“The point of logic is to give an account of the notion of validity: what follows from what”
(Graham Priest, Introduction to Non-Classical Logic, 2008)
Part I: Lesson
1. Logic in Acting
Logic is all about argument and persuasion. Consider the following conversation between two students:
Student A: Hey there! I just realized something really cool.
Student B: Let’s hear it.
A: I realized that color isn’t real.
B: Ok, I’m confused. Why do you think that?
A: Well, we see with our eyes, right? And seeing is just what happens when light hits our eyes and our brain processes it in a certain way. The reason something looks ‘blue’ is just because the light that bounces off of it is a different frequency than the light from something that looks ‘red’. Thus, different colors don’t have anything to do with the objects we are seeing, it’s just different kinds of light waves!
B: Well, what you say may be true, but I’m not sure that you’ve proved to me that ‘color isn’t real’.
A: What do you mean?? I just told you that color is just our brain’s response to light waves, so color is just something ‘in our brains’, or at least something that we think we see after our brain processes light in a certain way — the stuff we think has different colors might all in fact just be the same shade of gray; the only thing that matters is how our brain processes different light waves!
B: You still haven’t convinced me. I accept that our brains react differently to different light waves when we see color, but you haven’t given me a reason to think that this directly means that color isn’t real. Couldn’t we use our brains to get the sensation of different colors from light waves with different frequencies AND for it to be true that the objects are actually colored? Maybe the light waves just transfer the color from the object to our brains? If you want me to believe your conclusion, you are going to have to give me a better argument than that!
What’s happening here? Well, Students A and B are arguing about a conclusion: “color is (not) real”. To try to convince Student B, Student A presents an argument with premises: “seeing is light hitting our eyes”, “the frequency of the light determines the color we see”, (and other premises). Student B isn’t convinced, because they don’t think that the conclusion follows from the premises. That is, Student B thinks you can’t infer “color isn’t real” from the premises Student A offers.
[Activity]: Can you think of any discussions in your experience that are similar to the one between Students A and B? Share with the class/with a partner, explaining why you had a disagreement and what it was about. Could learning concepts like the bolded ones above help you in future discussions? If so, how?
[Activity]: What do you think is the point of logic? What do you think we study in logic and why? Share with a partner!
2. What Are We Doing When We Do Logic?
a. Premises, Conclusions, Inferences
Logic is the formal study of correct patterns of reasoning. Whenever we make a decision, pass a judgment, or express a position, we do so in a fairly patterned fashion. Whether explicitly or implicitly, we start from a set of assumptions and beliefs about something and we use them to arrive at some further belief. Our starting assumptions and beliefs are called premises, while our belief resulting from our starting assumptions is called a conclusion. The overall process of getting from premises to a conclusion is what we commonly call an inference or argument.
As we think about how we reason from premises to conclusions, it is very useful to cast each component of an inference as a declarative sentence. Declarative sentences are just those sentences which assert something in a way that allows them to be either true, or false (and not both). “All people are mortal.”, “Fire is cold.”, “All bachelors are unmarried.”, “Some tables are wooden.” are all examples of declarative sentences, each of which is capable of being evaluated as either true, or false. In logic, we are especially interested in seeing what kinds of inferences we use and in characterizing which of them are correct and why.
b. Logical Form
Inferential reasoning is fundamental to virtually every kind of decision-making practice in which we engage. As such, we certainly have some intuitive notion of what counts as a good inference and what does not. For example, it is very intuitively obvious that something goes wrong in the following:
Premise 1 (P1): All people are mortal.
Premise 2 (P2): Aristotle is a person.
Conclusion (C): Aristotle is not mortal.
In order to characterize what goes wrong in such instances, we use the notion of logical form. Every inference is reducible to some structural pattern, to some form; we wish to study these forms in order to establish which of them will lead us to good conclusions and which of them will not. In future lessons, we will develop a vocabulary to describe these logical forms. We will see that the above example is called a syllogism, and we will discuss the notions of validity, soundness, and truth.
[Activity]: In your own words, try to describe what you think is wrong with the above argument!
[Activity]: With a partner, come up with your own examples of arguments like the one above. Try replacing ‘All’ with ‘Some’, putting ‘not’ in front of ‘mortal’, replacing the words with others you like better – get creative! Discuss how each argument you come up with works (or doesn’t). Try to find patterns!
Part II: Examples
Let’s return to some examples of arguments in order to find out some more things about them. First, we will look at two instances that show how helpful it really is to take arguments expressed commonly and recast them in the premise-conclusion form. Consider the following sequence of statements:
1) It’s pretty obvious that all snakes are animals. Think about it. What’s a snake? It’s a reptile, right? I mean, not just a single snake, all of the snakes. You take any snake and it’s a reptile. But then what’s a reptile? Again, all the reptiles, not just a single one, you get it. Take any reptile – is it not an animal? Of course it’s an animal! So, all snakes are animals.
This is an unclear way to make an argument, at least compared to a formally structured one, but we don’t always formalize our arguments before making them; for example, the speaker above begins their argument with the conclusion – a very common practice. This is not to say that formalized arguments are somehow more correct or that our common speaking terms are defective; however, formalized arguments do make it easier for us to characterize what is going wrong (or not) in an inference than regular language does. Generally, when we’re looking to apply the tools of logic, a messier structure can be misleading and makes it harder to say anything about the formal components of an argument. The previous paragraph is just a roundabout way to express the following argument:
Premise 1 (P1): All snakes are reptiles.
Premise 2 (P2): All reptiles are animals.
Conclusion (C): All snakes are animals.
Reducing to premise-conclusion form is a useful tool to parse out what is really going on when a claim of any sort is being made. Now let’s consider another argument:
2) It’s pretty obvious that all tables are chairs. Think about it. What’s a table? It’s a piece of furniture, right? I mean, not just a single table, all of the tables. You take any table and it’s a piece of furniture. But then what’s a piece of furniture? Again, all the pieces of furniture, not just a single one, you get it. Take any piece of furniture – is it not a chair? Of course it’s a chair! So all tables are chairs.
If someone expressed this to us, we might be tempted to say, “That’s not an argument!”. We commonly say something like that when faced with particularly faulty reasoning, but from a logical perspective this would not be correct. The above paragraph is an argument, albeit a bad one – we can see how bad it is when we change it to a series of declarative sentences which can be evaluated as either true, or false, as follows:
Premise 1 (P1): All tables are pieces of furniture.
Premise 2 (P2): All pieces of furniture are chairs.
Conclusion (C): All tables are chairs.
Again, reducing the argument to premise-conclusion form makes it much easier to identify what is wrong with it – the second premise is false. The point is that an argument has a well-defined set of features, and any sequence of statements that fits them counts as an argument. Of course, it follows that certain sequences of statements are not arguments, like the following:
3) A self-taught musician, [Julian] Bream learned playing to radio dance bands with the lute his father bought from a sailor on London’s Charing Cross Road in 1947. As a child prodigy, his early recitals led to him being “acknowledged as one of the most remarkable artists of the post-war era”, according to the Royal Academy of Music. After studying piano and composition at the Royal College of Music, and completing national service, he became one of the most prolific and best-selling recording artists in classical music.”
(BBC News, Julian Bream: Classical Guitarist Dies Aged 87, 2020)
This sequence of statements does use some language that is frequently employed in argumentation and each declarative statement taken on its own is capable of being true or false. However, there is no discernible logical connection between them – there are no premises being leveraged towards a conclusion. Thus, this would not be considered an argument. It is merely a sequence of claims about the life of a beloved classical guitarist.
[Activity]: We have mentioned that declarative sentences are the objects of logical analysis. Recall that declarative sentences can be either true, or false. Considering this, do you think that statements such as questions or exclamations can be evaluated from a logical perspective? Why or why not?
Part III: Exercises
Part IV: Conclusion
Logic is the formal study of correct patterns of reasoning. When we engage in logical inquiry, we are not concerned with particular features about the content of the sentences we are analyzing. Instead, we are preoccupied with the logical form of the inferences under study. In order to characterize these inferences, we will need to develop a particular vocabulary that captures its features. This will be the topic of our next lesson, covering validity, soundness, and truth.