In this video I'm going to talk about translating sentences ...
from English into our symbolic notation.
To talk about translation
first we'll talk about the basic connectives,
our five main symbols.
Then about other words that don't map exactly onto the ...
five main connectives. "Unless"
and "neither" are two such words.
In a later video, we'll talk about complex ...
sentences that combine more than one connective.
First let's talk about "not."
If B stands for "Brad shaves"
then if we have "Brad doesn't shave"
the "not" is there in the middle of the sentence
and we get negation B.
Next let's talk about "and."
Let A be Angie is happy and B be "Brad shaves."
Then "Angie is happy and Brad shaves" is
A & B.
Let's look at another sentence,
"Angie is happy, though Brad shaves."
One thing we can say for sure about this is that it tells us two ...
facts, that Angie is happy and Brad ...
shaves. Both these things are true.
So here we also have A&B.
This statement also suggests that it's surprising that Angie ...
is happy, given that Brad shaves.
But we won't concern ourselves with the suggestion!
We just translate what's literally entailed by the ...
sentence, which comes out as A & B.
So "but," "though,"
"however," and all similar words can be ...
translated as &.
We also need to be on the lookout for hidden compound sentences.
Let's start with these sentences.
If we have "Brad and George shave"
that's the same as "Brad shaves and George shaves," which is B&G.
If we have "George shaves and dresses well"
that's the same as "George shaves and George dresses ...
well" which is G&D.
When a connective like "and" joins things that aren't ...
complete sentences, you have to expand it out to ...
complete sentences before you translate it.
Let's look at "or" now.
In this class, "or"
is always inclusive; it always means "or both."
With these abbrevations
"Brad or George shaves"
becomes B v G.
If you want to say the exclusive or,
"but not both," you have to spell it out.
So you would explicitly say "Brad or George shaves, but not both."
The "but" becomes &,
and we put a negation on "both Brad and George shave"
and we get (B v G) & ~(B & G)
Now let's look at "if," which is going to have some ...
more complications.
Say A is "Angie is happy" and B is "Brad shaves"
"If Brad shaves, then Angie is happy"
becomes B -> A.
This doesn't necessarily mean that if Angie is happy,
then Brad shaved! There could be other things ...
that make her happy.
You can also have "if" in the middle,
like "Brad shaves if Angie is happy."
To represent that, we have to untwist the ...
sentence so the "if" comes at the beginning.
We get "If Angie is happy, Brad shaves."
Which is A -> B.
Notice that the letters A and B are in the opposite order from ...
the original sentence, "Brad shaves if Angie is ...
happy." When we have the word "if"
in the middle of a sentence, we wind up switching the ...
order of the sentences around the arrow.
"Only if" is related to "if," but different.
One way to translate "only if" is as "not...
if not." You put negations on both ...
sides, and you change the "only if" to "if."
So if L is you win the lottery and T is you buy a ticket
"You'll win the lottery only if you buy a ticket"
means "You won't win the lottery if you don't buy a ticket."
Since "if" is in the middle here,
we have to switch the order of the sentences.
So it's "If you don't buy a ticket you won't win the lottery"
Or ~T -> ~L
Note that this does not hold vice versa!
It is certainly not necessarily true that if you do buy a ticket ...
you will win the lottery.
It's also possible to do "only if"
as a single arrow without using any extra negations.
Using the same abbreviations
let's look again at "You will win the lottery only if you buy a ticket."
This amonts to saying that if you won the lottery,
then you must have bought a ticket.
There's no other way for you to have won the lottery.
That becomes L -> T.
which is equivalent to ~T -> ~L.
(You can do a truth table to check that if you want!)
So when we have "only if" in the middle of a sentence,
you can translate it with an arrow and keep the sentences ...
in the same order! No extra negations are ...
needed if you do this.
Note again that this does not work in reverse.
This does not say that if you buy a ticket you win the ...
lottery. It must be L -> T, not T -> L.
We can organize this into a two-by-two table.
Whether you have "if" or "only if,"
you can translate it with a single arrow.
If it's "if" at the beginning,
you keep the sentences in the same order.
If it's "if" in the middle,
you switch the order. If you happen to see "Only if"
at the beginning, switch the order.
If you have "only if" in the middle,
it's the same order.
For a bit more on conditionals
there are some other synonyms for "if,"
like "Provided that," "Assuming that,"
and "On condition that."
All these behave like "if" no matter where they are in ...
the sentence, at the beginning or in the middle.
So "George shaves provided Brad shaves"
is just "George shaves if Brad shaves"--you turn the ...
"provided" into "if"
in the middle, where it is.
And since "if" is in the middle,
we switch the order, and get B -> G!
Our last basic connective is "if and only if"
given these sentences
"Brad washes his hair if and only if he showers"
becomes W <-> S. Unlike with the arrow,
the order doesn't really matter here.
In order to have a double arrow,
you're going to need "if and only if"
in full in your sentence.
If you just have "if" by itself,
or "only if" by itself,
you use a single arrow (and maybe some negations,
if you're using that way of translating "only if.")
Let's look at a couple of other phrases now.
First, "neither... nor."
With these abbreviations
"Neither Brad nor George shaves"
could be translated by taking the negation of Brad shaves ...
or George shaves
which would be negation parentheses B v G.
There's another equivalent formulation for this.
If neither one of them shaves, that means Brad doesn't ...
shave, and also George doesn't shave.
So that's ~B & ~G.
These are not equivalent to ~B v ~G,
though. That would mean at least one ...
of them doesn't shave, but it would leave open the ...
possibility that one of them does shave.
Finally, let's talk about "unless."
With these abbreviations
let's translate "You'll be cold unless you wear a jacket."
One way to handle it is by turning "unless" into "if not."
So we get "You'll be cold if you don't wear a jacket."
Since this has "if" in the middle,
we switch the sides, and we get ~J -> C.
Note that this isn't an if and only if.
We haven't literally said that you won't be cold if you do ...
wear a jacket. Maybe you wear a jacket but it's too thin.
We can also, surprisingly,
translate "unless" as "or."
To say that you'll be cold unless you wear a jacket is to ...
say that either you wear a jacket or you'll be cold.
This can be translated as J v C
or, since order doesn't matter ...
around a wedge, as C v J.
That's also what you'd get if you just put "or"
where the original "unless" was.
So, translating "unless"
as "if not" means you have to worry ...
about where the extra negation goes and which ...
order things go around the arrow.
If you just translate it as "or" you don't have to worry about ...
that. So that might be easier for you.