Logic 101 · Series

Introduction to Logical Symbols

The first set of tools that we need to reconstruct, raise, and evaluate deductive arguments are logical symbols. A good deductive argument must follow the Structural Principle of the Code of Intellectual Conduct, which says that deductive arguments should adhere to rules of argumentation that produce formally valid, sound arguments. That principle is true when we reconstruct others’ arguments as charitably as we can, and when we raise our own arguments.

As I’ve already discussed in previous posts, we have to identify the conclusion and premises of arguments and then place them into standard form in order to begin evaluating them. What we are going to learn how to do in this post is how to translate deductive arguments that we have placed in standard form into logical form so that we can see if they follow the rules of deductive logic. The logical form of an argument is the way that the argument’s assertions are organized and expressed in sentences that follow the conventions of a logical “language.”

A logical language (and there are many) attempts to capture what we say in conversational language in an expressible, symbolic form that we can evaluate. So, an argument is either deductively valid or invalid because, when we express it in the right logical language, it is organized in such a way that the conclusion cannot be false when the premises are true. That’s what it means to be a deductive argument, after all.

Sentences and Propositions

When I mentioned the “assertions” of an argument in relation to logical form just now, I had in mind a technical term called a proposition. Human language (in our case, English) is a tool that we use to express thoughts about reality. Many of the things that we might say about our world are either true or false. This sentence, for example, is either true or false:

(F) Water freezes at 32°F (at sea level).

There are four ideas in that sentence: water, freezes, 32°F, sea level. We can translate all of these ideas into another language (German, French, Greek, etc.) to express the same idea.

Well, the “idea,” or the truth-content behind (F), whether it is stated in English or some other language, is what I mean by a proposition. It is the proposition behind the sentence I’ve labeled (F) that makes it either true or false when it is expressed in any language we might choose to say it in. In other words, it isn’t the English words and their formation that makes the sentence true or false; it is what those English words express, when organized in particular ways, that we can say is either true or false.

Conditional Statements

Thinking back to our previous discussion of necessary and sufficient conditions, let’s explore what (F) might mean in its logical form. Assuming that (F) is true and assuming we are at sea level, it seems right to assert the following:

(1) If it is at most 32°F outside, then water in my fountain is frozen.

At the same time, it seems right to say the following:

(2) If the water in my fountain is frozen, then it is at most 32°F outside.

In other words (F) means that a temperature of 32°F at sea level is necessary and sufficient for the water in my fountain being frozen.

Now, let’s replace everything that was said in (1) and (2) with variables to see what’s being said. Let’s suppose the following:

O = it is at most 32°F outside (at sea level).
W = the water in my fountain is frozen.

This means that we can write (1) as: “If O, then W.” And we can write (2) as “If W, then O.”

Now we’re ready to learn our first logical symbol. Each time I introduce them, I will show you several symbols that often get used in logic for that kind of statement and then say which one I will stick to using on this website.

The way that conditional statements like (1) and (2) can be written without their English words are like this:

(1) O → W   OR   O ⊃ W
(2) W → O   OR   W ⊃ O

The symbol for conditional statements is either →or ⊃. I will exclusively use the first on this website.

Now, (1) and (2) are clearly both true, since (F) expresses a necessary and sufficient condition. When we have a condition that is both necessary and sufficient, we are dealing with a biconditional statement. Again, remember from the post on necessary and sufficient conditions that a biconditional is simply two propositions like (1) and (2) joined with the word “and.” So, we can write something like this to say that both (1) and (2) are true:

(3) O → W and W → O.

As it turns out, there’s a logical symbol that allows us to replace that English word “and” here. That symbol (called a conjunction) is either & or . I will exclusively use the first on this website. Now we can write (3) in a different way:

(3) (O → W) & (W → O).

I’ve inserted the parentheses to make it clear where each expression begins and ends.

Now back to biconditional statements. As it turns out, there’s an even shorter way to express (3) in logical form. This is by using the logical symbol ↔ or ≡. I will exclusively use the first on this website. So, we can rewrite (3) in the following way:

(4) O ↔ W

You can probably see by now the incredible usefulness of translating what we’ve written into logical form. It is much easier to analyze something like (4) than it is to analyze (1) and (2) separately. All that to say, (4) is the proposition that expresses the meaning of (F).

Before we move on, let’s recap the symbols that we have learned:

Conditional:  →, ⊃
Conjunction: &, ∧.
Biconditional: ↔, ≡

For each of these types of symbols, I will exclusively use the first one I have written when I use logical symbols of these types on this website.

Negation and Disjunction

Oddly enough, I have to tell you a little bit more about conditional statements before I introduce the next two logical symbols. A conditional proposition of the form P → Q means that it is either the case that P is false, or that Q is true (or both). (For now you’ll have to take my word for it, but I’ll be able to prove this once we discuss truth tables.)[1]

In other words, the proposition P → Q means the same thing as the following:

Not-P or Q.

If you suspect I’m about to introduce some logical symbols here, you’re on the right track! There’s a symbolic shorthand for the word “not” here (called a negation) and also for the word “or” (called a disjunction). Those symbols are:

Negation: ~, ¬
Disjunction: v

I will exclusively use the first symbol (~; tilde) for negation. A negation simply negates the truth of a proposition. If P is true, for example, then ~P is false. But if P is false, then ~P is true.

So, we can write the previous sentence (Not-P or Q) in logical form as: ~P v Q.

Technically speaking, each one of the symbols I’ve introduced so far are called logical connectives (or logical operators) because they connect (operate on) propositions together. The next two logical symbols are not logical connectives, but they are still important symbols in logic.

“Therefore” and “Contradiction”

Believe it or not, catching people in lies is an exercise in logical reasoning. When what a person says (let say it is ‘P’) comes against a reality that indicates the opposite (~P), then it has to be the case that what they previously asserted can’t be the case. In other words, a sentence that has the logical form P & ~P is always false, which is the definition of a contradiction.

I will explain this more in a later post, but finding contradictions is useful in reductio ad absurdum arguments. These arguments assume that something that has been said is true, and derives a contraction (P & ~P) from that statement. Any time we have a premise that is a contradiction, under it we can assert the following:

Therefore, [contradiction].

Now we have two more words that need a logical symbol attached to them:

Therefore:
Contradiction:

Summary

We now have the basic tools that we need to be able to place arguments in logical form after we have reconstructed them. Here are all of the logical symbols that I will use on this website:

Conditional:  →
Conjunction: &
Biconditional:
Negation: ~
Disjunction: v
Therefore:
Contradiction:

From here we can also move on to other important material. In the next post I will introduce you to eight basic rules of deductive logic. These rules will be ways of constructing premises and arguments in logical form that can always be used in deductive logic. Later on, I’ll tell you more about all of these logical symbols when we get to a discussion of truth tables.

 


Footnotes:

[1] For those already adept in logic: yes, I am aware of the paradoxes of material implication and I’m skirting over them on purpose. There’s a reason I’ll be teaching Aristotelian (categorical) logic as well.

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