Basics

Introduction

One definition of philosophy has it as a sort of activity, rather than a body of subject matter. This definition holds that philosophy is just “thinking hard about something.”

But this cannot be exactly right: a young earth creationist might think hard about how it was Noah fit the dinosaur eggs on the Ark alongside all of the other animals, but no one would say she is engaging in philosophy. So some other condition is necessary before we can declare a certain sort of thinking ‘philosophical.’

Probably, this other condition is thinking clearly.  If this is correct, then logic is absolutely essential to doing philosophy, for it is a cornerstone of reason. Studying logic teases out the inferential processes our brains are using all the time and helps us hone them, thus making us better thinkers. In addition, it  helps us learn about ourselves (or, more properly, our brains).

Take the principle of non-contradiction. This principle holds that no statement can be both true in false in the same sense, at the same time. Obviously, such precepts are fundamental to proper thinking and if we cannot even have these, any other sort of inquiry into the nature of the world becomes irrelevant.

Purpose

This section of the site, then, is dedicated to logic, which is necessary for philosophy. In particular, it focuses on ‘symbolic logic,’ which is a formal (almost mathematical) elucidation of deductive logic. All this means is that symbols are used to stand for statements (sentences which are either true or false) and logical connectives (such as ‘∧,’ which merely stands for the word ‘and’). By ‘deductive’ logic, we mean the sort whose arguments have conclusions which are necessarily true when their premises are true.

The point of all this is to study the ‘form’ of deduction rather than any content which might be deduced. In other words, we do not particularly care about what the statements which encompass an argument are actually saying; all that is our concern with symbolic logic is their relationship(s) with one another.

What an Argument Is

In philosophy, an ‘argument’ is merely a collection of statements such that certain statements (premises) are set up as reasons for believing another statement (the conclusion). Any philosophical paper worth the effort it takes to read it will have multiple arguments, usually interlocking. Philosophy is thus argumentative by nature: reasons are provided for believing or not believing certain statements about the world.

What an Argument Looks Like, Symbolized

Here’s a pretty basic argument:

   1. If Ned likes cheese, then he is a block.

   2. Ned likes cheese.

   3. Therefore, Ned is a block.

Now, when we turn these statements into ‘statement variables’ and symbolize the argument, it looks like this:

1. L → B
2. L…Therefore, B
3. B(2, 1, MP)

Each numbered line is a premise. For simplicity’s sake, this argument is only two premises long. The ‘Therefore’ denotes the conclusion of the argument. All steps following the premise with the ‘Therefore’ designator are steps to proving the conclusion correct. Proving a conclusion merely involves using various inferences to reach it with the information given. Think of it like a game, where the end-goal is the conclusion itself.

As you can see, symbols (letters) stand in for the original statements of the argument. In this case, L stands for ‘Ned likes cheese’ and B stands for ‘Ned is a block.’ The arrow between L and B in premise (1) is a logical operator which represents a ‘material conditional;’ all this means is that when L is true, then B is true. Finally, the content of the parentheses at the end of premise (3) include the inference used to produce the premise as well as which premises the inference was used on. In this case, we used Modus Ponens (MP) on premises (2) and (1) to produce the conclusion, B (and thus we won the ‘game’ of proving an argument correct in only one step!).

As the above argument illustrates, the content of the premises is completely irrelevant for logic. Rather, it is how these premises interact with one another that interests us.

What You’ll Find on this Site

For some ungodly reason, publishers of symbolic logic textbooks refrain from including complete answer sheets with their books. One purpose of the logic section of this site is to remedy this injustice: you will find complete solutions sets to the texts I have worked through. I think this is important, because learning symbolic logic is actually much easier when one sees how ‘difficult’ problems are solved. Actually, these problems are tricky rather than difficult; in any case, having their solutions can be a major help in learning logic.

  • A glossary

In addition, I will be creating a glossary of logical terms. This will be drawn, primarily, from the texts I work through. There is no canon for symbolic logic and different texts use similar terms slightly differently. Such variances in meaning will be noted.

Central to understanding logic is understanding (memorizing) the list of valid inferences one can make on various arguments. There are quite a lot, but most texts boil them down to an essential list of 20 or so operations which are most common. This site will make them readily available to the reader.


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