Introduction to Philosophy/Logic/Logic and Reason

Introduction to Philosophy > Logic > Logic and Reason

Many questions can be answered with logical reasoning alone. But even the simplest questions often have difficult answers, which may be hard to verify to be true. Let alone the more fundamental questions of life. How would one go about reasoning about such deep and sometimes vague questions?

After all, we are on a quest for the truth. In a world with many opinions, illusions of the mind, misconceptions, and perhaps even deliberate lies, how can we come to a fundamental understanding of the world, and share this understanding with others? Good reasoning should have a way to show, concretely and objectively, whether something is true or not, regardless of who writes it. That is, good reasoning should be logical.

Most of us will have pondered many questions in the past. But how do we know if the reasoning we use is correct? The study of logic allows us to understand how to construct correct lines of reasoning.

The most basic element of logic: the argument
A good first step is to consider what we would do if we were to try to convince someone else: we first subdivide the complexity of our big line of reasoning into the smallest possible chunks — the individual arguments that can each be individually assessed for correctness. This way, if our full line of reasoning is valid, each argument will logically support each other and lead to a convincing final conclusion.

It is important that we shouldn't skip over any arguments, even if these skips may seem "obvious". After all, each argument builds upon the last, and if we break one link in the chain, the entire line of reasoning falls apart.

As such, the types of arguments we will encounter are usually very simple. After all, each such argument should be a concrete logical statement that can be relatively easily, perhaps even trivially, verified to be true or false. A commonly cited example of a logical argument is the following:

This is a very simple argument, and the conclusion is perhaps plainly obvious. Indeed, that is a key feature: each argument should be as plain as possible, so that the chain of arguments is as strong as possible. After all, that would allow us to use the conclusion of this argument, that Socrates is mortal, in a next argument, to support some bigger conclusion.

Indeed, arguments form the most basic of building blocks within logic. Throughout the various pages of this chapter, we will explore arguments in great depth. In the next section, we will cover how to reason about individual arguments.

The principles of an argument
The example we just saw is called a syllogism, which is a form of deductive reasoning that uses two propositions to come to a single conclusion. In this case, the first two statements are the propositions, which are assumed to be correct. If they are indeed, then the logical conclusion, that Socrates is mortal, is true.

This makes it easy to see where a logical flaw occurs in such reasoning. If everything before the line is assumed to be correct, and the logic is valid, then everything after the line must be true as well. Importantly, there are only two ways in which such arguments may be flawed: either (one of) the propositions themselves are false, or the logical deduction itself (i.e., the step that occurs at the horizontal line) is false. For example, consider:

The logical part of this syllogism is valid: indeed the conclusion is true if the two propositions are. However, the proposition that all men are immortal isn't in fact true, and as such the complete argument, although valid, isn't sound.

Alternatively, consider:

Although both propositions are correct, the deduction isn't logically valid.



This methodical way of writing down arguments makes it much easier to verify their truthfulness. But by using natural language, like English, it becomes quite easy to make wooly arguments that sound correct, while they are actually incorrect.

To make it easier to see what the actual logic behind a syllogism is, a syllogism can be stripped of its content and generalized to:

This syllogism is logically valid. This means that you can substitute what you like for,   and  :

Importantly, with this simple building block, it becomes possible to create long chains of verifiably correct reasoning. There exist other types of arguments beyond syllogisms as well, which we will cover in detail in one of the later sections.

So far, we have assumed the propositions to be correct, without really formally checking them individually. Is it indeed the case that all men are mortal? And is Socrates really a man?

Although answering these questions with rigor is difficult, if the propositions are the conclusion of a different argument, we can chain those two arguments together. In practice, that is often very challenging, and often, there will remain propositions that cannot be formally verified.

The importance of correct reasoning
Just as it is important to astronomers that their telescopes are functioning properly, it is important to people in all disciplines that their processes of reasoning are working properly. Logic is something fundamental to all science, in the broadest sense of 'science'. But nearly all logicians will tell you that what they are doing is not just philosophy of science.

Though reasoning is carried out by human minds, most logicians would also say that what they are doing is not just psychology. Though it is carried out using language, and we shall be talking about grammar, syntax and semantics, most logicians would want to distinguish logic from linguistics and the philosophy of language.

From the time of Aristotle and the Prior Analytics, through to the middle ages and beyond, logic was studied in a largely verbal form. Indeed &lambda;&omicron;&gamma;&omicron;&sigmaf; (logos), from which the word 'logic' is derived, can mean 'word' or it can mean 'reason'. There were various forms with nice Latin names like modus ponens and modus tollens, and if you want to become a logician you will need to know what those mean. But certainly from the time of Frege onwards, i.e. late 19th Century, logic has become much more symbolic, and it starts to look like mathematics, with which it has much in common. But even here, many logicians would say that what they are doing is not just mathematics.

Or if it is mathematics, there is a big question about what mathematics is. Replies to this question are known as formalism, platonism and intuitionism. Both logic and mathematics are concerned with structure in a rather abstract way — in the case of logic it is the structure of reasoned argument, whereas with mathematics, it is often, though not always, the structure of numbers.

The stuff that looks like mathematics is called 'formal logic'. There is also a lot of stuff that isn't quite so mathematical that asks questions like: 'What are names?', 'What is truth?', and so on.

The problem with saying more about what logic is than what I have said at the beginning, is that you start to use logic to define logic. This is circular and self-referential. Logicians know that circularity and self-reference can cause problems and paradoxes. How and when they do and how and when they don't is I suppose interesting.