Against both my better judgment and the hue and cry of many, I will continue my semi-informed-by-past-years-of-studying “exposition” of predicate logic which I started here.  If I accomplish nothing else, I will give Burl something to complain about for the next week or so.

In the previous installment, we talked about how syllogistic statements about “all x’s” assert the truth of a conditional statement.  “All dogs bark” asserts that for all x’s, if x is a dog, then x barks.  Formally expressed, that’s:

∀x(Dog(x) –> Barks(x))

or something similar.  It doesn’t say anything about whether there actually are any dogs.  Additionally, the ‘For all…’ symbol – ∀ – doesn’t allow you to say anything about only some dogs.  Let us address that issue.

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Editor’s Note: Matt Teichman, our guest on the Frege episode, has been good enough to provide this primer on logic for our listeners who’ve not already had to sit through a class on it or who might need a refresher:

One of the things we said at the beginning the episode on Frege was that he is the father of modern logic.  But what is logic, anyway?  For those of you who never heard of logic before, here is a quick primer.

Logic is the study of what philosophers call valid reasoning.  In everyday conversation, valid just means ‘good’ or ‘appropriate.’  But in logic, valid has a special technical meaning.  To see what it means, let’s look at an example of an argument:

Argument 1:

All people drink water.
Matt is a person.
—————–
Therefore, Matt drinks water.

Argument is also a term we’re going to use in a special technical sense.  Here, rather than referring to e.g. a dispute between me and my wife about whether to get a new car, it refers to a line of reasoning meant to convince someone of something.  More specifically, we’ll think of an argument as a sequence of sentences broken down into two parts: first come the premises, then comes the conclusion.  (You can also have an argument with just one premise, or even an argument with no premises!  But all arguments need to have a conclusion.)  In the above example, the premises are All people drink water and Matt is a person, and the conclusion is Matt drinks water.

Anyway, let’s get back to validity.  An argument is valid just in case it is impossible for its premises to be true and for its conclusion to be simultaneously false.  Consider the above example: if all people drink water, and Matt is a person, then there’s simply no way that Matt could fail to drink water.  It’s absolutely, positively guaranteed.

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