It may sound surprising, but you already have all it takes to produce high-level logic: right behind your eyes.
In the first term of college, I was introduced to a subject called Logic. In the "I think, therefore I am" fashion, you might think this would be an easy subject. After all, we are all taught how to think throughout school and, since all of us "think", there would be nothing unfamiliar in the subject that could trick the less-than-alert mind.
Shockingly enough, in the same way people have a hard time with Math in college, Logic was the subject back then where students failed the most.
Why is Propositional Logic Hard?
What could be the reason for that? The only thing, in theory, that might prove troublesome was the amount of unfamiliar terminology involved in propositional logic.
The professors used Latin expressions like "modus ponens" and "modus tollens" to define some sort of processes of inferring a result, and these often caused people to complain about the "esoteric nature" of the subject.
We can call this metalanguage. In this case, it can be roughly translated as "fancy language to talk about a topic that makes it harder for the students to understand what the thing really is". And it has its charm in the sense of categorizing things.
Intense practice with examples, however, can help students learn much more quickly than filling up their hard drive with vocabulary they'll probably only use to pass the tests and then never use again – except in those word game puzzles.
This is the point where I get in. Having had good results in Logic back then, I was shocked with how my classmates talked about Logic on social media (sentences like "this is not a topic for first term – it's too hard", or "why do they expect me to learn Latin? This is the 21st century!" were common).
Fortunately for me, instead of focusing on the naming of each inference rule I heard of, I decided to focus on understanding what each one led to – which probably made me worry less about the topic than my peers.
Ergo… 😋 just kidding. SO, let me try to show you in this article some of these rules of propositional logic.
I will try to avoid the fancy metalanguage and present you with a couple of examples to help show you that you already think the way the Logic subject presents to you.
You just skip the part of using a fancy name and giving a complicated explanation for the process itself.
Modus Ponens, Modus Tollens, Hocus Pocus, Abracadabra...
Let's begin with the really tough ancient Latin names: Modus Ponens and Modus Tollens. Modus ponens is defined in Wikipedia as follows:
In propositional logic, modus ponens (/ˈmoʊdəs ˈpoʊnɛnz/; MP), also known as modus ponendo ponens (Latin for "method of putting by placing") or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q. P is true. Therefore, Q must also be true."
Now I don't know about you, but when I see phonetic symbols and words that I have no idea what they mean – but someone is telling me I should – my heart skips a beat.
In fact, we could go straight to the last two sentences here to get to the point. The first long and haunting sentence basically serves to tell you "heroes, there is a dark and menacing-looking forest ahead of you. Leave behind all hope, ye who enter."
The bottom line, though, is "when the first thing is true, the second is, too. Since we know that the first is true, what do you make of the second?"
See? After beating the Latin words and hacking and slashing through 'propositional', 'implication', 'antecedent', 'deductive', and 'inference', you just come to the conclusion something is true because another one is true as well. Great! Now moving on.
Example of modus ponens
As I mentioned before, I believe examples work better than fancy words, so let's come up with a simple example:
I'm from a country where there is no snow at all, ever. When I'm in the US during the winter, I love making snow angels. It's winter, and I'm in the US. What do you think I'll do when it snows?
If what you thought was "you'll make snow angels, duh", congratulations! You have just gone through the propositional logic we talked about earlier. And you did not have to say a prayer in Latin, use magic words, cast a spell, or anything else. 😊
Peas and Queues
Now you may ask, "ok, but what about the P and Q thing in the last two sentences of the definition you shamelessly CTRL-C-ed and CTRL-V-ed from Wiki?"
Good catch! These are representations of what we call propositions (that's why this is called "propositional logic", by the way).
A proposition is nothing more than a sentence. They could have said "sentence A" and "sentence B" instead. But someone in the past chose P and Q, just like in Math you'd go with X and Y.
Using the example above, being in the US in the winter (proposition P), to me, means (implies) that I must get to the ground and wave my arms and legs frantically to make what looks like the shape of an angel.
Now, you know I'm in the US in winter and it snows (I told you that "P" is true). Here is the moment where we conclude that I'll do what I say I do whenever the first sentence is true – I'll act according to the second proposition ("Q") and make it true as well.
Now that you know how silly the author of this article is and understand the first rule better, let's move on to the next rule, avara kedavra… I mean, modus tollens.
Here, Wikipedia "helps" us once again with a beautiful, wordy definition:
In propositional logic, modus tollens (/ˈmoʊdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away") and denying the consequent, is a deductive argument form and a rule of inference.
Modus tollens takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
If you did not love going through 'denying the consequent' and 'so is its contrapositive', you are a heartless human being.
Example of modus tollens
Again, here comes Mighty Mouse to save the day and tell you to focus on "If P, then Q. Not Q. Therefore, not P."
What this rule is saying is actually a complement of what the first one says. If the second sentence is not true, then the first probably isn't, either. Thus, if I'm not making snow angels now, what would you make of it?
Since we have paired the idea of being in the US in the winter and making snow angels, at least some part of the first proposition can't be true: either I'm not in the US or it's not winter.
Anyways, since, for the first proposition to be true, these two parts have to be, we can assume that the first sentence is, somehow, false.
And just like that you realize you already know two propositional logic rules without even having to consult your dictionary! 😃
In this article, we saw that it is possible to know (and practice) logic without even studying it. We also learned that we already know more logic than we might imagine and that the author of this article loves playing in the snow.
As a bonus, we practiced two rules of propositional logic, whose fancy names might scare you from even looking at them: modus ponens and modus tollens.
There are others, though. If you liked the way they were explained here and would like to see more logical rules explained in a form you will definitely understand – and possibly realize you already make use of – send your feedback to yours truly on Twitter. I'd love to walk you through the other rules as well – and maybe share more of my silliness through examples.
Happy coding! 😉