What is Logic?

Logic is a part of the study of human reason, the ability we have to think abstractly, solve problems, explain the things that we know, and infer new knowledge on the basis of evidence. Traditionally, logic has focused on the last of these items, the ability to make inferences on the basis of evidence, by evaluating the deductive validity of arguments. This section explains briefly what this means.

§ 1 Logic as the systematic evaluation of arguments

M: An argument isn't just contradiction.

O: Well! it CAN be!

M: No it can't! An argument is a connected series of statements intended to establish a proposition.

O: No it isn't!

M: Yes it is! 'tisn't just contradiction.

Monty Python,Argument Clinic

In logic, we use the word 'argument' to refer to the attempt to show that certain evidence supports a conclusion. This is very different from the sort of argument you might have with your family, which could involve screaming and throwing things. We are going to use the word 'argument' a lot in this book, so you need to get used to thinking of it as a name for a rational process, and not a word that describes what happens when people contradicts or disagree with each other. .

An argument in this technical is a set of statements intended to provide someone a reason to believe one of the statements in that set, which we call a conclusion. Suppose you are wondering if you friend Bob plays the guitar. You can't quite remember exactly what he plays, but you do recall that he uses a bow. Since you know that guitars are plucked and not bowed, you conclude that Bob indeed does not play the guitar.

We can reconstruct this line of thought as an argument, like the one to the right. We call the two sentences above line premises. The word 'therefore' signifies that the sentence below the line is the $conclusion$ of the argument. If you believe the premises, then the argument provides you with a reason to believe the conclusion. You might use reasoning like this purely in your own head, without talking with anyone else. Sometimes you might work things verbally. This is not a problem, because the business of logic is not to describe exact what's going on in your mind but to systemize the rational structure of thoughts.

1: Bob plays a bowed instrument.

2: All guitars are plucked and not bowed

Therefore, Bob does not play the guitar

This example showcases a core idea in logic: some inferences are truth-preserving. These inferences, if true, would guarantee the conclusion to be true as well. This kind of inferences is called $valid$. There are a number of different ways to define the notion of validity, but we focus on this one:

$$Validity: \text{an argument is valid if and only if it is impossible for the premises to be true and the conclusion false}$$

The important thing to see is that the definition above tries to get at what $would$ happen if the premises were true. It does not assert that the premises actually $are$ true. (It also does not assert that they are false.)This is why a valid argument is sometimes defined as one where the conclusion is true in every imaginable scenario in which the premises are true. This sounds pretty imprecise at the moment, but we will sharpen our understanding of validity as we go on.

Evaluating arguments based on their validity is called deductive reasoning, in contradistinction to inductive reasoning, where the value of an inference is based on its probability. In this class we concern ourselves exclusively with deductive reasoning. The techniques you would learn in a probability or statistics class are examples of inductive reasoning.

Representing Validity

One way in which we can think about validity is by representing an argument like the one above in a graphical form that accentuates its logical structure. First, consider a circle that contains all plucked instruments and another one that contains all bowed instruments.

Premise 1

states that whatever instrument Bob plays belongs to the bowed circle. This can be represent by a smaller circle situated within the bowed circle. Based on this premise alone, do we have good enough reason to think that Bob does not play the guitar?

Not really!

Since that premise alone does not give us enough information to where guitars would be in this diagram. Premise 1 only implies that Bob's instrument is in the bowed circle - but the overlapping area is still part of that circle! What if guitars can be both plucked and bowed? Then it would belong in the middle. Premise 1 $by$ $itself$ does not discriminate against that.

Premise 2 states that

guitars are not played with bows. This means that it is impossible for the guitar circle to be inside of the overlapping areas.

Diagrams like these are immensely useful at the beginning of the study of logic. We will use them for quite a bit in the first module. There are some nuances to them, however, so we will discuss them more thoroughly. Let us finish this section with a short exercise.

Reading Exercise: Determining Validity Intuitively

For this reading exercise, try to find out if the argument given is valid, in the technical sense defined above: if the premises is true, must the conclusion be true as well?

It might be difficult to think about these arguments just in your head. Just try your best - for this exercise you are not graded on correctness. You will get credit as long as you complete the set of 10 questions.

After you are done, you can look at the answers. These problems are randomly generated, so you can redo it for practice.

Click 'Next' to begin.

Consider this argument: 'No Huskies are Bulldogs. All dogs are Bulldogs. Conclusion: no Huskies are dogs.' To begin, it's obvious that something is off with the argument: one of the premises 'all dogs are Bulldogs' is clearly false. Does that mean the argument is invalid?

Not necessarily! Remember an argument is valid $if$ the premises were true, the conclusion would be true as well. So there is a hypothetical nature to validity - we are to ask if the premises are true: we are asking what would happen if they were. So this argument is indeed valid - IF all dogs are Bulldogs, then there is no way for Huskies to be dogs.

Because of this, there is another concept for valid arguments that contain true premises called soundness. An argument is sound when it is valid AND has only true premises.

To be sure to keep this in mind when doing this reading exercise. You only have to do it once to get credit for it. You are not graded by your performance, but you are encouraged to do well on them. Once you are finished, correct answers will be shown. It would be a good idea to check and see what you have gotten wrong.

I hope by now you have gotten a sense of what validity means and how it could be something that is difficult to think about intuitively. Most of what we do here is to learn formal tools that allow us to evaluate the validity of complex arguments more efficiently and reliably. Venn diagrams, the topic of the next section, is one of them.

§ 2 Euler and Venn diagrams

In 1880 English logician John Venn published two essays on the use of diagrams with circles to represent categorical propositions, like the ones in our Bob example. Venn noted that the best use of such diagrams so far had come from the brilliant Swiss mathematician Leonhard Euler, but they still had many problems, which Venn felt could be solved by bringing in some ideas about logic from his fellow English logician George Boole. Although Venn only claimed to be building on the long logical tradition he traced, since his time these kinds of circle diagrams have been known as Venn diagrams. While they might not be as elegant and powerful as some of the tools we will learn later on, they are still useful in evaluating arguments.

Consider the diagram below that represents the claim that all guitars are plucked instruments. Outside of college logic classes, you may have seen people use a diagram like this to represent a situation where one group is a subclass of another. You may have even seen people call concentric circles like this a Venn diagram. But Venn did not think we should put one circle entirely inside the other if we just want to represent 'All X is Y.' Thus, technically speaking what we have here is an Euler diagrams, a precursor of the Venn diagram.

What's wrong with Euler diagrams?

Venn pointed out that the circles in an Euler diagram like ours don't just say that 'all guitars are plucked instruments', since area of the plucked circle not covered by the guitar circle implies that there non-guitar plucked instruments. While our background knowledge tells us that this is true, the goal of the diagram is to accurately represent the statement in question, not to impose external information. In logic we do not want to assume any proposition that is not explicitly stated. This statement is unclear about whether $all$ guitars or $only$ guitars are plucked. (What's the difference?)

Making ambiguity explicit

So it should leave it open whether the guitar circle is smaller than or the same size as the plucked circle. The problem however is that Euler diagrams cannot express this relation clearly - either the two circles are of the size, or one is smaller. There is no way to express the ambiguity needed here. Furthermore, it is confusing to put one of the circles directly on top of the other, as shown by the diagram.

Venn's suggestion:

to represent just the content of a single proposition, we should always begin by drawing partially overlapping circles. The advantage of this is that we always have spaces available to represent the four possible ways the terms can combine: Area 1 represents things that are plucked instruments but not guitars; area 2, things that are plucked instruments and guitars; area 3, things that are just guitars; and area 4 represents things that are neither plucked instruments nor guitars.

Representing impossibilities

We can then mark up these areas to indicate whether something is there or could be there. We shade a region of the diagram to represent the claim that nothing can exist in that region. For instance, if we say 'All guitars are plucked instruments,' we are asserting that nothing can exist that is in the guitar circle unless it is also in the plucked instruments circle. So we shade out the part of the guitar circle that doesn't overlap with plucked instruments.

Representing ambiguous entities

What if I want to say: Mary plays a plucked instrument? Now we are dealing with $existence$, and not just $possibilities$. To assert that a certain thing exists, we put an X in a region, so there should be an X somewhere in the plucked circle. Should we put the X in the overlapping area? We certainly don't know if Mary plays the guitar, but we also don't know if she does not play it. So we have another ambiguity here. To represent this we put the X right on the border, signifying that the statement is indifferent in respect to whether the instrument is also a guitar. (If Mary turns out to play something that is both a guitar and plucked, then the X would go in the overlapping area.)

Reading Exercise: Basic Venn Diagrams

§ 3 Aristotelian logic and categorical statements

The premise 'all guitars are plucked instruments' is an example of what logicians call a categorical statement. For most of the history of logic in the West, the focus has been on arguments that rely extensively on those statements called categorical syllogism . Aristotle began the study of this kind of argument in his book the Prior Analytics (c.350 BCE).

A categorical syllogism is a two-premise argument composed of categorical statements. There are actually all kinds of two-premise arguments using categorical statements, but Aristotle only looked at arguments where each statement is in one of the 'moods' of categorical statement. Each mood is a combination of two quantities and two qualities. A quantity of a categorical statement can be either universal - applying to everything in a category, particular - at least one thing in a category. Each categorical statement is also said to have either an affirmative or a negative quality. 'All guitars are plucked' is an instance of a universal affirmation - it makes a positive claim that everything that in the category of guitars. 'No guitar is bowed' would be a instance of universal negation. Some guitars are bowed would be an instance of a 'particular affirmation', and so on.

Basic Categorical Statements in Venn Diagrams

If a region of a Venn diagram is blank - if it is neither shaded nor has an x in it - then it could go either way. Maybe such things exist, maybe they do not. Notice that when we draw diagrams for the two universal forms, we do not draw any x's. For these forms we are only ruling out possibilities, not asserting that things actually exist. This is part of what Venn learned from Boole. The proposition, 'All guitars are plucked instruments,' denies the existence of any guitar that is not a plucked, but it does not assert the existence of some guitar that is a plucked. That probably reads like gibberish - guitars obviously do exist. So what's the deal?

The reason for this is to accommodate categorical statements about things that don't exist yet makes perfect sense, for instance 'All unicorns have one horn.' This seems like a true statement, but unicorns don't exist. Perhaps what we mean by 'All unicorns have one horn' is that if a unicorn existed, then it would have one horn. But if we interpret the statement about unicorns that way, shouldn't we also interpret the statement about dogs that way? Really all we mean when we say 'All dogs are mammals' is that if there were dogs, then they would be mammals. It takes an extra assertion to point out that dogs do, in fact, exist.

The issue we are discussing here is called existential import. A sentence is said to have existential import if it asserts the existence of the things it is talking about. Until Boole, universal-affirmative statements were often interpreted as having existential import. You might find that more intuitive, but if you interpret all universal-affirmative statements with existential import, they are always false when you are talking about things that don't exist. So, 'All unicorns have one horn' is false in the traditional interpretation. On the other hand, in the modern interpretation all statements about things that don't exist are true. 'All unicorns have one horn' simply asserts that there are no multi-horned unicorns, and this is true because there are no unicorns at all. We call this vacuous truth. Something is vacuously true if it is true simply because it is about things that don't exist. Note that all statements about nonexistent things become vacuously true if you assume they have no existential import, even a statement like 'All unicorns have more than one horn.' A statement like this simply rules out the existence of unicorns with one horn or fewer, and these don't exist because unicorns don't exist. This is a complicated issue that will come up again starting in later sections when we consider conditional statements in this module, and predicate logic later.

A categorical syllogism consists of two categorical statements as premises, and one as conclusion. The logical relations between these statements are based certain terms they share. For instance, consider this argument:

P1. All mammals are vertebrates.

P2. All dogs are mammals.

C. All dogs are vertebrates.

Notice how the statements in this argument overlap each other. Each statement shares a term with the other two. Premise 2 shares a term with the conclusion and another with Premise 1. Thus there are only three terms spread across the three statements. Each of the three statements can take one of four categorical mood. This gives us $4 \times 4 \times 4,$ or 64 possibilities. In addition to varying the kind of statements we use in an Aristotelian syllogism, we can also vary the placement of the terms involved. The combination of 64 moods and 4 figures gives us a total of 256 possible Aristotelian syllogisms. Most of these are valid, but a good chunk of them are. We won't go through every single syllogism like a good medieval logician, but we should be able to analyze them using Venn diagram.

Three terms:

Since each syllogism involves three terms, we need 3 overlapping circles.

Premise 1:

All mammals are vertebrates, so we grey out the the area of the mammal circle that does not overlap with the vertebrate circle.

Premise 2:

All dogs are mammals, so we grey out the the area of the dog circle that does not overlap with the mammal circle. Note that part of the circle was greyed already due to premise 1.

Does the conclusion 'all dogs are vertebrates' follow?

Upon examining this Venn diagram, we can see that if we greyed out areas in accordance to the premises, the only white area left in the dog circle also overlaps with the vertebrate circle. This represents the idea of an valid argument as one with premises that, if true, would make the conclusion true as well.

§ 4 Wrapping up

Reading Exercise: Definitions Review Quiz

Finish this section by completing the concepts review quiz. For this quiz, you have to answer all questions correctly to get credit. However you may try as many times as you like.

After familiarizing yourself with the ideas in the section, you should do the logical exercise ('logicise') 'Venn Diagrams and Syllogistic Validity,' where you will be asked to build Venn diagrams to determine the validity of categorical arguments.

We actually will not go any further into categorical arguments. While categorical syllogism is taught in many contexts and important for historical reasons, it is very limited compared to the formal system we will learn in this class. Nevertheless, I hope it gave a you taste of the sort of stuff we will be doing for the rest of the course. But the ideas involved in categorical logic, such as existence and categories are still very important, especially in our later study of predicate logic in module 2. Before we go there, however, we will learn about the foundation of modern logic - the formal language of sentence logic.