Propositional Logic Introduction

Chapter 1 9 mins

Learning outcomes:

  1. A brief history of logic
  2. What is propositional logic
  3. What are propositions

What is logic?

Logic is a really broad term , that can specifically refer to different things. But whatever it refers to, it all boils down to the following:

Logic is the science of reasoning.

The intuitive sense that we all have and use all day long in reasoning, on arguing on things, is what logic is.

If we formalise all these ways we think and reason on different things, we get formal logic. But first let's take a ride back in the past...

Going back in ancient times of the Greeks, Aristotle was the first person to formalise the ideas of reasoning. He wrote the first treatises on formal logic. Due to his contribution to logic and thinking, as we study them today, he is known as the 'father of logic'.

Now Aristotle essentially worked on logic in a linguistic manner. That is, he never really came up with a rigorous set of symbols to blend into his ideas of logic.

However, his formalisation of reasoning was solid enough to influence other people to develop it into the modern logic we know today.

In the 18th century, mathematician Gottfried Liebniz conceived of having symbols in formal logic to ease working with it. Although, he did a lot of work in this area, it largely remained confined to draft papers; and likewise unseen.

In his book History of Western Philosophy, the logician Bertrand Russell claimed that Leibniz had curated logic in his writings to an extent which was achieved some 200 years later; it's just that those writings were never published.

In the 19th century, with their tireless work, mathematicians George Boole and Augustus De Morgan gave a systematic, mathematical and symbolic approach to logic. They came up with symbols to represent concepts in logic, which although arcane today, ultimately led to the construction of the symbols we use today in mathematical logic.

George Boole is considered an extremely noteable personality in logic. He is known as the 'father of symbolic logic'. The term 'Boolean', which refers to true or false values, was created in his honor. He was the one who created Boolean algebra.

So altogether in the 19th century, the till-then mere studies of formal logic took a sharp turn, got a symbolic and mathematical touch, and became a principal subject in the area of mathematics.

Based on this new approach to logic, the foundation of mathematics which were believed to be weak, were also formalised.

Today, what we study in mathematical logic, is much what was developed in the 19th century by an array of logicians including, but not limited to, George Boole, Augustus De Morgan, Charles Sanders Pierce, Gottlob Frege, Giuseppe Peano, Bertrand Russell, Alfred North Whitehead and so on.

So as you might realise by this point, logic has its roots way way back when the Greeks and other ancient civilisations studied it, and from that point on elapsing all this long span of time, found its way in modern-day mathematics, and guess what — modern-day computer science.

What is propositional logic?

Propositional Logic, or PL, is the most fundamental area of logic. That's why it's sometimes also referred to as zeroth-order logic — everything else (in logic) begins from here.

Simply put, it's the area of logic that deals with propositions.

Now what is a proposition?

Well, it's really simple.

A proposition is a declarative sentence.

For example, the sentence 'I love cats' is a proposition. It's a simple declarative sentence.

By that means, propositional logic is the area of logic that deals with declarative sentences.

And to be good in PL, we have to fully and truly understand its most fundamental concept — propositions.

Although, the definition above for propositions is correct, it's incomplete. Let's first see some examples of propositions before having a look over the complete definition.

Below shown are a couple of sentences. Your task is to determine whether each one is a proposition (a sentence in which some assertion is made; some fact, or opinion is stated) or not.

  1. Python is a programming language.
  2. JavaScript was created by Brendan Eich.
  3. What time is it?
  4. Please pass the book to me.
  5. Can I go the washroom?
  6. ::45 + 5 = 10::
  7. ::x + 5 = 10::

Starting from the first one, it's indeed a declarative sentence. So is the second one. Both these sentences are clear-cut facts (which may be true or false, but it doesn't matter as to what are they) and when we know we are working with facts, we know we are working with propositions.

The third sentence is not a declaration. It's rather a question, and questions are generally not regarded as propositions. Hence, it's not a proposition.

The fourth sentence is also not a declaration. It's rather an imperative/command. You're asking someone to pass the book to you. Akin to questions, imperatives are also not declarative sentences, and likewise not propositions.

The fifth sentence is again a question and likewise, not a proposition.

The sixth sentence is a declaration. We're saying that ::45 + 5:: is equal to ::10::. We know that this is not true, but who cares. It is a declarative sentence, and therefore a proposition.

The seventh statement is a declarative sentence, but notice one thing. All the declarative sentences we saw above, could be termed as either true assertions or false assertions. But we can't say the same for this seventh sentence. ::x:: is undefined and likewise we don't know whether ::x + 5 = 10:: is true or false. Currently, it is neither true, nor false.

It turns out that declarative sentences that are neither true, nor false aren't considered propositions.

Likewise, here's the partial completion of the notion of a proposition:

A proposition is a declarative sentence that is either true, or false.

Although this definition is now more comprehensive in explaining what exactly is a proposition, it still misses on one key idea.

Let's clarify that idea with one example...

Consider the statement: 'This sentence is false.'

Do you think that this statement is a declarative sentence?

Well, yes it is.

Can this statement have a truth value? In other words, can you say that it is true, or say that it is false?

Let's reason...

If we say that this sentence is true, we are saying that whatever is written in the sentence is true. However, the sentence then says that it is false. Hence, it is both true and false at the same time!

You don't need to understand the intuition behind the sentence 'This sentence is false'. It represents one of the most confusing things in logic, which we'll see later on in the following text.

Conversely, if we say that this sentence is false, then we are saying that 'This sentence is false' is false. If 'This sentence is false' is false then its opposite would be true. That is, we are asserting that the sentence is true. Once again, the sentence is both true and false at the same time!

If a declarative sentence has none of the truth values (true or false), as we saw in the seventh sentence above, it's difficult for us to combine it with other sentences and then do reasoning based on it.

Similarly, if a sentence has both truth values, once again it would be strenous to combine it with other declarative sentences that have only one truth value at any particular instance, and then do reasoning with it.

Likewise, we remove the possibility of such dual-natured sentences creeping into propositional logic. They are known by a special name — paradoxes.

A paradox is a statement that is both true and untrue at the same time. It is a self-contradictory statement.

Anyways, now that we have finally filtered out all the things that are not propositions, we could consider the complete definition of a proposition:

A proposition is a declarative sentence, that is either true, or false, but not both (at the same time).

Let's see how well do we now know about the concept.

Determine which of the following sentences are propositions.

  1. I love cakes.
  2. Give me the donut.
  3. Who came first in the exams?
  4. ::x + y = 5::
  5. What is your problem?
  6. Alex is playing cricket.
  7. Let's play football.
  8. ::1 \times 5::.
  9. ::2 \times 10 = 21::.
  10. Auditions are happening in Room 1.
  1. Yes
  2. No; it's an imperative.
  3. No; it's a question.
  4. No; it's doesn't have any truth value.
  5. No; it's a question.
  6. Yes
  7. No; it's an imperative.
  8. No; it's not a declarative sentence.
  9. Yes
  10. Yes

Moving on

So this is it for the introduction to propositional logic. Its elements are propositions which are declarative assertions, either true, or false, but not both. Simple!

In the next chapter, we'll discover what are atomic propositions, what are compound propositions and how can we use propositional variables to simplify working with propositions.