# 7.2: Propositions and their Connectors

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Wouldn’t it be great to have a system that tracked *all* of these logical entailments? Maybe we could devise a way of taking English sentences (and sentences in other natural languages as well!) and stripping them down to just their logical structures so that we could explore the properties of just the logical structures of statements and arguments.

And since the Greek word for form or structure is “logos”, let’s call this system “Logic”, yeah?

Okay so Logic is the study of the logical forms of statements, sets of statements, and arguments. Propositional Logic is just a particular sort of logic. What makes it special?

Aristotelian Logic was discussed in the previous chapter. There, we focused on “terms” or “categories” and the various ways they can be related to one another. One category might totally include another, partially include another, or totally or partially *exclude* another. Here are some examples:

- The category “mammal” totally includes the category “dog”, meaning that every dog is included in the category mammal—or simply: every dog is a mammal.
- “Animal” partially includes “cats” in that some animals are cats, but not all of them are.
- “Gem” partially excludes “diamond” because not all gems are diamonds: there are some gems that are not diamonds.
- “Human” totally excludes “dinosaur” in that no humans are dinosaurs and no dinosaurs are humans.

These are relations between *categories*, hence the name “Categorical Logic.” Propositional Logic, alternatively, focuses on the relations between different entities: propositions or statements. Sometimes it is called “Sentential Logic” instead because it is the logic of how *sentences/statements/propositions *relate to one another.

If we’re going to represent the logical form of a statement in English, we need to decide first *what* we want to symbolize (represent using symbols) and then second *how* we are going to symbolize those things. That is, we must decide what aspects of ordinary language is part of the logical form or is *logically relevant*. Once we have made that decision, we can symbolize the logically relevant bits and then leave the rest out. We’d then be left with *only* the logically relevant aspects of the original statement.

Once we have done this symbolization, we can then study the logical form on its own apart from the particular content of the statement we’re interested in.

We’ll learn 5, yes *five *different logical symbols and that’s all. It’s a bit like learning a language with only five words: translation will be a bit tricky sometimes, but you don’t have to spend much time developing your vocabulary.

As a sneak preview, the five logical symbols that we’ll learn correspond roughly to the following English words or phrases:

*And *

*Not *

*Or *

*If...then... *

*If and only if...will... *

First, since the task here is to learn about how to symbolize an English sentence into propositional logic—how to extract the logical form from a sentence, that is—we should learn the most basic operation: symbolizing an atomic proposition.

Atomic propositions are simple propositions that *have no logical content*. Here are some examples:

*I’ve been swimming every day this week.*

*There is a bird nesting in the attic.*

*The Order of the Phoenix is a secret order from the Harry Potter book series that sought to undermine the goals of Voldemort.*

Atomic propositions are sometimes longer, sometimes shorter, but they always have one thing in common: they make one unified a claim about the world that might be true and might be false. There are no relations between propositions (relations like ‘and’, ‘or’, or ‘if...then...’) nor are there any negations or ‘not’s’—in short, there is no logical content in an atomic proposition.

To translate an atomic proposition is quite simple. Replace one whole atomic proposition with a single variable called a “sentence letter”. Here is how you would symbolize the previous examples:

*S *

*B *

*O *

I just chose what I took to be the** most significant word** from each proposition and then used the first letter of that word as the variable. So I chose “swimming”, “bird”, and “Order”. We do this so it’s easier to tell which variable symbolizes which English sentence.

That’s all there is to symbolizing a simple or atomic proposition.

## “And” AKA Conjunction

Ever heard of “conjoined twins”? These are twins that are stuck together. That name comes from the word “conjunction” or “conjoint” (remember conjoint premises from Chapter 3?). A conjunction is a sticking together or, simply, an “and”. Imagine the following exchange:

*I am the director of marketing *

*Nice to meet you, I’m from an investment firm. We’re very interested in your company. *

*In that case: full-disclosure, I also work for the investment firm that owns a majority share in this company. *

*Wait, you’re the director of marketing **and* you work for their investment firm?

This final reply is a *conjunction*: a joining together of two otherwise separate statements. It makes perfect sense what an “and” or conjunction does: it says “both A and B are true.” The statement as a whole turns out to be false if either of the conjoined propositions turns out to be false.

Here are a few examples of conjunctions:

*I’ve always been a quiet person, but when I get on stage I like to be loud.*

*No one will ever love you and you’ll die alone!*

*Lincoln was shot in Ford’s Theatre, but Kennedy was shot in a Lincoln Continental made by Ford. You do the math.*

The first statement is only true if it is true that A: This person has always been a quiet person, and B: When this person gets on stage they like to be loud. If either of those statements ends up being false, then the whole statement above is false.

The second statement is only going to be true if A: no one will ever love the person they’re talking to, and B: the person they’re talking to will die alone. If someone ends up loving them but they die alone anyways, the second statement above is still false since one of the *parts* or *conjuncts* is false.

One part of a conjunction—one of the conjoined statements—is called a **“conjunct”**

How do we symbolize a conjunction? First, you’ll need to learn to recognize the indicator words that tell us that we’re dealing with a conjunction. Some really common ones are:

*And *

*But *

*However *

*Yet *

Remember from grammar or English class the categories “coordinating conjunction” and “subordinating conjunction”? Well a logical conjunction is usually marked by an English word from those categories.

Once we’ve recognized that we’ve got a conjunction, then we remove that indicator word and replace it with a conjunction symbol. There are two common conjunction symbols and we’re going to learn both so that if your instructor shows you a video or sends you to a website that uses a different one, you’ll be aware that different people use different symbols. Here are the two conjunction symbols:

\(\bullet\) | \(\wedge\) |

The one on the left is called a *dot* and the one on the right is called a *wedge*. There is **no difference in meaning between the two**. They mean the exact same thing. You’ll have to train yourself to see them as equivalent. A lot of logic requires training yourself in these ways—it’s like learning a new language!Okay, so now that we have our symbols, we can work through an example:

*I’m not much of a romantic, but I wanted to ask for your hand in marriage.*

###### Solution

There’s a bit more going on here than a conjunction, as you’ll see when we work through the next symbol, but for now we only know about conjunctions and so we’ll only symbolize the conjunction.

The first step is to identify the indicator word—the word that indicates that we’re dealing with a logical conjunction. Can you find it?

Yep! It’s the “but”. We simply replace that word with our symbol:

*I’m not much of a romantic *\(\bullet\)* I wanted to ask for your hand in marriage. *

And then we take out the rest of the content and replace each simple proposition with a **Sentence Letter**—an upper-case letter that stands in as a variable for any simple proposition.

Because the only thing we know how to translate at this point is the “but”, we’d remove the rest and replace it with variable letters:

N \(\bullet\) M

Why did I choose “N” and “M”? I chose the N for “not” as in “not much of a romantic” and the M for “marriage”. You simply choose a significant word from the simple proposition and then use that as the variable letter. There is a bit more too it than that in that you have to watch out for a few things, but we’ll discuss that after a bit more on the other symbols.

The last thing to do is to add parentheses around the outside. Every dot (\(\bullet\)) or wedge (\(\wedge\)) gets its own set of parentheses. Again, there’s a grammar to propositional logic, but we’ll discuss that in due time.

(N \(\bullet\) M)

There, we’ve done it! We’ve noted that this sentence is a conjunction between two simple propositions in terms of its logical form.

## “Not” AKA Negation

**Not so fast! Isn’t weird to use “N” to stand for “not romantic?”**

Good catch, student, good catch. It feels a bit weird to translate “I’m not romantic” as a simple proposition when “I’m romantic” feels like the simple proposition that’s being *denied* here. It would be good if we could capture the relationship between “I’m romantic” and “I’m not romantic” using a logical symbol. Otherwise we just translate them as simple propositions and the result is just R and N. There’s no clear logical relationship between R and N.

Maybe we could introduce a logical symbol that stands for “not” so we can capture this relationship? Let’s use these two symbols:

~ | \(\neg\) |

The left one is called a *tilde* and it’s on the top left of your keyboard. The right one is called a *hook* and has to be found in the “insert-symbol” dialogue box in Microsoft Word. Again, different people use different symbols, but these are the two most common. They again mean the same thing as one another—there’s no difference between them for our purposes here.

Now, we’d take a simple proposition—a proposition without *any* *logical content* and symbolize it using a variable letter. So “I’m romantic” becomes “R”. Then if there’s a negation, a denial, or a “not” in the sentence, we’ll use a tilde or hook to symbolize the idea that there’s a logical negation in the ordinary language sentence:

\(\neg\)R

We might read this “not R”. It means something like “It’s not the case that the simple proposition R is true.” If we take one step back, we can see what the negation is doing a bit more clearly:

\(\neg\)(I’m romantic).

We might read this “It’s not the case that I’m romantic.” I’m not romantic. The negation symbol marks for us that a simple positive proposition “I’m romantic” is being *denied* here.

Here are some examples of sentences with negations in them:

*I’m not the person you think I am *

*[ It’s not the case that I am the person you think I am]*

*The doctor isn’t in right now, may I take a message? *

*[ It’s not the case that the doctor is in right now]*

*None of our kids have ever failed a class. *

*[ Not one of our kids has failed a class]*

Notice how in the last one the simple proposition “one of our kids has failed a class” sounds “negative” in that failing a class is a bad thing. Logical negation isn’t all types of “negativity.” Instead, logical negation is when a statement is being *denied*.

Also notice the second example here has an “isn’t” in it. The “not” is contracted. This makes it sometimes easy to miss a negation. Keep an eye out for “n’t” as in “mustn’t” “isn’t” “can’t” “shouldn’t” and of course “whomst’ven’t” ;).

Let us revisit our example from the previous section on conjunction:

*I’m not much of a romantic *\(\bullet\)* I wanted to ask for your hand in marriage. *

Once we notice that there is a negation in the left conjunct (the left proposition), we can symbolize that negation *before* we eliminate the rest with a simple proposition variable:

\(\neg\) *I’m a romantic *\(\bullet\)* I wanted to ask for your hand in marriage. *

Now that we’re truly down to simple propositions with no logical content, we put our variables in. Again, we just choose what we think is the most significant letter in the sentence:

\(\neg\) R \(\bullet\) M

Finally, remember that every conjunction (\(\wedge\) or \(\bullet\)) gets a set of parentheses and that those parentheses go *outside of each conjunct*:

(\(\neg\) R \(\bullet\) M)

Remember that this symbolization would be equivalent since we have equivalent symbols:

(~ R \(\wedge\) M)

Note also that you wouldn’t want to symbolize it this way:

\(\neg\) (R \(\bullet\) M)

Where the hook (\(\neg\)) is on the outside of the parentheses. This means something entirely different. Something like “I’m not both romantic and want to propose marriage.” This is not what the original sentence said, so it’d be an inaccurate symbolization.

## “Or” AKA Disjunction

We’ve now covered two logical symbols: \(\bullet\) and \(\neg\) (and their equivalents \(\wedge\) and ~). We can symbolize and/but/however (conjunction) and also not (negation). What happens if you see sentences like the following?

*I’ll either strike out this inning or get a runner into home base to score the winning run. *

*Look, we know that the Republican or the Democrat will win the race, so there’s no point in voting for a third party. *

*The world will end unless we do something about global warming. *

In each of these sentences, there is a *disjunction* between two possibilities presented. The idea seems to be that one of the options will happen no matter what. We will do something about global warming *OR the world will end. A Republican will win OR a Democrat will win. I’ll strike out OR I’ll score the winning run. The following symbol symbolizes a logical disjunction: *

\(\vee\)

One theory is that this symbol started because in Latin “vel” means “or” and so the “v” at the beginning became the symbol for “or” in propositional logic. Sounds plausible enough. This symbol is sometimes called a “vee” or simply a disjunction symbol. We won’t learn an equivalent for it.

Disjunctions are similarly simple in that there are really only two words that it translates:

*Or *

*Unless *

Often the “or” is in a phrase using “either...or...”, but that’s basically it: don’t see an ‘or’ or ‘unless’? Then it’s almost certainly not a disjunction.

Remember that a disjunction means that “at least *one* of the disjuncts is true”.* *

One of the things being disjoined by a disjunction—one side of the disjunction—is called a **“disjunct”**

Translation for disjunction works almost the exact same way as conjunction. Identify the two propositions being connection via disjunction, then put the disjunction “\(\vee\)” between them and enclose in parentheses all the way outside each disjunct:

*The world will end unless we do something about global warming. *

*(The world will end) *\(\vee\) (*we do something about global warming) *

E \(\vee\) D

(E \(\vee\) D)

All done! Just remember that every “\(\vee\)” gets its own set of parentheses.

## “If...then...” AKA Implication AKA Hypothetical AKA Conditional

*If you’ve followed along well so far, then you’ll do just fine in the propositional logic unit. *

Notice how this sentence has an “if...then...” grammar. Since if...then...has a very clear logical definition, we can use a symbol to capture this structure and include it in our symbolization of the logical structure of this proposition. Put another way: we can capture the “if...then...” relationship using a logical symbol. Here are the symbols we’ll use:

\(\rightarrow\) | \(\supset\) |

The left one is called an *arrow* (for obvious reasons) and the right one is called a *horseshoe* (for still fairly obvious reasons). The right one comes from a specialized logic used primarily in mathematics called “set theory”. As a result, many people who teach intro to logic use the set theoretic symbol for an implication relation. I’m partial to the arrow for reasons that will become clear when we introduce the final symbol below, but it’s still important to be aware of both symbols in case you come across the other common symbol somewhere.

Another way of thinking of the sentence at the beginning of this subsection is as one stating something like “Following along well so far **implies** a high likelihood of success in the propositional logic unit.” Or maybe “**hypothetically**, if you are following along well, then you’d do fine in the propositional logic unit.” Or finally perhaps “On the **condition** that you are following along well so far, you’ll do fine in the propositional logic unit.” These different possibilities illustrate why we might call a statement like the one at the beginning of this subsection an implication, a hypothetical, or a conditional.

If I say “If you eat a peanut butter and jelly sandwich, you won’t be hungry anymore” I’ve said something *conditional* or *hypothetical*. I haven’t said “you will eat a PB&J” and I haven’t said “you won’t be hungry anymore.” Both of these claims could easily be false. What I instead said is that ** IF **you were to eat a PB&J,

**you won’t be hungry anymore. There’s a conditional relationship between them.**

*THEN*Here are some examples:

*If he’s not going to the dance with you, *

*that implies that he’s going with Sheandra!*

*If I took out Stannis’ army at Winterfell, *

*then I’d have no other rivals to my claim as Warden of the North*

*If you don’t do your homework, you won’t be allowed to go out tonight.*

*You’ll do well on the exam only if you study*

*A necessary condition for being admitted *

*is submitting a properly filled out application with all required documents*

*A sufficient condition for eating is *

*putting food in your mouth, chewing, and swallowing*

Symbolizing a conditional is fairly straightforward. Identify the antecedent—often the proposition between “if” and “, then”—and the consequent—often the proposition following “, then”. Once you’ve identified them, symbolized them (probably using a single variable letter, but the antecedent or consequent could always be more complex than that). Then add an \(\rightarrow\) or a \(\supset\) and enclosed in parentheses.

The first part of a conditional—what comes before the arrow/horseshoe—is called the **“antecedent”** and the second part is called the **“consequent.”**

Let’s take our first example: “*If he’s not going to the dance with you, that implies that he’s going with Sheandra!” Notice how there’s a “not” in the first part (the antecedent), so we’ll need to capture that when we’re symbolizing. Here is the sentence with the logical words taken out and symbolized: *

\(\neg\)* he’s going to the dance with you *\(\rightarrow\)* he’s going with Sheandra *

If it’s not the case that he’s going with you, then he’s going with Sheandra. Notice that the “that implies that” is an indicator word for a logical implication relation. Once we’re sure that the only English that’s left is parts of simple propositions, then we can go ahead and symbolize the simple propositions using variable letters:

\(\neg\)Y \(\rightarrow\) S

Finally, you’ll enclose the whole thing—antecedent, arrow, consequent—in parentheses:

(\(\neg\)Y \(\rightarrow\) S)

Be careful not to do this:

\(\neg\)(Y \(\rightarrow\) S)

If you don’t enclose the negation, you get something very different. The statement immediately above symbolizes a statement like “it’s not the case that if he’s going to the dance with you, he’s going with Sheandra.” But that’s not what our original statement said. Even worse, “\(\neg\)(Y \(\rightarrow\) S)” is logically equivalent to (Y and \(\neg\)S), which means “he is going to the dance with you and he’s not going with Sheandra.” But that’s definitely not what our original statement said or implied. So the lesson here is to always take care to enclose the *whole antecedent* (or disjunct, or conjunct) when putting the parentheses around an operator like a \(\rightarrow\), \(\vee\), or \(\bullet\).

There’s a bit more to say about each of these symbols, but for now let us press on. We’ll address some of this more in-depth detail in the following section (6.3 More Thoughts on Symbolization.)

## “If and only if...” AKA Equivalence

The final logical symbol in the basic set is sometimes called “material equivalence” or “biconditional.” Here are some examples:

*A person is Commander-in-Chief of the United States military if, but only if they are the President of the United States*

*When and only when you clean your room will you be given your allowance*

*If and only if I win the election will I make you my Chief of Staff*

*Owning a home is a necessary and sufficient condition for being a homeowner *

We might not see this one quite as commonly in ordinary language, but it does arise sometimes.

Here’re the symbols we use for equivalence or biconditional:

\(\leftrightarrow\) | \(\equiv\) |

The left one is called a “double arrow” and the right one is called a “triple bar”. Now you can see why I prefer to use the arrows rather than \(\supset\) and \(\equiv\) : there’s a clearer relationship between the symbols and that relationship is reflected in the logic of the symbols. Let’s look at an example. The second one from above actually means something like:

*If I win the election, I will make you my Chief of Staff; *

*but only if I win the election, will I make you my Chief of Staff *

Which we would symbolize in the following way:

((W\(\rightarrow\)C) \(\bullet\) (C\(\rightarrow\)W))

Notice how there’s an arrow going W to C and another going C to W? Well, the double arrow does all of this work without something as long and complex as this. We can just write:

(W \(\leftrightarrow\) C)

And it means the exact same thing. These two logical formulas are logically equivalent. It’s as if this formula means:

W\(\rightarrow\)C

W\(\leftarrow\)C

Seeing this relationship is key to understanding what equivalence is as a logical relationship.

Warning, though: “\(\leftarrow\)” is NOT a symbol in propositional logic. We only use it to illustrate things in textbooks, never when actually symbolizing or otherwise doing logic.

A final note about “if and only if” before pressing on: sometimes logicians and philosophers will use the word “iff” as shorthand for “if and only if.” When you see “iff” think “if and only if” and then think \(\leftrightarrow\) or \(\equiv\).

Okay, so now that you have a basic introduction to the five logical symbols, we can think a bit more technically about how to symbolize English sentences in propositional logic. Remember: the goal here is to have a sort of diagram of the logical structure of sentences and arguments so that eventually we can manipulate the symbols in that “diagram” to find out more about the logical structure.

## Logical Words and Operators Summary

Here is a table of the symbols you’ll see and the English logical words and phrases they are meant to symbolize. Note that there are more than one symbol for the same logical operator. This is because some textbooks use different symbols and It’s important to be aware of both symbols so you have access to a wider variety of practice problems and other resources (including these textbook chapters).

Table \(\PageIndex{1}\)

Not, it is not the case that, it is false that |
\(\neg\) (hook), ~ (tilde) |

And, yet, but, however, moreover, nevertheless, still, also, although, both, additionally, furthermore, along with, in addition to, ... |
\(\wedge\) (wedge), \(\bullet\) (dot), & (ampersand) |

Or, unless |
\(\vee\) (vee) |

If...then..., only if, implies, given that, in case, provided that, on condition that, sufficient condition for, necessary condition for |
\(\rightarrow\) (arrow), \(\supset\) (horseshoe) |

If and only if, iff, is equivalent to, necessary and sufficient condition for |
\(\leftrightarrow\) (double arrow), \(\equiv\) (triple bar) |