My question isn't exactly how to do a specific problem; it is to ask
you if logic is a type of thing where either you get it or you don't.
I recently had to drop symbolic logic because I just couldn't get it!
Especially when we started doing derivations with rules of replacement
like modus pollens. Derivations for SD+ are the most confusing, but I
can't even get SD. Is logic something where either you get it
or you don't?
Thanks for writing to Dr. Math.
Symbolic logic is something that you can master.
The hardest thing about symbolic logic is learning how to work with
the symbols. Once you know what all the symbols stand for, the
logic should come more easily.
I'll try to give you a bit of a crash course in basic symbolic
logic using an approach that I hope will help. Another place you
can turn to is the Logic
section of the Dr. Math archives.
[TOP]
Philosophers and Logicians
First, I'd like to do a bit of philosophizing as a way to
lead into the logic. Philosophers and logicians have a lot of
overlap in what they do. Many logicians are also philosophers, and
all philosophers are logicians to some extent (some much more so
than others). Given that there is such a connection between
philosophers and logicians, I find it striking just how radically
different the fields are. Philosophers are interested in finding
deep truths about the world, be they epistemological, metaphysical,
ethical, etc. Logicians (qua logicians) are only interested in
using a set of rules to manipulate arbitrary symbols that have no
relevance to the world.
The (sometimes difficult) marriage between philosophy and logic
comes from the fact that everyone in the world (except, I would
argue, people who are commonly called "crazy") accepts the truths
proven by logic to be universally true and unquestionable.
Philosophy needs logic because in order to establish that a
philosophical doctrine is true, one needs to show that the doctrine
is universally and unquestionably true. One needs to, in other
words, make a demostration that everyone would accept as proof that
the proposition is true. To do that, the philosopher needs logic.
[TOP]
Shorthand Sentences
Logicians, though, are very lazy people. They don't like to write
long derivations in English because English sentences can be fairly
long. So what they do instead is a kind of shorthand. If you give
a logician a sentence like
he will pick a letter and assign it to that sentence. He
now knows that the letter is just shorthand for the sentence. The
way I learned logic, capital letters are used for sentences (with
the exception of U, V, W, X, Y, and Z; I'll get to those later).
So let's just start at the beginning of the alphabet and use the
letter "A" to represent the sentence "The dog is brown." While
we're at it, let's use the letter "B" to represent the sentence "The
dog weighs 15 lbs."
In addition to saving time and ink, this practice of using
capital letters to represent whole sentences has a couple of other
advantages. The first is that to a logician, not every word is as
interesting as every other. Logicians are extremely interested in
the following list of words:
and
or
if...then
if and only if
not
They call these words "connectives." This is because you can
use them to connect sentences that you already have together
to make new sentences.
When you write in English, those words don't stand out; they just
get lost in the middle of sentences. Logicians want to make sure
the words look special, so they take the whole rest of the sentence
(the part they don't care about) and use a single letter to
represent that. Then their favorite words stand out. Let's rewrite
our earlier example about the dog using our logician's shorthand:
ASSUMING: A
AND ASSUMING: B
I CONCLUDE: A and B
The other advantage of using capital letters to represent sentences
is that you ignore all the information that isn't relevant to
what you're trying to do. For the derivation
I did above, it didn't matter that the sentences were both about
some dog. It didn't matter that they were about weight or color.
They could have just as easily been sentences about how tall the dog
is or about a cat or a person or a war or whatever. And if we can
do the derivation for A and B, then we can do the same exact
derivation for C and D or E and N or any other sentences we like.
[TOP]
Parentheses
I know that a lot of books and instructors claim that it is okay to
drop the outermost parentheses in a sentence. I've done it myself
many times. And 95% of the time it won't cause you trouble if
you're careful. But let's say we started with this 'sentence'
and decided to negate it. Well, the way to negate a sentence is to
stick a ~ on the front, so let's do that:
But wait! What we did there was just negate the A. We wanted to
negate the whole sentence. If we were really sharp, then we might
notice that somebody had given us an illegitimate sentence
that was missing parentheses, and so we would add the
parentheses before adding the ~, to get:
which is what we wanted.
It seems silly to make such a big deal about parentheses when we're
dealing with simple sentences, but when you're doing a 30-line
derivation and you're tired, it's easy to make a mistake just
like that on line 17 and get yourself into real trouble. It's
better to just remember the simple rule and always add parentheses
when you have a two-place connective.
. . .
Let's take a deep breath and then go quickly over what
we have so far.
[TOP]
Using Connectives
Connectives are logical terms,
| ^ | (and) |
| v | (or) |
| -> | (if...then) |
| <-> | (if and only if) |
| ~ | (not) |
which you can add to a sentence.
A simple sentence is one that has no connectives. For example: A
(the dog is brown).
A complex sentence is a sentence which is made up of one or more
simple sentences and one or more connectives. Some examples are:
(A ^ B)
(A v B)
(A -> B)
(A <-> B)
~A
You can use connectives on complex sentences just as you can on
simple sentences. Let's introduce a new simple sentence "it is
raining," and let's call our new sentence C. We now have a lot
more sentences that we can make. (Keep in mind, we have no idea yet
which of these sentences are true or false; we also don't yet know
how these sentences relate.) For example:
(C ^ B)
~C
(B v C)
(C v B)
(B -> C)
(A -> C)
(C -> A)
(B <-> C)
(~B <-> C)
~(B <-> C)
C
~~C
((A ^ B) v C)
(((A ^ ~B) v ~C) -> (~(A v B) <-> C))
These can get a little complicated. That last sentence is
especially scary looking; we'll come back to it in a little while.
For now, here is a quick run-down of how to use connectives
to make complex sentences from simple ones.
| To make this complex sentence |
Do this |
We say |
|---|
| ~C |
Stick a ~ on C. |
~C is the negation of C. |
| (B v C) |
Use a v to join B and C. |
(B v C) is the disjunction of B and C. |
| (B ^ C) |
Use a ^ to join B and C. |
(B ^ C) is the conjunction of B and C. |
| (B -> C) |
Use a -> to join B and C. |
B implies C.
(B -> C) is a conditional or implication. |
| (B <-> C) |
Use a <-> to join B and C. |
B implies C and C implies B.
(B <-> C) is a biconditional. |
[TOP]
Complicated sentences
Now let's take a closer look at the most complicated sentence on our
list and see if we can make it more manageable. The way to analyze
a complicated sentence is to start at the outside and work your way
in.
The outermost parentheses on this ugly sentence
(((A ^ ~B) v ~C) -> (~(A v B) <-> C))
are used to connect these two sentences
((A ^ ~B) v ~C)
(~(A v B) <-> C)
with a
So the way to build our ugly sentence is to start with these two
less ugly sentences:
((A ^ ~B) v ~C)
(~(A v B) <-> C)
and connect them with the main connective
We can then analyze each subsentence if we like.
I told you before that simple sentences are represented by the
capital letters A through T, and that U, W, X, Y, and Z are saved
for something else. (I rarely use V because it looks to much like
the symbol for 'or'.) U, W, X, Y, and Z are used as shorthand for
other sentences in logic (some books use italic letters and others
use Greek letters, but since I only have plain text to work with, I
use the end of the alphabet). I call these "sentence variables."
So, just as we can take the English sentence
and use the capital letter D to represent it, we can take the
sentence in logic
and use the capital letter U to represent it.
[Note: it is also legal to use capital variables to stand for simple
sentences. So you can take the simple sentence
and use the letter Z to stand for it.]
This can be useful in analyzing complicated sentences. For example,
if we have the scary looking sentence
((((A ^ ~B) v (B <-> C)) -> (~(C v D) ^ ~(~A -> ~~D))) v A)
we can start using sentence variables to stand for subsentences. So
if U stands for
Then we have
(((U v (B <-> C)) -> (~(C v D) ^ ~(~A -> ~~D))) v A)
and if V stands for
then we have
((V -> (~(C v D) ^ ~(~A -> ~~D))) v A)
If W stands for
we have
((V -> (W ^ ~(~A -> ~~D))) v A)
and if X stands for
we have
and if Y stands for
we have
So we know where our main connective is. And by substituting back
in for the sentence variables, we can recreate our sentence in
managable chunks.
It is very important to keep track of what sentence variables stand
for when you're doing this kind of substitution. This can be a
major source of error if you're not keeping close track of what
every letter stands for.
. . .
Now that we know all the details of the language of symbolic
logic, it's time to actually do symbolic logic.
The first step with every sentence is to identify the main
connective. The reason is simple:
In symbolic logic, the main connective of a sentence
is the only thing that you can work with.
Let's look at our complicated sentence from earlier
(((A ^ ~B) v ~C) -> (~(A v B) <-> C))
is fundamentally an implication between these two subsentences:
((A ^ ~B) v ~C)
(~(A v B) <-> C)
There is no
in either subsentence, but the sentence as a whole is still first
and foremost an implication because of what its main connective is.
So when you're trying to figure out how in the heck you can work
with this ugly sentence
(((A ^ ~B) v ~C) -> (~(A v B) <-> C))
you need to remember that it is an implication and treat it just as
one.
[TOP]
What the Connectives Mean
Here's a quick course on what the connectives mean. (I assume you
have some familiarity with them.)
| The sentence |
is TRUE whenever |
is FALSE whenever |
|---|
| A |
A is true |
A is false |
| ~A |
A is false |
A is true |
| (A ^ B) |
A is true and B is true |
A is false; or B is false; or both A and B are false |
| (A v B) |
A is true; or B is true; or both A and B are true |
A is false and B is false |
| (A <-> B) |
A and B are both true; or A and B are both false |
A is false and B is true, or A is true and B is false |
| (A -> B) |
A is false; or B is true; or A is false and B is true |
A is true and B is false |
This last one is a little weird, so let's think about it. If we
translate it back into English, we get
If the dog is brown then the dog weighs 15 lbs.
How would we go about proving that this sentence is false?
Let's say that the dog is brown and the dog weighs 15 lbs. Does
that disprove the if...then statement? Certainly not!
What if the dog is brown but the dog weighs 25 lbs.? That does
disprove the statement.
What if the dog turns out to be white? Then we cannot disprove the
inference because it only makes a prediction about a brown dog. If
the dog isn't brown, then we can't test the prediction.
So the only way to make the sentence
false is to make A true and B false at the same time.
Given any other values of A and B, the sentence comes out true.
[TOP]
12. v Elimination
The last rule is a little tricky. It's "v elimination." It says if
you have
and you have
and you have
Then you are entitled to
[Most of the time this means that when you have a disjunction that
you don't know what to do with, you have to derive an implication
for each side of the disjunction before you can go on.]
The rule is hard to do with derivations, but it is actually not too
hard to understand if you take an example.
Let's say we know
(A v B)
[The dog is brown or the dog weighs 15 lbs]
And we know
(A -> E)
[If the dog is brown, then the killer is a man]
And we know
(B -> E)
[If the dog weighs 15 lbs, the killer is a man]
Then we don't have to bother figuring out whether A is true or B is
true; either way we are entitled to
Continuing Our Last Derivation
Let's continue our last derivation to get a demonstration of
"velim."
{
1) A [assumption]
2) A [repetition of 1]
}
3) (A -> A) [->intro on 1-2]
4) ((A -> A) v B) [vintro on 3]
{
5) (A -> A) [assumption]
{
6) C [assumption]
7) C [repetition of 6]
}
9) (C -> C) [->intro on 6-7]
}
10) ((A -> A) -> (C -> C)) [->intro on 5-9]
{
11) B [assumption]
{
12) C [assumption]
13) C [repetition of 12]
}
14) (C -> C) [->intro on 12-13]
}
15) (B -> (C -> C)) [->intro on 11-14]
16) ((A -> A) v B) [repetition of 4]
17) ((A -> A) -> (C -> C)) [repetition of 10]
18) (B -> (C -> C)) [repetition of 15]
19) (C -> C) [velim on 16, 17, 18]
[Not the most efficient way to prove (C -> C), but it is valid.]
There are a lot of other rules people try to tell you, but anything
you can do with those, you can do with these 12 rules.
[TOP]
Mundane Rules: What Do You Have?
Now that we have the steps for doing derivations, let me
try to explain that confusing business about discharging
assumptions. I'm going to approach this from a slightly different
angle this time.
Of the 12 rules I gave you, 8 are pretty straightforward. They are
what I would call the "Mundane Rules." The way Mundane Rules work
is: they say "if you have X and Y and Z, then you are entitled to U."
The tricky thing with Mundane Rules is knowing what you "have."
You "have" any sentence that is written down on a line of the
derivation except those which are closed off in curly brackets (which
are gone forever once the brackets close).
Being "entitled" to something just means that you can legally write
it down as the next line of the derivation.
The easiest Mundane Rule is repetition:
If you have
then you are entitled to
Another Mundane Rule is ^ introduction:
If you have
and you have
then you are entitled to
Simple enough. (I went into more detail on _why_ this is a sound
rule in the last e-mail.)
Another pretty easy Mundane Rule is ^ elimination:
If you have
then you are entitled to
Or, if you prefer, you are also entitled to
So far so good.
Another Mundane Rule is -> elimination:
If you have
and you have
then you are entitled to
This is actually the same thing as Modus Ponens, so you can call it
that if you prefer. Since I don't speak Latin, I prefer calling it
"-> elimination" because that is more descriptive of what the rule
is doing.
Another Mundane Rule is <-> introduction:
If you have
and you have
then you are entitled to
This one is a little tricky to explain. Let's assume somehow we have
Under what conditions could that be true? There are 3 possibilities:
X is true and Y is true
X is false and Y is true
X is false and Y is false
Also, we have
That can only be true under these conditions:
X is true and Y is true
X is true and Y is false
X is false and Y is false
Since we "have" both of these sentences, then they must
both be
true. So under what conditions are they both true? Well, only
these two:
X is true and Y is true
X is false and Y is false
Which are exactly the conditions for:
Which means we are entitled to write that.
Another Mundane rule is <-> elimination:
If we have
and we have
then we are entitled to
OR:
If we have
and we have
then we are entitled to
Another Mundane Rule is v introduction:
If you have
then you are entitled to
and you are also entitled to
This is a little tricky too. We "have" X, which means X must be
true. Now we can just, out of the blue, pick any sentence we
like and put it into a disjunction with X. Why can we do
that? Well, let's say we pick a FALSE sentence. Is that still okay?
Yes, it is! Even if Y is false, the disjunction with X is still
true, so we haven't written a false sentence, and we are still
okay.
The last Mundane Rule is v elimination:
If you have
and you have
and you have
then you are entitled to
So much for the Mundane Rules. Mundane Rules are useful in
derivations because they let you move from one step to the next.
They tell you what you can do with the sentences you have. They
also can give you a hint as to what you need to do next. For
example, if you have
and you want to eliminate the v, but you don't have
yet, then you'd better go get those two sentences.
The problem with the Mundane Rules is that they only let you play
around with sentences you already HAVE. You can't get anything NEW
out of them.
So far we've gone over the 8 Mundane Rules. There are 12 rules in
total. Of the 4 remaining, one is a 'Special Rule' and three
are 'Fun Rules'.
[TOP]
Three 'Fun' Rules
The three Fun Rules all have this form:
If you assume
and then, on that assumption, you derive
You can discharge the assumption you made at X and then you are
entitled to
The first Fun Rule is -> introduction:
If you assume
And then, on that assumption, you derive
You can discharge the assumption you made at X and then you are
entitled to
This is, in my opinion, the most important and fundamental rule in
logic. It is the foundation of all logic. [It's also really
important for derivations. If you look up at the Mundane Rules, a
lot of them require you to have sentences of the form (X -> Y) to
apply them.]
The justification is that if you assume (but don't prove)
and then, on that assumption, you derive
then you HAVE NOT proven that the floor is wet, but you have PROVEN
(no assumptions required) that
(A -> P)
[If the dog is brown, then the floor is wet.]
The last two Fun Rules are closely related. One is ~ introduction:
If you assume
And then, on that assumption, you derive
and
you can discharge the assumption you made at X; then you are
entitled to
[You may have been taught this rule as a reductio ad absurdum.] The
idea is that if assuming X leads you to a contradiction, then there
must've been something contradictory ABOUT X ITSELF. So X must be
false. If X is false, then by definition ~X is true: no ifs, ands,
buts, or assumptions about it.]
The last Fun Rule is ~ elimination:
If you assume
and then, on that assumption, you derive
and
you can discharge the assumption you made at ~X and then you are
entitled to:
The idea here is basically the same. ~X is contradictory and
therefore false, so X is proven true.
I have another nickname for ~ elimination. It is what I call
the "Fallback Rule." With every other rule, the way it works is by
getting what you want by introducing the main connective or by using
what you have by eliminating the main connective. But take a look
back at what I just did with ~ elimination. I just proved X is
true. There's no way to tell by looking at X that you can prove it
by eliminating a ~, but you can. [Actually, ANY sentence you like
can be proven with ~ elimination, but it is sometimes hard to do.]
So this is where Step 5 of the derivations comes from. If you are
trying to prove some sentence X, the first thing to try is to try to
introduce the main connective of X. But if you run into a dead end
doing that, then assume ~X and try to derive a contradiction.
Those are the rules reviewed and better organized
so that they make sense, and the difficult bit about discharging
assumptions is (I hope) a little clearer.
One other thing to watch out for. Some logic problems ask you to
prove that a certain sentence is a logical truth. On those
problems, you have to discharge all your assumptions and prove that
the sentence is true with no assumptions (that is, write it without
indenting and outside of all the curly brackets). I'll do an
example of a derivation like that in a minute.
[TOP]
Deriving a Conclusion
Other logic problems give you a list of "givens" or "hypotheses" and
ask you to derive a conclusion from them. In those problems, what
they are saying is that you need to assume the hypotheses, but not
discharge those assumptions. Let me give you an example:
Given:
Prove:
Here we go:
{
1) A [assumption, given]
{
2) (A -> B) [assumption, given]
{
3) (~B v C) [assumption, given]
4) A [repetition of 1]
5) (A -> B) [repetition of 2]
6) B [->elim on 4&5]
Now what do we do? We have a disjunction on 3 that we don't know
what to do with, so we need to eliminate it. But in order to
eliminate it, we need to get (~B -> something) and (C -> something).
Let's first work on getting (~B -> something).]
{ [Note: this assumption isn't given,
so we're going to have to discharge it]
7) ~B [assumption]
Let's see if we can derive C. If we derive (~B -> C) then we'll be
most of the way to finishing the problem. How can we derive C?
Well, we should try to introduce the main connective. But wait!
There is no main connective. C is just a simple sentence. So what
can we do? I guess we have to do step 5, try ~elimination.
{ [another assumption we'll have to discharge]
8) ~C [assumption]
9) B [repetition of 6]
10) ~B [repetition of 7]
} [Closes off lines 8-10]
11) C [~elim on 8-10]
Up to this point we haven't closed off any assumptions. That means
that all of the lines up to this point were sentence that we "have"
and can use. But now we just closed off lines 8-10 by discharging
the assumption at 8. That means that lines 8-10 are gone, they are
off-limits and illegal forever. The good news, though is that we
derived C, so now we can discharge the assumption we made at line 7.
} [Closes off 7-11]
12) (~B -> C) [->intro on 7-12]
This may seem like a bit of sleight of hand, like I'm trying to pull
the wool over your eyes. How can I use the assumption I made at
line 7 as part of the contradiction? I just did a ~elimination to
prove C, but there was nothing contradictory about ~C itself; the
contradiction was that I had B and then I assumed ~B. This is the
familiar refrain "anything can be proven from a contradiction."
Once I assumed ~B, I could've proven (~B -> anything-I-want), I
chose to prove (~B -> C) because I eventually want to get C.
Now we have made some progress on eliminating the disjunction we had
on line 3: (~B v C). We have (~B -> C), now we need (C -> C), so
let's go get it.
{
13) C [assumption]
14) C [repetition of 13]
Notice that I can repeat 13 because I have not yet discharged that
assumption. However, I cannot repeat the C on line 11 because I
closed off line 11 already, so it is gone forever.
} [Closes off 13-14]
15) (C -> C) [->intro on 13-14]
16) (~B v C) [repetition of 3]
17) (~B -> C) [repetition of 12]
18) (C -> C) [repetition of 15]
19) C [velim on 16,17,&18]
Not all of those repetitions were necessary, since we "had" those
lines already (they hadn't been closed off), but I added them for
clarity.
You should go back and double-check the derivation to make sure that
I never broke the rules by using a line that was closed off and that
I didn't break any other rules. Also, make sure that I
discharged all the assumptions except the three I was given at the start.
[TOP]
Deriving a Sentence
As I said before, the other type of problem is where you are
handed a sentence and told to derive it. In this problem, you can
make any assumptions you need, but you have to discharge all of them
and end up with the sentence you're looking for at the end. This
usually involves finding the main connective of the sentence you're
supposed to prove and then introducing it (sometimes you have to
find other sentences too: for example if the main connective is ^,
you need to prove each half of the sentence and then do an ^intro).
Sometimes trying to do this will get you to a dead end, and then
you may try to assume the negation of the sentence you're trying to
get and see if you can find a contradiction.
Let's do an example. Let's try to prove
(((~A ^ ~C) v (~C <-> B)) -> (B -> ~C))
The main connective is a ->, so let's introduce it. To do that we
need to assume the left and derive the right.
{
1) ((~A ^ ~C) v (~C <-> B)) [assumption]
Here we have a disjunction, so we need to eliminate it. That
means we need to find two entailments. It would be great if we had
((~A ^ ~C) -> (B -> ~C)) and ((~C <-> B) -> (B -> ~C)), so let's try
to get those. First, let's work on ((~A ^ ~C) -> (B -> ~C)).
{
2) (~A ^ ~C) [assumption]
3) ~A [^elim on 2]
4) ~C [^elim on 2]
We actually don't need line 3, but it's good practice to get both
sides of an ^ while you can just in case you might need them later.
Now we have ~C, but what we want is (B -> ~C), so let's work toward
getting that.
{
5) B [assumption]
6) ~C [repetition of 4]
} [closes 5-6]
7) (B -> ~C) [->intro on 5-6]
} [closes 2-7]
8) ((~A ^ ~C) -> (B -> ~C)) [->intro on 2-7]
So now what we need to finish the velimination on line 1 is to
derive ((~C <-> B) -> (B -> ~C)).
{
9) (~C <-> B) [assumption]
So now we need (B -> ~C).
{
10) B [assumption]
11) ~C [<->elim on 9-10]
If you wanted to, you could make this a little more clear by
repeating (~C <-> B) and then doing the <->elim, but it's not
necessary since we have both (~C <-> B) and B we are entitled to ~C.
} [closes 10-11]
12) (B -> ~C) [->intro on 10-11]
} [closes 9-12]
13) ((~C <-> B) -> (B -> ~C)) [->intro on 9-12]
Now we can do our velimination on line 1 because we have
((~C <-> B) -> (B -> ~C)) and ((~A ^ ~C) -> (B -> ~C)). If you
want, for clarity, you can repeat line 1 and line 8, but it's not
necessary.
14) (B -> ~C) [velim on 1,8,13]
}
15) (((~A ^ ~C) v (~C <-> B)) -> (B -> ~C))
As always, you should go back and double-check the derivation once
you're done.
The hardest thing about doing derivations is figuring out what to do
next. When you have a lot of random rules with Latin names to
choose from, it's difficult. This set of rules helps you to know
what to do by either introducing what you're trying to get or
eliminating what you have.
My advice to you is to try to do some of the derivations in your
book or that you had for your class using these rules. Any
derivation is possible with them. It takes a lot of time to learn logic
and have it sink in, but if you take it slowly enough and practice,
it will become easier. Get comfortable with these 12 basic rules and
the 5-step method for doing derivations.
[TOP]
Rules with Latin Names
Many (probably most) places, you don't learn these 12 rules with
logic. Even though I think these rules make the most sense and allow
a straightforward approach to solving any problem in symbolic logic,
more advanced students may want to study the rules such as Modus
Tollens and DeMorgan's Law.
The twelve rules I've presented here are systematic and
straightforward, and all of them move by baby steps. The way
I think of these rules (in some cases this is not historically
accurate) is that logicians noticed that when doing derivations,
they often repeated the same steps over and over. Eventually,
someone decided that rather than doing these same five or ten steps,
you can take shortcuts.
Modus Tollens serves as an instructive example. Let's say I have:
And I have:
And I am trying to get:
Here's what I'd have to do. Since I'm trying to find ~A, I'll do a
~introduction on A:
{
1) (A -> B) [assumption, given]
{
2) ~B [assumption, given]
{
3) A [new assumption]
4) B [->elim on 1 and 3]
5) ~B [repetition of 2]
} [closes off 3-5]
6) ~A [~intro 3-5]
We have to do this so often that we just call these five steps
"Modus Tollens." Modus Tollens is a shortcut rule. There are several
others too, some more involved.
The following is a list of the major rules,
together with a justification of why each of them is valid and a
short example of how you might use some of the more challenging ones.
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1. Modus Ponens
This is one of the most straightforward laws in logic. It states
that if you have
and you have
then you are entitled to:
This is just what we've been calling "-> elimination."
The reason it works is that we are given (X -> Y). Which means
that X cannot be true at the same time Y is false. So if X is true
(which is the other given), then Y must be true as well, so we are
free to conclude Y is true.
Example:
"If it is raining, then there are clouds" and "it is raining"
together imply "there are clouds."
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5. Hypothetical Syllogism
The rule here is that if you have
and you have
then you can conclude
Here's why:
We know that "if X is true, then Y is true." And we know that "if Y
is true, then Z is true." But we don't know anything about whether
any of the letters are actually true or not.
Let's assume (or hypothesize) for a second that X is true. Then, by
modus ponens, Y is true. And then by modus ponens again, Z is true.
So: If we assume X is true, then we conclude Z is true. Since we
didn't know X was true, we cannot take Z home with us, but we can say
that "If X was true, then Z would be true." This is equivalent to
saying "If X, then Z" or (X -> Z).
Example: "If it is raining, then there are clouds" together with "if there are
clouds, then the sun will be blocked" imply "if it is raining, then
the sun will be blocked."
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9. Switcheroo
(I've heard that this was actually named after a person, but I don't
know that for certain.)
This is a shortcut rule which states that if you have
then you can interchange that with
To understand why, let's think about (~X -> Y). This says that ~X
cannot be true at the same time that Y is false. Or, to put that
another way, X cannot be false at the same time Y is false.
So (~X -> Y) can only be false when X and Y are both false.
Similarly, the only way for (X v Y) to be false is to have X and Y
both false. So the two expressions are true unless X and Y are both
false, so they have the same "truth conditions" and are therefore
equivalent (i.e. interchangeable).
Example: "My dog is fat, or my cat is fat" is equivalent to "If my dog is
thin, then my cat is fat."
(This one is hard to wrap your mind around, but think about what must
be true/false about the world in order to make each statement true or
false and it should eventually become clear.)
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12. Rule of Joining
This is just what we've been calling "^ introduction.""
The problem with shortcut rules is that they're easy to misuse.
In my opinion, the best way to learn them is to practice with the
twelve systematic rules and if you find yourself doing the same steps
over and over, you may have found a shortcut rule.
If there's a rule you don't understand, try to use the twelve
systematic rules to figure out how the rule works. Once you see the
steps in deriving the rule and you know why it is a valid
shortcut, you won't have any trouble using it. And remember, if you
get stuck and don't know what to do, you can always fall back on
the twelve systematic rules.
Another topic which comes up often in logic is how to translate
complicated English sentences into logical notation. There are some pages in the
Dr. Math archive which can help with that:
This is really just an introduction to logic. There are more detailed
systems that allow you to systematize properties of objects, other
possible worlds, the relation between knowledge and belief, and even the
logic of obligations. You can find more on how to go beyond the
basics here:
Best of luck,
- Doctor Achilles, The Math Forum
Symbolic Logic 
The Math Forum is a research and educational enterprise of the Drexel University School of Education.