Section 6: Formal languages

Here is a famous valid argument:

Socrates is a man.
All men are mortal.
.˙. Socrates is mortal.

This is an iron-clad argument. The only way you could challenge the conclusion is by denying one of the premises— the logical form is impeccable. What about this next argument?

Socrates is a man.
All men are carrots.
.˙. Socrates is a carrot.

This argument might be less interesting than the first, because the second premise is obviously false. There is no clear sense in which all men are carrots. Yet the argument is valid. To see this, notice that both arguments have this form:

\(S\) is \(M\).
All \(M\)s are \(C\).
.˙. \(S\) is \(C\).

In both arguments \(S\) stands for Socrates and \(M\) stands for man. In the first argument, \(C\) stands for mortal; in the second, \(C\) stands for carrot. Both arguments have this form, and every argument of this form is valid. So both arguments are valid.

What we did here was replace words like ‘man’ or ‘carrot’ with symbols like ‘M’ or ‘C’ so as to make the logical form explicit. This is the central idea behind formal logic. We want to remove irrelevant or distracting features of the argument to make the logical form more perspicuous.

Starting with an argument in a natural language like English, we translate the argument into a formal language. Parts of the English sentences are replaced with letters and symbols. The goal is to reveal the formal structure of the argument, as we did with these two.

There are formal languages that work like the symbolization we gave for these two arguments. A logic like this was developed by Aristotle, a philosopher who lived in Greece during the 4th century BC. Aristotle was a student of Plato and the tutor of Alexander the Great. Aristotle’s logic, with some revisions, was the dominant logic in the western world for more than two millennia.

In Aristotelean logic, categories are replaced with capital letters. Every sentence of an argument is then represented as having one of four forms, which medieval logicians labeled in this way: (A) All \(A\)s are \(B\)s. (E) No \(A\)s are \(B\)s. (I) Some \(A\) is \(B\). (O) Some \(A\) is not \(B\).

It is then possible to describe valid syllogisms, three-line arguments like the two we considered above. Medieval logicians gave mnemonic names to all of the valid argument forms. The form of our two arguments, for instance, was called Barbara. The vowels in the name, all As, represent the fact that the two premises and the conclusion are all (A) form sentences.

There are many limitations to Aristotelean logic. One is that it makes no distinction between kinds and individuals. So the first premise might just as well be written ‘All \(S\)s are \(M\)s’: All Socrateses are men. Despite its historical importance, Aristotelean logic has been superceded. The remainder of this book will develop two formal languages.

The first is SL, which stands for sentential logic. In SL, the smallest units are sentences themselves. Simple sentences are represented as letters and connected with logical connectives like ‘and’ and ‘not’ to make more complex sentences.

The second is QL, which stands for quantified logic. In QL, the basic units are objects, properties of objects, and relations between objects.

When we translate an argument into a formal language, we hope to make its logical structure clearer. We want to include enough of the structure of the English language argument so that we can judge whether the argument is valid or invalid. If we included every feature of the English language, all of the subtlety and nuance, then there would be no advantage in translating to a formal language. We might as well think about the argument in English.

At the same time, we would like a formal language that allows us to represent many kinds of English language arguments. This is one reason to prefer QL to Aristotelean logic; QL can represent every valid argument of Aristotelean logic and more.

So when deciding on a formal language, there is inevitably a tension between wanting to capture as much structure as possible and wanting a simple formal language— simpler formal languages leave out more. This means that there is no perfect formal language. Some will do a better job than others in translating particular English-language arguments.

In this book, we make the assumption that true and false are the only possible truth-values. Logical languages that make this assumption are called bivalent, which means two-valued. Aristotelean logic, SL, and QL are all bivalent, but there are limits to the power of bivalent logic. For instance, some philosophers have claimed that the future is not yet determined. If they are right, then sentences about what will be the case are not yet true or false. Some formal languages accommodate this by allowing for sentences that are neither true nor false, but something in between. Other formal languages, so-called paraconsistent logics, allow for sentences that are both true and false.

The languages presented in this book are not the only possible formal languages. However, most nonstandard logics extend on the basic formal structure of the bivalent logics discussed in this book. So this is a good place to start.

Summary of logical notions

~An argument is (deductively) valid if it is impossible for the premises to be true and the conclusion false; it is invalid otherwise.
~A tautology is a sentence that must be true, as a matter of logic.
~A contradiction is a sentence that must be false, as a matter of logic.
~A contingent sentence is neither a tautology nor a contradiction.
~Two sentences are logically equivalent if they necessarily have the same truth value.
~A set of sentences is consistent if it is logically possible for all the members of the set to be true at the same time; it is inconsistent otherwise.