Propositional logic is a logic that concerns itself with propositions. But what is a proposition? A proposition is a statement that can be evaluated to be true or false. A proposition cannot be both truth and false, nor can it be neither true nor false. The truthhood of statement is what we call it’s truth-value.
Examples of Propositions:
Alice ate the pizza.
1 + 1 = 3
Nonexamples:
(Propositions cannot be questions because questions cannot maintain truth values)
x + y = z
(The truth value of this statement depends on the values of x, y, and z. Thus we cannot determine the truth value of this statement and cannot be a proposition.)
Before I introduce how logic actually works, I would like to make a few remarks. Propositional logic is a language, a symbolic language like mathematics. It differentiates from natural languages like English and French. This is because there is a lot of ambiguity and misunderstanding in natural languages. By using a symbolic syntax, we can articulate clearly and directly. It is especially important to maintain clarity and directness in argument analysis. However, it is important to be able to apply logic to natural languages. In the same way we can juxtapose English words to Spanish words, we can juxtapose English words to Propositional Logic. In the same sense, we will call it a translation.
Now we can finally get to the fun stuff.
(M1) An infinite supply of literals or atomic propositions denoted by
As we are working in propositional logic, we concern ourselves mainly with propositions. This means propositions are our smallest working element, which is why we call them atomic propositions. In the same sense, let us observe this argument:
(1) Alice is reading a book.
(2) Bob is eating a sandwich.
(C) Alice is reading a book and Bob is eating a sandwich.
Although not the most complex argument, it is a good starting point (extra credit: Is it a valid argument?). Now let us translate this argument to the language of Propositional Logic. We can easily see all three of these statements are propositions through our definition of a proposition earlier. Also stated earlier, our smallest working elements are propositions. Thus, our translation will simply be:
(1) Alice is reading a book. => p
(2) Bob is eating a sandwich. => q
Now you’re probably thinking, “what? That’s it?”. Yes, that is it. We simply translate each unique proposition to a corresponding unique atomic proposition literal. Although we can technically use any letter to represent our propositions, as long as they are unique, logicians tend to start from p. Now you might be thinking about the conclusion:
(C) Alice is reading a book and Bob is eating a sandwich. => r ?
Do we simply assign it to r? Not quite. Note that the conclusion is actually built of two separate propositions, namely [1] and [2], and simply conjoined with the word and. The most important rule in the translation process is keeping track of unique propositions and their corresponding unique atomic propositional literal. So we must keep in mind what p and q represent. Thus, our conclusion can be translated to:
(C) Alice is reading a book and Bob is eating a sandwich. => p and q
Another example:
(1) It is raining.
(2) If it is raining, then I will bring an umbrella.
(C) I will bring an umbrella.
This time [2] is built of two separate propositions. We can break down this argument like so:
(1) It is raining. => p
(2) If it is raining, then I will bring an umbrella. => if p then q
(C) I will bring an umbrella. => q
There are two things you should notice by now. One, we are not translating the entire sentence. Two, the translation is not entirely symbolic. We will fix that next. There are key words and linguistic patterns during our translation, some you may have already noticed. They are primarily not, and, or, if-then, if and only if. These are not all of them, but are the main ones and best for explaining the next concept.
(M2) Connectives:
Unary connectives:
Negation [¬]
Binary connectives:
Conjunction (∧)
Disjunction (∨)
Material Conditional (→)
Material Biconditional (↔)
You’re probably wondering what all of that means. These symbols and operators are what we call connectives. As they sound, they are used to connect our propositions.
[1] Negation – As mentioned earlier, all propositions have a truth value. Negation is most like the English word not. It simply inverts the truth value of the statement. If a statement were true, is then negated, it then becomes false and vice-versa. For example, given the statement,
Alice is reading a book.
We can get that statement’s negation in the form:
Note that this is still a proposition, and only one of these can be true. If Alice is indeed reading a book, then the first statement is true and the second false. As we translated earlier, if
The statement’s negation in propositional logic is:
Alice is not reading a book. => ¬p
[2] Conjunction – A conjunction is most similar to the English word and. A conjunction is true only when both of its component propositions are true, and false otherwise. Returning to the first example above, "p and q" can be represented as p ∧ q in propositional logic. For example,
is true only when both propositions
Alice is reading a book
and
are true, It is not sufficient for only one to be true.
Conjunction is a binary connective because it requires two propositions. For example, it does not make sense to say
Alice is reading a book and
and end the sentence there. There must be a second clause.
[3] Disjunction – This corresponds mostly with the English word or. A disjunction is true when either when either or both of its component propositions are true, and false otherwise. Again, it is a binary connective for the same reasons found in the conjunction. Thus, the statement
Alice is reading a book or Bob is eating a sandwich
is true when
or
is true. It is sufficient for only one to be true. Note that we are using the inclusive definition of disjunction, compared to the exclusive definition. I will explain the difference in another post.
[4] Material Conditional – This is usually just called a conditional or an implication. I insisted on calling it material conditional because there are other forms of conditionals, like causal conditional. However, these are outside the scope of propositional logic, and will not be discussed further here. From now on, I will refer to the material conditional as conditional or implication.
Anyways, a conditional is mostly resembled by the English phrase if-then. A conditional is true when either the if-clause is false, or when the then-clause is true. A conditional is false only when the if-clause is true, and the then-clause is false. Most of this is intuitive. Let’s look at this example
I will explain the truth values of the conditional in what I believe is most intuitive. Suppose the first proposition,
is true, and the following proposition,
I will bring an umbrella
is also true, then it makes sense for the entire conditional to be true. This is a simple conditional in English. If A happens, then B will also happen. Since A happens, and B happens as well, then the if-then declaration must also be true.
The second case is if both propositions are false. This is also intuitive to an extent. Contrary to case one, Supposing the first proposition,
is false, and the following proposition,
is also false, then it makes sense for the conditional to be true. Essentially, I did not lie to you about bringing an umbrella because it is not raining in the first place. Working with the negation I introduced earlier may help clear this up
If it is not raining, then I will not bring an umbrella.
In a sense, whether or not I will bring an umbrella depends on whether or not it is raining. If it is raining, then I bring an umbrella. If it is not raining, then I don’t. This holds for the intuition here, but not as well in a different case. Therefore, I will not define it as so, but is a good way to think about it for now.
The third case is the only false case for the conditional, which is when the first proposition is true, and the following proposition is false. Again, the example
is true, but
is false. If someone told you, If it is raining, then I will bring an umbrella, and it happened to be raining, however you do not see them carrying an umbrella, you would most likely feel lied to. You would say that statement is false. The first proposition is true, but the second proposition did not follow as it is supposed to.
The last case is hardly intuitive at all. If the first proposition is false, and the second proposition is true, how could the conditional be true? The example
is false, but
is true. This is why in my case two explanation, I denied that bringing an umbrella is dependent on whether or not it is raining. In fact, it is not at all. Whether or not it is raining, I can still bring an umbrella. Thus, in a sense, we do not really care whether or not it is raining (we obviously do, or case two would fall apart), but whether or not we are bringing an umbrella.
Another way to think about it is the then-proposition being more important. We only care about whether or not I am bringing an umbrella. If I am bringing an umbrella, then all is great and rain no longer matters. However, if I am not bringing an umbrella, we must see why I am not. Well, if it is not raining, I did not make a promise to bring an umbrella anyways, so I did not lie to you. Thus, my conditional statement is still true. However, if I promised you that I would bring an umbrella when it is raining, and it is in fact raining, then I would have lied to you and my statement is false.
I do hope most if not all of this explanation made sense to you, and that it is a somewhat intuitive in regards to English if-then statements. If not, then let me know down in the comments what part you’re having trouble understanding and I will try to clarify for you.
If none of the conditional was intuitive to you, there is one last way of understanding it. That is to simply accept the connective as an operation that is true in certain cases and false in a certain case. We as logicians have defined the conditional as such, just like how mathematicians have defined the + sign as the summation of two numbers. I do think this is a cheap-shot at understanding the conditional and is much better to understand the intuitive stance of an if-then statement, but to each their own.
[5] Biconditional – This corresponds mostly with the English phrase if and only if. It is quite similar to the conditional, but more constrained. Assuming the same example,
Case four no longer works here. I can no longer just bring an umbrella, even if it is not raining, because as stated, I will only bring an umbrella only if it is raining. The easiest way to think about the biconditional is it is true only when its input propositions have the same truth values and false otherwise. If the first proposition and second proposition are both true, then the biconditional is true. If they are both false, then the biconditional is again true. The biconditional is false when the truth values are different.
The last thing to explain is the parentheses. These simply allow us to appropriate the scope and have proper grammar in our language. Just like in mathematics, they tell us what statements to look at first and prioritize.
That is it for propositional logic! It is comprised of those atomic propositions and connectives. However, it is a powerful tool. I do realize this post is getting a bit lengthy, so I would like to cut it off here as there is a lot of information to digest. I wanted to restrain this post to just an introduction to the mechanics. I will be writing a part II, with more working examples, technical depth, and stronger explanations.
Let me know if there is anything here that was confusing, so I can add it to that post. Also let me know if there was anything here you want me to talk about more in depth. Feel free to toss those questions and comments at me.