lisa: Would you mind running through some of the basic principles of such a system? Or would that take too long?

John: I am happy to do that for you Lisa, but right now let us now take a break, and in the next video I will go through those principles.

t and edit me. It's easy.

Transcript of Logic 3

John: What is referred to as logic in universities consists of two recursively defined classes of statements. Let us refer to the first of these classes as k. We will give the other one a name in due course.

Lisa: ok.

John: details aside, the membership of k is defined as follows. k contains any given atomic statement.  An atomic statement is one that does not consist of other statements. An example is “snow is white.” Another example is “grass is green.”

Lisa: I follow. So “the sky is blue” is atomic, but “the sky is blue and the moon is round” is non-atomic.

John: Correct. and k is generated by a recursion that maps those atomic statements onto non-atomic statements. That recursion consists of several different rules. One of them is as follows.  when a statement p belongs to k and a statement q belongs to k, then the statement “p and q” also belongs to k.

Lisa: I follow. If you put an “and” between members of k, the result is another member of k.

John: Exactly. Another one of those rules is that if p belongs to k, then not not p also belongs to k. In other words the negation of the negation of any member of k is a member of k.

Lisa: I think I understand. If snow is white belongs to k, then so does snow is not not white

John: Exactly.

Lisa: And I bet another such rule is:  If the statement “Smith has two cars” belongs to k, and if the statement “Smith is rich is a consequence of Smith has two cars” also belongs to k, then the statement “Smith is rich” is a member of k.

John: Yes, that last one is sometimes referred to as transitivity.

Lisa: So all of these rules jointly define a recursion whose inputs are atomic statements and whose outputs are non-atomic statements.

John: Correct. It would take around an hour to go through the details, but yes, that’s pretty much it. Anyway, this class of statements is known as the propositional calculus.

Lisa: So the propositional calculus is an algorithm that tells you how to transform atomic statements into non-atomic statements.

John: Correct. It is that simple. The only qualification is that the propositional calculus is undefined for statements about relations between classes. So statements such as “all whales are mammals”, “some people are literate”, and “most dogs are friendly” fall outside the scope of the propositional calculus.

Lisa: What do we call such statements?

John: We refer to them as quantified generalizations.

Lisa: So “no people play the banjo” and “all penguins are birds” are quantified generalizations.

John: Correct.

Lisa: You mentioned two classes of statements, one of them being the propositional calculus. Does the other one concern quantified generalizations?

John: Yes. The study of quantified generalizations is known as the predicate calculus. So the predicate calculus studies relations of logical dependence involving quantified generalizations.

Lisa: Can you be more specific?

John: Yes, but let us take a break. I will go into specifics in the next video.

Transcript of Logic 4

Lisa: John, last time you said that you would discuss specifics relating to the predicate calculus. I would like you to start by giving me an example of an inference rule that belongs to the predicate calculus.

John: Here is one such inference rule. The statement “everything is round” entails “Smith is round.” In general. “everything has phi” entails “alpha has phi”, for arbitrary alpha.

Lisa: I see.

John: And “Smith is round” entails “something is round.” In general, “alpha has phi” entails “something has phi.”

Lisa: Ok. I think I get it. Another such rule would presumably be “if everything that has phi also has psi, and if everything that has psi also has chi, then everything that as phi also has chi.”

John: Correct.

Lisa: So the predicate calculus is an algorithm for deriving quantified generalizations from atomic statements and also from other quantified generalizations.

John: Exactly. And that’s pretty much it. The only nuance is that in order to apply the predicate calculus to statements of ordinary language, it is necessary to re-parse those statements.

Lisa: Please explain.

John: “Smith smokes” has the same grammatical structure as “nothing smokes.” Each consists of a subject-term followed by a predicate term. But whereas “Smith smokes” entails “something smokes,” “nothing smokes” is incompatible with “nothing smokes.”

Lisa: I think I follow. Please continue.

John: We obviously want the rules of the predicate calculus to recognize this difference. At the same time, if those rules are to be general, they have to be defined for statement-forms, in the absence of specific information about statement-contents. A system of logic that cannot work unless loaded with specific information about the meanings of “nothing smokes” and “Smith smokes” and other individual statements is completely useless. In order to be useful, a system of logic has to be defined for a few easily identified statement forms. But as we have just seen, if we use the forms had by statements of natural language, the results are not what we want.

Lisa: And how do we fix that problem?

John: By reparsing quantified generalizations. “nothing smokes” becomes “for any x, x does not smoke.” “no whale smokes” becomes “for any x, if x is a whale, then x does not smoke.” “something smokes” becomes “for some x, x smokes.” And “something green smokes” becomes “for some x, x is green and x smokes.”

Lisa: So if we reformat quantified generalizations according to these rules, then the predicate calculus gives us the right results, is that the idea? If we do all of this re-parsing, then those rules tell us that  “something smokes” is a logical consequence of “John smokes” but not of “nothing smokes”, is that right?

John: Yes, exactly.

Lisa: it seems to me that there is a serious problem with the predicate calculus. in order to get the predicate calculus to generate right results, we have to seriously restructure the statements we are targeting, and we therefore have to know a lot about those statements, including their logical properties. But if we already have all of that information, we don’t really need the predicate calculus.

John: you are right lisa. In order to format statements in such a way that the predicate calculus generates the right results, we have to know how to reformat them. But if we know how to reformat them, then we already pretty much know whatever it is it that the predicate calculus is going to tell us about those statements.

Lisa: Is this paradox based on an error or does it point to an actual limitation of the predicate calculus?

John: It is not based on an error, and it does point to a severe limitation on the part of the predicate calculus. It shows us that more is lost than gained by using the predicate calculus.

Lisa: Please clarify.

John: If the truths involved in using an algorithm are more complex than those generated by the use of it, then that algorithm is a failure, even if it yields the right results.

Lisa: I think I understand. It is harder to know how to reformat statements in such a way that the predicate calculus tells us what follows from them than it is to figure out what follows from them in a purely ad hoc manner.

John: Exactly. And there is a larger point here. There are two conditions that a mathematical assertion must satisfy. First, it must be true. Second, it must be feasible to use it. The predicate calculus does not satisfy the second condition.

Lisa: Is that true of systems of logic in general?

John: No it is not, since any viable program or app embodies a system of logic. The problem is that such systems of logic are too context-specific to be of general interest.

Lisa: I think I understand. The specific systems of logic that are sufficiently context-independent to be studied in a vacuum are of little use. Therefore the ones that are worth studying cannot be taught in a vacuum, and they arise naturally in the course of investigations that do not ostensibly concern logic.

John: Exactly. My designing the curriculum used at this university involves my implicitly constructing systems of logic on a daily basis. But what makes those systems of logic useful is precisely that they are too context-specific to be interesting or even intelligible to people not embedded in those contexts.

Lisa: And the result is that what is referred to as logic in universities is not real logic and is also not very substantial.

John: Exactly.