Preface

Note: as I’m rewriting this intro a few parts in, I must confess that the tutorial series took a bit of a weird turn: I realized that I would have to come up with examples to elucidate concepts. This turned into me indulging in whatever I found funny at the time, so be advised.

Welcome to my crash course in symbolic logic. While I strive to make this tutorial series accessible to anyone, the topics discussed will be explained at a level roughly corresponding to a first or second-year undergraduate course. The reason I decided to take on this project is that logic can seem like a daunting topic. It certainly did for me. But it really doesn’t have to be. For a topic like symbolic logic, having good educational resources, such as a patient professor and a concise textbook, make a HUGE DIFFERENCE. 

While having an awesome tutor and disciplined practice are key, I want to help reduce the need to buy an expensive textbook. You see, I have this sneaking suspicion that academic publishers (not the authors, mind you!) are cleptomaniacs who love charging poor students hundreds of dollars to buy books chock full of filler material. So, the aim of this tutorial series is to provide an additional resource covering the basics of symbolic logic in a distilled form. It should be noted that learning symbolic logic largely means doing symbolic logic. I will have a couple of examples per chapter for you to get a general idea, but I highly encourage you to find/construct your own examples. If you’re a student, think of this tutorial as your friend’s class notes: it’s not an exhaustive explanation of all concepts, but it may be the thing you need to finally get that ‘aha!’-moment. 

To help us with examples, I will introduce you to a set of characters for each part of the series. 

Cast of characters

Wow, you’re in luck! Daniel and Tony are here to help us for this part of the series. I am truly touched by their commitment to our education.

What is symbolic logic?

In order to answer this question, it makes sense to first ask more broadly what logic is. Logic is the study of reasoning. It is a normative endeavour: it deals with questions about how we should reason, and evaluates what constitutes good vs. bad reasoning. It should be noted that a logical argument’s goodness/badness is purely evaluated on form and not meaning: the semantics of a logical argument can be completely absurd and empirically untrue and yet still be logically valid and vice versa. 

Symbolic logic narrows down this enterprise a little further. It is a system that codifies reasoning into formal rules and symbols. There are many different kinds of symbolic logic, each governed by a specific set of rules and assumptions. Some types of logical systems include paraconsistent logics, predicate logics, modal logics, and higher-order logics. The types of symbolic logic we are going to discuss in this tutorial series are sentential logic (AKA propositional calculus) and predicate logic (AKA first-order logic). The type of logics that we’re going to look at are said to be bivalent, i.e. they only have 2 values: true and false. They also abide by a rule known as the law of the excluded middle, meaning that they are necessarily either true or false — there is no in-between.

Basic Concepts

Propositions & Arguments

A proposition (AKA statement, sentence), in logic, is a statement that can be evaluated as either true or false. 

An argument, in logic, is a set of propositions containing premises and a conclusion. The premises are supposed to contain the evidence/reasons for the conclusion. Logic is all about the relation between an argument’s premises and its conclusion. This is where the evaluation happens! One of the most simple forms of a logical argument is called a syllogism. Such an argument contains two premises and a conclusion. Here’s an example of a syllogism:

Deduction & Induction

Arguments can be either inductive or deductive:

An inductive argument is one where the premises at best suggest the conclusion and you cannot be absolutely sure. Any type of probability statement is an inductive one. Any type of empirical science (science based on observation) is engaging in inductive reasoning: a scientist observes a circumstance for a finite number x times; they notice a trend in result y; they (inductively) conclude that y is true. The induction stems from the finiteness of the observation: because the evidence (premise) for the scientist’s conclusion is based on a finite set of observations, she cannot absolutely conclude that y is true — because there is always the possibility that a future observation of the same circumstance will yield a contradictory result. This is known as the problem of induction. For instance: if I said that the sun rises in the morning and sets in the evening, you would say that this was right. This is an inductive argument, because there is an assumption that this will always be the case — which is not something anyone could guarantee. After all, some freak cosmological event may occur which destroys the sun, thus falsifying the argument…

A deductive argument is one where the premises are meant to provide absolute evidence to guarantee the conclusion. There is no need to infer any other information: all you need to make infer the truth or falsehood of the argument is found in the premises and the conclusion. This is the type of reasoning that underlies symbolic logic and other formal systems such as mathematics. The syllogism above is an example of a deductive argument. We will only deal with deductive reasoning in this tutorial series. It should be noted, however, that some logics (modal logics) deal with possibility - but they do so in an a deductive way. If you’re interested in learning more about modal logics after this tutorial, I suggest looking into the work of Saul Kripke.

Form, instance & validity

As mentioned before, logical evaluation of an argument only concerns itself with the form of the argument. As such, what makes an argument valid or invalid is a question of form. Consider these 2 arguments:

These two arguments are instances of the same argument form: [p or q; not q; therefore p]. There are infinitely many instances of the same form, with an infinite array of subject matters. The form is determined by the rules of the logical system, which consist of the truth functions of logical connectives (such as ‘and’, ‘or’, ‘not’, ‘then’). If this sounds like gibberish right now, don’t worry. We’ll get into logical connectives and evaluating truth values as soon as we have established some conceptual scaffolding. 

An argument instance is valid if and only if (iff) it has a valid form. An argument form is valid iff there are no instances of that form where all premises are true and the conclusion is false. In other words, validity is when there is no way for the conclusion to be false if both premises are true; the truth of the premises guarantee the truth of the conclusion. That being said, an argument can be valid even if one or more of its premises are false. An argument is said to be sound if its form is both valid, and all premises are also true. 

An argument form is invalid if there in fact is an instance where all premises are true and the conclusion false. This is where the power of counterexamples comes in: if you want to prove to a friend that their argument is invalid, all you have to do is to provide a counterexample, which is to say another instance of the same form where the premises are true and the conclusion is false.

Counterexample example

Let’s say your friend Daniel, an esteemed intellectual and Rick & Morty enthusiast, argues the following:

Daniel thinks his argument is bulletproof: not only does he think it’s valid but also sound. After all, the premises are true, and the conclusion is also true. Pretty solid, right? He thinks he’s “WReCkeD you with LOGIC”. But you’re convinced that the form of the argument is actually invalid. In fact, you’re sure that your friend is committing a formal logical fallacy known as affirming the consequent. You observe the form of the argument: [if p then q; q; therefore p]. You have just read this tutorial and realize that if you can find another instance of the same argument form, where premises are true and conclusion false (i.e. an invalid instance), you can make your friend see the error of his ways. You’re reminded of your uncle, Tony, who has a very long neck. Like, really long - everyone in town knows about it. In fact, this was a source of great anguish when you were a child because all the other kids at school — including Daniel — would tease you for having a long-necked uncle. You realize that Daniel will readily acknowledge the truth of the proposition that Tony has a long neck. You start thinking about other things that have long necks but aren’t human like Tony: guitars, giraffes, dinosaurs, snakes (for philosophical purposes a snake is basically just one long neck)... “Okay,” you say, “now I just gotta piece together an argument of the same form using this information. This is what you come up with:

PArt one wrap-up

What we have learned so far

In this section of the tutorial we have learned that logic is the study of (primarily deductive) reasoning, and that symbolic logic is the systematization of reasoning through symbols and formal rules. Logic is normative, meaning that it evaluates reasoning based on whether it’s good (valid; sound) or bad (invalid). These in turn are evaluated based on argument form and not argument content. We learned that one way to show that an argument is invalid is to provide a counterexample, i.e. another instance of the same argument form, where the conclusion is false but all the premises are true.

Where are we going next?

It should be noted that the examples of arguments used here are mainly meant to get the intuition pumps working. I have just expected you to intuit the truth/falsehood of premises and conclusions. We haven’t actually gone through how to assert truth and falsehoods. This is where we will be heading next. In the next tutorial, we will introduce sentential logic (AKA propositional calculus) and its Boolean operators. We will stick to sentential logic for quite a while.