Glances Ahead: More to Think About

III. Logic and Natural Language
 
       The logical tools found at this website can easily convince us that the following
syllogism is a valid argument:

1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.  

We might then want to ask what form the argument takes, and see how we can represent it symbolically. We let the statement variable p stand for the first premise, the variable q stand for the second premise, and the variable r stand for the conclusion. None of these statements is a conjunction, implication, disjunction, or negation, so it seems as though our translation into statement variables is the simplest translation we can find. However, the translation tells us that our original valid argument is one of this form:    

1. p.
2. q.
3. Therefore, r.

       We can clearly see that this argument is not valid. Then what has gone wrong? The problem is that the original sentences had a structure to them, and that structure is lost when they are translated into statement variables. Note that Socrates is the subject of both the second premise and the conclusion. Also, both the first premise and the conclusion talk about something being mortal. Finally, both premises involve men - either making a statement about men in general, or telling us that someone is a man.
       The validity of this argument must lie in the fact that all three sentences are talking about the same three things: Socrates, being a man, and being mortal. Translating these three clearly related sentences as p, q, and r is definitely inadequate for our purposes. We want to be able to show that this argument is valid. However, the logical tools we have so far do not tell us how to retain the structure of the sentences in our translation. For this, we must turn to predicate logic.    
        In predicate logic, we can distinguish two kinds of variables. Predicate variables are generally uppercase letters, and object variables are generally lowercase. For example, using M to stand for the predicate ‘being a man’, and s to stand for Socrates, we would symbolize the sentence “Socrates is a man” as M(s). So if we were to use a to stand for Aristotle, translating the sentence “Aristotle is a man” as M(a), we can notice that there is something that both Socrates and Aristotle have in common. This fact would not be captured if we had used our previous technique of allowing a single letter to stand for an entire sentence.    
        A full translation of the syllogism into predicate logic would involve some symbolism that hasn’t been introduced yet - existential and universal quantifiers. The former allows us to represent existence claims, such as “There is a dog in the room”, while the latter allows us to express general claims, such as “All dogs have four legs.” I won’t go into too much more detail than this here, but I will provide a translation of the original argument. Let s represent Socrates, M represent the predicate “being a man”, and L represent the predicate “being mortal”. We then have the following:  

1. All men are mortal. 8x(M(x) ! L(x))
2. Socrates is a man. M(s)
3. Therefore, Socrates is mortal. L(s).

       Even predicate logic does not capture all the structure of our ordinary English sentences, though. Often, even when it seems obvious how we translate a sentence, something can be lost in the translation. Here’s an example of a sentence that seems easy to translate: ”I went outside and opened my umbrella.” We don’t need predicate logic to translate this sentence, so we’ll stick to our original method for now. Let p stand for the statement ”I went outside”, and q stand for the statement ”I opened my umbrella”. It seems clear, then, that the translation would be p ^ q. We’ll see why it’s not quite so clear-cut.    
        A simple fact about conjunctions is that p ^ q is always equivalent to q ^ p. You can verify this using truth tables if you like. But switching the conjuncts around in our original sentence gives us this: ”I opened my umbrella and went outside.” This last sentence doesn’t seem to mean quite the same thing as the original, though. We translated the “and” in my first sentence as a simple conjunction, but it actually suggested something more than that. It seemed to suggest a temporal order to the events. First, I went outside. Then, I opened my umbrella. However, the second translation seems to suggest that I did those things in the reverse order. So even cases that seem simple can be deceptively complicated, as ordinary sentential logic doesn’t have the means to represent a temporal order to events.    
       Now, the fact that there are some things about natural language that logic can’t quite capture doesn’t mean that logic is somehow inadequate. It’s a valuable tool, but in using it, we should be aware that it has some limitations. So I’ll conclude with some things you might want to think about with respect to this issue. What we just saw is that there is structure to natural language that logic either does not capture or captures with difficulty. How much do we want logic to imitate the structure of natural language? We probably don’t want it to be a new natural language. After all, the rigid structure and precision of logic is what makes it so useful. Yet one thing it is used for is to analyze our natural language arguments. Where does the balance lie?

I. Introduction to Symbolic Logic: Using Truth Tables

II. Philosophical Analysis

IV. The Law of the Excluded Middle