7.3: More Thoughts on Symbolization
- Page ID
- 223897
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Grammar of Propositional Logic
The five symbols are sometimes called “logical operators”. There’s a reason for this that we’ll get to in section 6.4, but for now be aware that when I say “operator” I am referring to one of the five logical symbols (or equivalent symbol pairs) that we’ve learned so far.
Logic has a grammar or syntax just like English has a grammar. If I was to say “the milked I stars goats those under,” my sentence wouldn’t make any sense—and not because I’ve chosen words that don’t mean anything in that combination. Indeed, if I put them in a grammatical order, then suddenly I’ve got a sentence that makes sense (even if it’s a bit strange): “I milked those goats under the stars.” Logic is similarly sensitive to the order we put symbols in—one mistake and suddenly it doesn’t make sense.
Here are the rules of grammar for Propositional Logic—called its “syntax”. One way to think about a syntax is that it’s a set of rules for what counts as a “Well-Formed Formula” or a properly put-together string of symbols. Sometimes logicians call these WFFs (pronounced by logicians “woofs”) or simply “formulas” (after all, we don’t tend to talk about non-well-formed formulas).
1. Any atomic proposition (sentence letter) is a formula.
Examples:
- G
- F
- X
- Z
2. Sticking a negation to the left of any formula (any well-formed formula at all, no matter how long or complex) produces a new formula.
Examples:
- ~G
- ~(F\(\rightarrow\)D)
- \(\neg\)[D\(\leftrightarrow\)(R\(\bullet\)(Q\(\vee\)X))]
3. The other four operators (other than negation) are called “Connectives”. Putting a connective between any two well-formed formulas and then enclosing the whole in parentheses (or brackets) produces a new well-formed formula.
Examples:
- (F\(\rightarrow\)D)
- (~R\(\bullet\)~(Q\(\vee\)~X))
- [D\(\leftrightarrow\)(R\(\bullet\)(Q\(\vee\)X))]
End of list. These are the only strings of symbols that count as formulas (well-formed formulas, that is): atomic sentence letters, formulas with a negation to the left of them, and connectives with formulas on either side (and parentheses outside of them).
Here are some rules that follow from this definition of the syntax of propositional logic:
1. A Negation is not a connective and a connective must have 2 “arguments” or formulas on either side.
Examples of Non-Formulas:
- D~S
- [D\(\neg\)(R\(\bullet\)(Q\(\vee\)X))]
2. Every connective needs a set of parentheses or brackets
Examples of Non-Formulas:
- Q\(\vee\)~X
- D\(\leftrightarrow\)(R\(\bullet\)(Q\(\vee\)X))
3. Never put two formulas adjacent to one another
Examples of Non-Formulas:
- ~G(F\(\rightarrow\)D)
- \(\neg\)[D(R\(\bullet\)(QX))]
4. Never put a negation to the left of a connective or close parenthesis.
Examples of Non-Formulas:
- (F~\(\rightarrow\)D)
- [D\(\leftrightarrow\)(R\(\bullet\)(Q\(\vee\)X))\(\neg\)]
To expand a bit on the structure of the operators, it’s helpful to note that negation is a monadic operator, whereas each connective is a dyadic operator. What this means is that a negation takes one (monadic as in ‘mono’ as in “mono-a-mono” or one-on-one) input whereas the connectives or dyadic (dyadic as in ‘di’ as in two as in “dilemma” or “two options”) operators take two inputs. So a negation looks like this:
~____________
Where any well-formed formula at all can go in the blank spot. And the connectives look like this:
(______ \(\bullet\) ______)
(______ \(\vee\) ______)
(______ \(\rightarrow\) ______)
(______ \(\leftrightarrow\) ______)
Where, again, any WFF—anything from “A” to “[(~A v ~B) \(\bullet\) ~(A \(\bullet\) B)]”—can go in the blank spots.
Examples follow. Notice all the different sorts of formulas that can go in the blank spots:
[(~A v ~B) \(\bullet\) ~(A \(\bullet\) B)]
[(P \(\rightarrow\) ~Q) \(\vee\) (~P \(\leftrightarrow\) Q)]
[Z \(\rightarrow\) ~((D\(\bullet\)~X)\(\rightarrow\)(G v H))]
[((G v H) \(\leftrightarrow\) ~Q) \(\leftrightarrow\) ~D]
It’s not hard to see why the dyadic operators are called “connectives”: they connect formulas together in different ways.
Tricky Conditionals
It’s quite easy to make mistakes in understanding the various conditional propositions and the order in which one should symbolize them. It’s worth spending a moment to learn how to properly order conditional propositions.
The first thing to note is that there are essentially three separate conditional phrases that indicate different conditional relationships:
If...(then)...
Only if
If and Only if
Remember now and for the rest of your days that these are different: they mean different things and they have different logical structures. Thus, we must use different logical symbolization patterns to translate them.
“If”
First, if you run into a sentence with an ‘if’ by itself (no ‘only’ before it), then the rule is always to rewrite the sentence in your head or on paper as an if...then... sentence. For example, if you come across the sentence:
People won’t help me if I don’t have any money
You should first notice that there’s a ‘not’ at the beginning and hidden in “don’t”. Keep that in mind for later. Second, you should notice that there is an “if” in the middle of the sentence. This means we need to switch up the order of the sentence until it has the structure “if...then...”. Here is the result:
If I don’t have any money, then people won’t help me
Then we make (again, we can do this all in our heads) the negations more explicit:
If not I have any money, then not people will help me
Then I’m ready to symbolize:
(\(\neg\)M\(\rightarrow\)\(\neg\)H)
I chose ‘M’ for “money” and ‘H’ for “help”.
If, alternatively, you come across a sentence like:
If I’ve been kind to you, then you must return the favor.
You should notice that it’s already in “if...then...” format and so doesn’t need to be rearranged. Just symbolize as is:
(K\(\rightarrow\)R)
Another way of putting this rule is that whatever comes after a solitary ‘if’ is the antecedent of the conditional, whether that ‘if’ is at the beginning or middle of the sentence.
“Only if”
Second, if you come across an “only if”, the rule is to (either on paper or in your head) draw an arrow over the phrase. That arrow will be pointing at the consequent. You may need to rearrange the sentence to make sense of it. Here’s an example:
You’ll get into Stanford only if you keep your grades up.
Solution
Draw an arrow over the words “only if”:
You’ll get into Stanford \(\overrightarrow{\text{only if}}\) you keep your grades up.
And if it, like an arrow requires, has a proposition on both sides, then you’re golden:
(S\(\rightarrow\)G)
‘S’ for ‘Stanford’ and ‘G’ for ‘grades’. Alternatively, what happens if the ‘only if’ is at the beginning of the sentence like the following?
Only if you check your oil and tire pressure regularly will you get to keep your car.
Again, just as before, we draw an arrow over “only if”:
\(\overrightarrow{\text{Only if}}\) you check your oil and tire pressure regularly will you get to keep your car.
But this is not like before because there isn’t anything to the left of the arrow. This is bad, since arrows need a proposition on either side. Where do we get that other proposition? Well, usually we’ll identify the word ‘will’, which acts a bit like ‘then’ in an “if...then...” sentence. Then we take what comes after the ‘will’ and move it to the front—that’s our antecedent.
you will get to keep your car \(\overrightarrow{\text{Only if}}\) you check your oil and tire pressure regularly
Now we’re ready to symbolize, but before we do, notice that there’s an ‘and’ in the consequent.
(C\(\rightarrow\)(O\(\wedge\)T))
‘C’ for “you get to keep your car”, ‘O’ for “you check your oil regularly”, and ‘T’ for “you check your tire pressure regularly”.
“If and Only If”
If we see “if and only if” (or “iff” for short), then we ignore what we’ve learned about dealing with ‘if’ and ‘only if’ separately. Instead, we simply use a double arrow or triple bar (\(\leftrightarrow\) or \(\equiv\)). Again, we might find that we need to rearrange the pieces a bit. Here’s an example where we need to do some rearranging:
If, but only if, you get a job will you be able to afford a new car
Solution
The rule here is to draw a double arrow over whatever phrase means “if and only if”. Here, it’s the first phrase:
\(\overleftrightarrow{\text{If, but only if}}\), you get a job will you be able to afford a new car
Notice how, again, we’ve nothing to the left of our connective symbol. But we need something over there. So we, again, look for ‘will’ and then move the last bit to the front before symbolizing:
you will be able to afford a new car \(\overleftrightarrow{\text{If, but only if}}\), you get a job
(C \(\leftrightarrow\) J)
Summary
 |
 |
 |
| Translates as... | Translates as... | Translates as... |
| (A\(\rightarrow\)B) | (B\(\rightarrow\)A) | (B\(\leftrightarrow\)A) |
Why are they symbolized like this? Remember that an arrow means “if the antecedent (left) happens, then the consequent (right) will happen”. Let’s rehearse some of our examples.
People won’t help me if I don’t have any money
The claim here is that if this person doesn’t have money, then people won’t help them. That is, they won’t be poor and find people to help them. They’ll need money if they want help. Thus the “I don’t have money” is the antecedent.
You’ll get into Stanford only if you keep your grades up.
This isn’t claiming that you will get into Stanford if you keep your grades up. Of course not! Stanford is among the most selective schools in the world, so even if you do extremely well, there’s still no guarantee. The claim is something more like “if you don’t keep your grades up, then you won’t get into Stanford.” Keeping your grades up is a necessary condition for getting into Stanford. You won’t get into Stanford without also keeping your grades up. So if you happen to get into Stanford, it follows that you must have kept your grades up. Hence “you keep your grades up” is the consequent.
If, but only if, you get a job will you be able to afford a new car
This is essentially claiming two things: if you get a job, you will in fact be able to afford a new car; and if you can afford a new car, you must have gotten a job. It’s a biconditional or equivalence.
Necessary and Sufficient Conditions
Translating from statements about Necessary and Sufficient Conditions is relatively straightforward. Necessary conditions go to the right of an arrow. Sufficient conditions go to the left of an arrow. Necessary and Sufficient Conditions get a double arrow/triple bar. Here are some examples:
If you can turn in a completed term paper by the end of finals week, that’d be sufficient to pass the class.
Solution
What’s the sufficient condition? Right-o, “turning in a completed term paper by the end of finals week”. Where does the sufficient condition go? Right-o again! To the left of the arrow:
Turn in completed paper by the end of finals week \(\rightarrow\) pass the class
(C \(\rightarrow\) P)
‘C’ for “Turn in completed paper by the end of finals week” and ‘P’ for “pass the class”. Here’s another example:
Getting an A in participation is necessary for getting an A in the class
Solution
What’s the necessary condition? Righty-O-Daniels, “getting an A in participation”. And where do necessary conditions go? Righty Righty Mighty Tidy! To the right of the arrow:
getting an A in the class \(\rightarrow\) Getting an A in participation
(C \(\rightarrow\) P)
‘C’ for “getting an A in the class” and ‘P’ for “getting an A in participation.” Finally:
Being President is necessary and sufficient for being Commander-in-Chief
What’s the necessary and sufficient condition? Being President. Where does the condition go?
Really, it doesn’t much matter since (P\(\leftrightarrow\)C) is logically equivalent to (C\(\leftrightarrow\)P). Mostly we just keep these statements in the order in which they are written:
(P\(\leftrightarrow\)C)
‘P’ for “Being President” or “One is president”, and ‘C” for “being Commander-in-Chief” or “one is Commander-in-Chief”.
Tricky Ands, Nots, and Nors
I can’t go to the dance with both you and your friend. Neither you nor your friend are welcome at my house. Either you or your friend won’t be able to go to the dance because you’ll have to stay here and clean up this mess. I’m not going to the dance with you and I’m not going to the dance with your friend. Each of these statements has a different logical form, but some of them have a special logical relationship with one another. Let’s take similar statements and think about how to translate them.
1. Neither Baixa (pronounced Bye-ee-sha) nor Ramón will be president of the club next year.
- We translate “neither-nor” statements as “not or”:
- \(\neg\)(B \(\vee\) R)
2. I’m not going to fire either Xia or Bo.
- We translate “either not A or not B” statements as:
- (\(\neg\)X \(\vee\) \(\neg\)B)
3. Nia and Goran can’t both go in our car with us.
- We translate “not both” statements as:
- \(\neg\)(N \(\wedge\) G)
4. You can’t be here and your friend can’t be here.
- We translate “not A and not B” statements as:
- (\(\neg\)Y \(\wedge\) \(\neg\)F)
If I tell you that you can’t date Tamik and you can’t date Peter, then it follows that you can date neither Tamik nor Peter. Can you see how the following two formulas are logically equivalent?
(\(\neg\)T \(\wedge\) \(\neg\)P)
\(\neg\)(T \(\vee\) P)
Similarly, notice how the following makes perfect sense: “You’ll have to choose which to not have tonight: no ice cream or no cookies—you can’t have both ice cream and cookies.” So these two are logically equivalent:
(\(\neg\)I \(\vee\) \(\neg\)C)
\(\neg\)(I \(\wedge\) C)
These equivalences have a special name: De Morgan’s Theorems. They’re named after a logician named Augustus De Morgan. We’ll go over this in a bit more depth when we get to Natural Deduction in Chapter 6.
For now, keep in mind that these are equivalent, but always translate in the way that best captures the ordinary language sentence. Don’t translate using the alternative equivalent forms. Here are some more examples:
- Either you won’t pass the class or you won’t fail the midterm.
- (\(\neg\)P \(\vee\) \(\neg\)F)
- Neither of these dogs is going to work for me.
- \(\neg\)(A \(\vee\) B)
- Notice how I have to capture the sense in which this statement is about neither “dog A” nor “dog b”, even though it doesn’t say as much. We have to capture the fact that it’s a “neither-nor” statement even though the disjuncts don’t quite show up in the English sentence.
- Neither you nor I will be graduating on time.
- \(\neg\)(Y \(\vee\) I)
- We’re not both going to be able to go out tonight.
- \(\neg\)(Y \(\wedge\) I)
- Again, the disjuncts “you are able to go out tonight” and “I am able to go out tonight” aren’t really explicit in the English sentence, but we still have to capture the fact that this is a “not both” statement, and so we have to fill in the subjects “you” and “I”.
- That window isn’t clean and that other one over there isn’t clean.
- (\(\neg\)T \(\wedge\) \(\neg\)O)
- I went with T and O for “That window” and “the Other window”.
Exclusive vs. Inclusive “Or”
When you’re on an airplane and the attendant comes by, asking “would you like cream or sugar in your coffee?”, it’s okay to say “yes, I’d like both, please.” It would be odd if the attendant said, “I’m sorry, I said would you like cream *OR* sugar, I can’t give you both.”
Alternatively, if you’re voting, you often get the choice between a variety of candidates. Let’s simplify it to a choice between 2 candidates. You can vote for either one or the other candidate, but you cannot vote for both. If you do, your vote won’t count at all.
These are both cases where “or” is used, but the first one is what we might call “inclusive” in that both are allowed. Alternatively, the second one is called “exclusive” in that only one or the other is allowed.
The way our “or” operator in standard propositional logic works is as an inclusive or: a disjunction is true if both of its disjuncts end up being true. Here’s an example:
Either the maid or the butler is the killer
Solution
If we interpret this exclusively, then the killer cannot be both the maid and the butler. It must be one or the other. If we interpret this inclusively, then the killer could be the maid, it could be the butler, or it could be both of them conspiring. Central point: there are two ways of understanding an English ‘or’, and the propositional logic symbol ‘\(\vee\)’ stands for the inclusive or.
There are two ways of understanding an English ‘or’, and the propositional logic symbol ‘\(\vee\)’ stands for the inclusive or.
Given that ‘\(\vee\)’ means inclusive or, how would we go about translating exclusive or or XOR? Well, one answer would be to simply invent a new symbol. Some folks already have. Here are a few different options:
| \(\veebar\) | \(\dot \vee\) |  |
\(\oplus\) |
In this text, we won’t use these symbols. We’ll stick to our original five symbols instead. In principle you could use as many symbols as you like, but it makes certain things harder, so we’ll just stick with our five.
Using just our five symbols, though, we can translate an exclusive or. How might we do that? Think through it step-by-step. An exclusive or is saying something like:
Either A or B will happen, but not both A and B
How do we translate “Either A or B”? Simple:
(A \(\vee\) B)
‘but’ becomes ‘\(\bullet\)’, but how might we translate “not both A and B”? Again, we’ll go step-by-step. How do we translate “both A and B”? Simple:
(A \(\bullet\) B)
And since it’s “not both,” we simply put a \(\neg\) in front of it:
\(\neg\)(A \(\bullet\) B)
So when we stick all of this together, we get:
(A \(\vee\) B) \(\bullet\) \(\neg\)(A \(\bullet\) B)
Either A or B, but not both A and B. Oh! And don’t forget the brackets that go with that middle ‘\(\bullet\)’:
[(A \(\vee\) B) \(\bullet\) \(\neg\)(A \(\bullet\) B)]
As a rule, when you see an ‘or’ in a practice problem, the way to symbolize it will be with a simple ‘\(\vee\)’, but occasionally it’s clearly meant to communicate the logical content of the exclusive or, so you’ll need to use this formula above.
Framing Words
English grammar has some really helpful features that reveal the logical form of sentences. These are what we might call “framing words.” Here are a few English sentences to look at:
Either you hand over the money or this will end in violence
If you both pass your exams and you get a summer job, then we’ll help you out with buying your new car.
Neither will you pass go nor will you collect $200.
Each of these sentences has its logical form built into it’s grammatical structure in a fairly explicit way. Let’s explore each in turn.
The first sentence has an “Either...or...” structure. When we see this, we can almost always treat it this way:
Either __[One disjunct]___, Or ___[the Other disjunct]___
So our first sentence:
Either you hand over the money or this will end in violence
Will be interpreted in the following way:
\[\underbrace{\text{Either you hand over the money}}_{\fbox{One Disjunct}} \text{ or } \underbrace{\text{this will end in violence}}_{\fbox{Other Disjunct}}\nonumber\]
We simply take whatever comes between the “either” and the “or” and call that one disjunct, and then we take whatever comes after the “or” and call that the other disjunct—even if they are logically complex. That is, even if there are logical words between the “either” and the “or”, we can still count all of that as one disjunct. Here’s another example to illustrate this latest point:
Either you both hand over the money and don’t call the police, or this will end in violence.
Notice how there’s a “both...and...” between the ‘either’ and the ‘or’? That’s okay, we’ll just treat the ‘both...and’ as, you guessed it, a conjunction; and then we’ll treat that whole conjunction as one disjunct of the overall disjunction. Here’s how it looks:
\(\big(\)([you hand over the money] AND [you don’t call the police]) OR
[this will end in violence]\(\big)\)
So our final logical formula looks like this:
((M \(\wedge\) ~P) \(\vee\) V)
And keep in mind that because we have equivalent symbol pairs, here is an equivalent formula that means exactly the same thing:
((M \(\bullet\) \(\neg\)P) \(\vee\) V)
Okay, now let’s look at our second example from above:
If you both pass your exams and you get a summer job, then we’ll help you out with buying your new car.
\(\big( \)([you pass your exams] AND [you get a summer job]) IMPLIES
[we will help you buy your new car]\(\big)\)
And the final logical formula looks like this:
((E \(\bullet\) J) \(\rightarrow\) C)
Let’s look at our final example really quickly:
Neither will you pass go nor will you collect $200.
“neither...nor” works essentially the same way as “either or” except that it’s negated, so whatever comes between ‘neither’ and ‘nor’ is one disjunct and whatever follows ‘or’ is another disjunct. Here is how this particular example works out:
NEITHER ([you will pass go] NOR [you will collect $200])
Here’s the final logical formula:
\(\neg\)(G \(\vee\) C)
What are the general rules that allow us to make full use of framing words? You’re in luck, I’ve compiled at least a great deal of them. Here they are:
|
Only if [consequent], will [antecedent]. [antecedent] only if [consequent]. |
|
[equivalent] if and only if [equivalent]. If and only if [equivalent], will [equivalent]. |
|
If [antecedent], then [consequent]. [consequent] if [antecedent]. |
|
Both [conjunct], and [conjunct].
|
|
Either [disjunct], or [disjunct].
|
We can use these framing words to translate even very complex sentences from ordinary language into a propositional logic formula.
Here’s an example:
If and only if a person is both arrested at the border and is carrying narcotics, will they be either both deported and have their visa revoked if they are not a citizen or both incarcerated and prosecuted if they are a citizen.
Solution
Woah! I need to take a nap!
Take your nap if you need it, but we do have all of the tools necessary to make sense of what’s going on here. Remember that we are just using symbols to represent the logical form of the ordinary language sentences here, and you already have some intuitive understanding of the logical form because you are an English speaker, so we’re just trying to capture what you already intuitively know. As always, if we go step-by-step we won’t get overwhelmed.
First: what’s the overall grammatical structure of the sentence? In other words: what are the framing words that set up the frame for the whole sentence?
Well, it starts with “if”, so “if...then”?
Remember, dear student, that “if and only if” is a different thing from “if”, so we have to learn to identify them and treat them differently.
Oh, so “if and only if...will?”
Yeppers. Good work. So we already know that the main operator of the sentence—the logical operator that belongs to the outer most parentheses—is... what?
A double arrow thingy
Yep, or a triple bar (\(\leftrightarrow\) or \(\equiv\)). So the overall structure is like the following:
( ??? \(\leftrightarrow\) ???)
What goes in place of those question marks? Well, the first thing to find out is where each equivalent starts and stops. We look between the “if and only if” and the “, will” for one equivalent:
a person is both arrested at the border and is carrying narcotics
And then we look after the “, will” for the other equivalent:
they be either both deported and have their visa revoked if they are not a citizen or both incarcerated and prosecuted if they are a citizen.
Now we can just go equivalent-by-equivalent. Let’s start with the first one—it’s shorter. We see a “both...and...” here, which tells us we’ll be working with a...what?
Conjunction!
Yes indeedy. Well done! So that’s fairly easy to do. The left equivalent will be a simple conjunction, which means the whole formula as far as we’ve gotten will look like this:
((A \(\bullet\) N) \(\leftrightarrow\) ???)
Now for the right equivalent, which is quite complicated. Here I’ve put the logical words in bold:
they be either both deported and have their visa revoked if they are not a citizen or both incarcerated and prosecuted if they are a citizen.
Again, we’ll just work step-by-step. We see and “either”, so that means we’ll be looking for our “or”. What comes between those two is one disjunct or half of a disjunction. Here’s that disjunct:
both deported and have their visa revoked if they are not a citizen
To reorient, we’re working on the left disjunct of the right equivalent, here:
((A \(\bullet\) N) \(\leftrightarrow\) ( \(\underset{\uparrow}{???}\) \(\vee\) ??? ))
Okay, what do we see there? A “both...and...”, which tells us that we’re again working with a conjunction. What else? An “if”, which tells us we’ll have a conditional to sort out. Remember that when we see an ‘if’ in the middle of a sentence, we need to mentally rewrite the sentence as an “if...then...” sentence. Technically, there’s a bit of ambiguity in the grammar here. I’m just going to default to the most likely reading, which is “if they are not a citizen, then they will both be deported and have their visa revoked.” So our left disjunct looks like this:
((A \(\bullet\) N) \(\leftrightarrow\) (\(\underset{\uparrow}{(\neg \text{C} \rightarrow (\text{D} \bullet \text{V}))}\) \(\vee\) ??? ))
Now we need to focus on that last disjunct where the question marks are. That’s the part that comes after the final ‘or’ in our example. It reads:
both incarcerated and prosecuted if they are a citizen
This will work essentially the same as the last disjunct. Rewrite as an “if..then” and then note that the consequent is a “both...and...” (a conjunction). Then symbolize:
((A \(\bullet\) N) \(\leftrightarrow\) (\((\neg \text{C} \rightarrow (\text{D} \underset{\uparrow}{\bullet} \text{V}))\) \(\vee\) (C \(\rightarrow\)(I \(\bullet\) P))))
Every time I choose a new Sentence Letter, I’m always checking the rest of the formula to ensure that I’m not reusing a letter that actually signifies a different proposition. I am sure to reuse that ‘C’, though, since there’s a clear logical relation between “x is a citizen” and “x is not a citizen”—one is a negation of the other and so our logical has to capture this fact.
That’s it. We’re all done! Good job! That was tough, but you can see how when we go step-by-step, we’re able to do this accurately and without much headache.
On an unrelated note, some cases use the same framing words as we have learned, but aren’t as clear. Here are a few examples:
Both you and your sister will need to clean your rooms before you can go out with friends tonight.
Either of you can drive the car today, but not your friends.
Think about what each is saying and then try your best to capture the idea in a symbolization. For instance, the first one seems to be saying something like:
Both you will need to clean your room before you can go out with friends tonight, and your sister will need to clean her room before she can go out with friends tonight.
There is a sense in which what is being said here is somewhat different than I have written above. It might mean something like:
Only if both of you clean your rooms will both of you get to go out with your friends tonight.
Which might break down into:
Only if you clean your room will you get to go out with your friends tonight AND only if your sister cleans her room will she get to go out with her friends tonight.
This might capture what’s being said fairly well, but we’ll stick to doing something a bit less sophisticated: we’ll stick to trying to capture the surface content of the English sentences and so we’ll focus only on explicit logical indicator words. So if we stick with the original reformulation, we get something with a logical structure like this:
[you will need to clean your room before you can go out with friends tonight]
AND
[your sister will need to clean her room before she can go out with friends tonight.]
It’s a simple conjunction: the bracketed sentences don’t have any internal logic to them—they’re just descriptions of the world, true or false. With that in mind, the symbolization is fairly straightforward:
(Y \(\wedge\) S)
‘Y’ for “you will need to clean your room before you can go out with friends tonight” and ‘S’ for “your sister will need to clean her room before she can go out with friends tonight.”
Now let’s take the second example from above:
Either of you can drive the car today, but not your friends.
This has an “Either...Or...” structure to it, but the actual disjunct propositions are hidden in the grammar a bit. If you think through it, you’ll find that the first part (before “, but”) is a disjunction between:
You can drive the car today OR (the other) You can drive the car today
This sentence is grammatically strange for our purposes in a few ways: it’s second person plural, meaning it’s a statement to two “you”s, and the two disjuncts are a bit hidden. It has an ‘either’ that isn’t followed up with an ‘or’, and so the disjunction isn’t as clear as it is when you have that nice frame to work with.
Now that we’ve got the hard part figured out, we can go ahead and symbolize the whole sentence:
[(Y \(\vee\) O) \(\wedge\) \(\neg\)F]
You can drive the car today OR (the other) You can drive the car today, BUT your friends can NOT drive the car today.