
Bivalence.
While there are 3valued and manyvalued logics, remember that our logic
is 2valued (or bivalent).
Therefore, "She was not unhappy" must be translated as if it were
synonymous with "She was happy."
If you dislike this restriction, then you dislike bivalence and will
have a reason to use a 3valued or manyvalued logic.

Exclusive disjunction.
Remember that  in our notation expresses inclusive
disjunction: p  q means that either p is true or
q is true or both.
The exclusive disjunction of p and q asserts
that either p is true or q is true but not
both.
The natural, but longwinded, way to express exclusive disjunction, then,
is (p  q) & ~(p & q).
Notice, however, that exclusive disjunction is really saying that
p and q have different truthvalues; if one of them is
true, then the other isn't, and vice versa.
The way to say they have different truth values is to deny their
equivalence: ~(p ⇔ q).
If you like, you can introduce a symbol for this: p ⊕ q.
For example, when a menu says "cream or sugar", it uses an inclusive
"or", because you may take one, the other, or both.
But when it says "coffee or tea", it uses an exclusive "or", because
you are not invited to take both.

Conjunction.
We express conjunction with many words other than "and", including "but,"
"moreover," "however," "although", and "even though".
In English these expressions sharply contrast the two conjuncts, saying
in effect "if you believe the first conjunct, then you will be surprised
by the second."
But they still assert conjunction.
The contrast between the conjuncts is not logically relevant; validity
never turns on it.

Sometimes "and" does not join whole propositions into a compound
proposition.
Sometimes it simply joins nouns: "Bert and Ernie are brothers"
(or are they just roommates?).
This cannot be paraphrased, "Bert is a brother and Ernie is a
brother," for that does not assert that they are brothers to each
other.

Sometimes "and" joins adjectives: "The leech was long and wet and
slimy."
This, however, can be paraphrased, "The leech was long, and the
leech was wet, and the leech was slimy."

Unless.
Sometimes "unless" should be translated as inclusive disjunction, and
sometimes as exclusive disjunction.
For example, "I'll go to the party unless I get another offer" means
that I'll go if nothing else comes along.
In many contexts it also means that I might go anyway; the second offer
might be worse.
So I'll go or I'll get another offer or both (inclusive disjunction).
Consider by contrast, "I'll go to the party unless Rufus is there".
In many contexts this means that if I learn Rufus is going, then I'll
change my mind and not go.
So I'll go or Rufus will go but not both (exclusive disjunction).
For symbolizing exclusive disjunction, see Tip 2,
above.
Because there is no hard and fast rule, paraphrase the English before
translating.
(I thank Susanna S. Epp for helpful correspondence on the subtlety of
"unless".)

And/or.
When people say "and/or", they seem to mean inclusive disjunction.
"Bring a scarf and/or a hat" means "bring a scarf or bring a hat or
both", and so is translated simply, S  H.

Neither, nor.
"Neither p nor q" means that both p and q
are false.
Therefore translate it ~p & ~q or ~(p  q).
These two formulas are equivalent by DeMorgan's Theorem.

Not both / both not.
Many students confuse "not both" with "both not".
If p and q are "not both" true, then we are denying
their conjunction: ~(p & q).
One of them may be true, just not both.
So this is equivalent to ~p  ~q.
On the other hand, if p and q are "both not" true,
then we are denying each of them; they are both false: ~p & ~q.
Neither of them may be true; so this is equivalent to ~(p  q).
The best cure for confusing "not both" with "both not" is familiarity with
DeMorgan's Theorems.
(For the related distinction between "not all" and "all not", see
Tip 33, below.)

Material implication.
p ⇒ q translates a wide variety of English expressions, for
example, "if p, then q", "if p, q", "p implies q", "p entails q", "p
therefore q", "p hence q", "q if p", "q provided p", "q follows from p",
"p is the sufficient condition of q", and "q is the necessary condition
of p".
The least intuitive is "p only if q". See the next two tips.
 But be careful.
Some nearly synonymous expressions, like "q because p", "q since p",
"because p, q", "since p, q", and even some instances of "p therefore
q", are not genuine cases, or at least not merely cases, of material
implication.
They may seem so because they make p into a condition or reason for
q.
But in "if p, then q" we are noncommittal about the truth of p,
whereas most speakers who assert "q because p" and its variants are
asserting the truth of p.
To capture this aspect of the proposition's meaning, use conjunction,
"q & p".
To capture the implication claim as well, use both conjunction and
material implication, p & (p ⇒ q).

Necessary and sufficient conditions.
We say that p is a sufficient condition of q when p's
truth guarantees q's truth.
By contrast, q is a necessary condition of p when q's
falsehood guarantees p's falsehood.
In the ordinary material implication, p ⇒ q, the antecedent p
is a sufficient condition of the consequent q, and the consequent q is
a necessary condition of the antecedent p.

Satisfy yourself of this by reflecting on modus ponens and
modus tollens.
Given p ⇒ q, modus ponens tells us that the truth
or presence of p suffices to give us q.
Hence the antecedent is the sufficient condition of the consequent.
Similarly, modus tollens tells us that the falsehood of
q (the truth of ~q) guarantees us the falsehood of p (the truth of
~p).
Hence the consequent is the necessary condition of antecedent.

Or, satisfy yourself of this by reflecting on an example: "If
Socks is a cat, then Socks is a mammal."
Being a cat is a sufficient condition of being a mammal.
Being a mammal is a necessary condition of being a cat.

The fact that material implication expresses sufficient and necessary
conditions in this way can be a great help in translation.
Ask yourself about a difficult English sentence: what is being
asserted to be the sufficient condition of what here?
What is being asserted to be the necessary condition of what here?
When you find the sufficient condition, make it the antecedent.
When you find the necessary condition, make it the consequent.

If p is both necessary and sufficient for q, then we must say
p ⇔ q (material equivalence).

Only if.
We translate "p only if q" as p ⇒ q.
This is surprising to many people because "if" usually cues the
antecedent.
Rather than say that "if" sometimes cues the consequent, it is better
to say instead that "only if" differs from "if", and "only if" cues the
consequent.
If you understand necessary and sufficient conditions, this translation
should make more sense: "p only if q" clearly asserts that q is a
necessary condition of p.
The necessary condition of something is the consequent of that something
in a material implication (Tip 9).
Modus tollens assures us that p ⇒ q asserts that p
is true only if q is true, or that q is the necessary condition of p.
For under modus tollens, from p ⇒ q and ~q
we can validly infer ~p.

If and only if.
Tip 10 showed why "p only if q" is translated
p ⇒ q.
It should already be clear why "p if q" should be translated
q ⇒ p ("if" cues the antecedent).
So if we say "p if and only if q" we are asserting both p ⇒ q
and q ⇒ p, which amounts to p ⇔ q.

Remember that p ⇔ q means that p and q have the same
truthvalue, not necessarily the same meaning.
So it may be correct to translate an English sentence into
p ⇔ q even if its components differ in meaning.

Many logicians, mathematicians, and philosophers abbreviate "if
and only if" as "iff".
Hence "p iff q" should be translated as p ⇔ q.

Just when.
Sometimes in English we say that p is true "just when" q is true.
(Or perhaps this locution is only common among logicians and
mathematicians.)
This means that p is true when and only when q is true, or that p if and
only if q, and should be translated p ⇔ q.

Even if.
"P even if q" means "p whether or not q" or "p regardless of q".
Therefore one perfectly acceptable translation of it is simply "p".
If you want to spell out the claim of "regardlessness", then you could
write "p & (q  ~q)".
The two translations are equivalent.
(A proposition conjoined to any tautology has the same truthvalue as
the original proposition.)

Truthfunctionality.
All our operators are truthfunctional.
So if an operator in English is not truthfunctional, don't translate
it with one of our operator symbols.
If the English operator has multiple meanings, one truthfunctional and
others not, then only translate it with one of our operator symbols if
you want the truthfunctional core of meaning and are willing to discard
the rest.
Two examples:

"And" in English sometimes expresses temporal succession, not just
conjunction.
"She cursed like a sailor and hung up."
Here the "and" should be translated as & because it does
express conjunction; but our translation will no longer make it
clear which act was performed first.
One function of the "and" in English is to do so, but that function
is not truthfunctional and cannot be captured by our operators.
"And" sometimes also functions as a slovenly substitute for the
infinitive, as in "Try and make me."
Here the meaning is even further from truthfunctional conjunction.

Conditional statements in English (if..., then...) often express
causation or definition rather than implication.
For example, we use "if..., then..." to express a definition when
we say, "if it's ice, then it's frozen", and to express causation
when we say, "if we put ice in boiling water, then it will melt".
But the implies operator expresses neither definition nor causation,
only implication.
Moreover, not all senses of logical implication in English are
truthfunctional.
In particular, implications in which the antecedent and consequent
must be "relevant" to one another are not truthfunctional.
The horseshoe only expresses truthfunctional implication.
(For more on what the horseshoe operator expresses, see
Tips 810, above.)

Punctuation.
Parentheses and brackets are the punctuation marks of our logical
language.
With a few exceptions, whenever you have two or more operators in a
single compound expression, you will need parentheses in order to prevent
ambiguity.
For example, p ⇒ q  r is ambiguous.
It could mean (p ⇒ q)  r, or p ⇒ (q  r), which are
not equivalent.

Negation is the only one of our operators with a predefined scope:
namely, the first meaningful proposition (simple or compound) to its
right. Hence ~p ⇒ q is not ambiguous; it
means (~p) ⇒ q, not ~(p ⇒ q).
Similarly, ~(p ⇔ q) ⇒ r is not ambiguous; it means
(p ⊕ q) ⇒ r, not
((~p) ⇔ q) ⇒ r, nor
~((p ⇔ q) ⇒ r), nor any of several other
variations.

Disjunction is both commutative and associative.
So (p  q)  r is equivalent to (q  p)  r and
p  (q  r).
As a result, omitting parentheses from disjunctive strings does not
create ambiguity.
One may write p  q  r.
The same is true of conjunctive strings.
(However, strings that mix disjunctions and conjunctions
require parentheses to prevent ambiguity.)

We could prevent this kind of ambiguity if we stipulated a hierarchy
of operators in which higher operators override lower ones.
In computer programming languages this is called "operator
precedence".
There are customary rules of operator precedence in algebra, but
none in logical notation.
Hence, we must use parentheses.

Unnecessary parentheses are inelegant but not illogical.
When you mean p, you could write (p) or
((p)) without affecting the truthvalue of the statement
or the validity of any argument in which the statement occurred.
The same is true with p ⇒ q, (p ⇒ q),
((p) ⇒ (q)), and so on.

Omitting necessary parentheses is a common mistake in translation.
Make one last check before you think your translation is finished.

Specific form.
Any list of three English sentences could be translated by three
proposition symbols: p, q, r.
But if the original sentences had some common structure, showing perhaps
that one of them followed deductively from the other two, then this
translation would obscure that fact.
So we follow the rule to reveal as much structure as our notation permits.
In propositional logic, this means never to use simple variables to
translate compound sentences.
Our translations will give us a formal representation of the original
English whether we follow this rule or not, but only when we follow this
rule will our translation give us the specific form of the
original.
In propositional logic, a translation yields the specific form of the
original when we can restore the original by substituting simple
statements for each distinct propositional variable in the translation.
If we must substitute compound statements for symbols to get the original,
then we know we have left some structure untranslated.
Sometimes an argument does not use all the structure we can express with
our notation.
In those cases, we could omit some in our translation and still capture
all that on which validity depends.
For this sort of exception to the rule, see Tips 49
and 50 below.

Implicit and explicit relations.
Many propositions assert relations implicitly that sometimes must be
made explicit in translation: "Martha's paper had been graded" tacitly
asserts that "Martha's paper had been graded by the underpaid
teacher."
We could translate the sentence
monadically
 graded(X) = the teacher graded Martha's X

dyadically
 graded(X,Y) = X graded Martha's Y

triadically
 graded(X,Y,Z) = X graded Y's Z

In the first case, we pack both the teacher and Martha into the
graded predicate.
In the second case we pack just Martha into it.
In the third case, we pack neither of these people into the
graded predicate, and use it only to denote the act of grading,
which involves three individual objects (teacher, student, assignment).
How much we make explicit with our notation, and how much we pack away,
is entirely up to us.
For the sake of proofs, we should make explicit all the structures on
which validity depends (Tip 50), but no more than
that (Tip 49).

Order of quantifiers.
If an expression with all its quantifiers stacked at the left side
(e.g. in prenex normal form, Tip 43), contains
more than one quantifier, then does it matter in what order they appear?
If it does matter, which quantifier goes first and when?

When the quantifiers are of the same type, then their order does
not matter.
For example:
 ∀X ∀Y (a(X,Y) ⇒ b(X,Y))
⇔
∀Y ∀X (a(X,Y) ⇒ b(X,Y))
 ∃X ∃Y (a(X,Y) ⇒ b(X,Y))
⇔
∃Y ∃X (a(X,Y) ⇒ b(X,Y))
But when the quantifiers are of different types, then their order
does matter.
We will soon see the rule telling us which order to use.
But first a note on what this rule corresponds to in English:

Grammatical interlude.
In English, "John loves Mary" and "Mary is loved by John" express
the same proposition.
One is in the active voice, the other in the passive.
In English we use voice and word order as clues to determine who
is the lover and who is the beloved  or, in a manyplace predicate,
which individual occupies which position in the relation.
We need a comparable clue in our formal language.

When quantifiers are of different types, their order matters.
Follow this rule: when order matters, the first quantifier
quantifies the subject of the sentence; the others quantify the
objects of the verb.
For example, let our universe of discourse be human beings:
 ∀X ∃Y loves(X,Y) =
Everyone loves someone (not necessarily the same person)
 ∃Y ∀X loves(X,Y) =
Somebody (a particular person) is loved by everyone
 ∀X ∃Y loves(Y,X) =
Everyone is loved by someone (not necessarily the same person)
 ∃Y ∀X loves(Y,X) =
Somebody (a particular person) loves everyone
For the sake of further analysis, take the second example above,
∃Y ∀X loves(X,Y) =
Somebody (a particular person) is loved by everyone.
By putting the Y quantifier first, we are making the
individual in the Y position  the beloved, not the lover
 the subject of the sentence.
We now have a choice.
We can make the beloved the subject of the sentence, which requires
the passive voice, "Someone is loved by everyone."
Or we can shun the passive voice by reassigning the beloved to the
object position, "Everyone loves the same person."
So we can say that the Y individual is only the subject of
the sentence if we are willing to use the passive voice.

Here are 10 examples.
1
 Everything attracts everything.
 ∀X ∀Y attracts(X,Y)

2
 Everything is attracted by everything.
 ∀Y ∀X attracts(X,Y)

3
 Something attracts something.
 ∃X ∃Y attracts(X,Y)

4
 Something is attracted by something.
 ∃Y ∃X attracts(X,Y)

5
 Nothing attracts anything.
 ∀X ∀Y ~attracts(X,Y)

6
 Nothing is attracted by anything.
 ∀Y ∀X ~attracts(X,Y)

7
 Everything attracts something.
 ∀X ∃Y attracts(X,Y)

8
 Something is attracted by everything.
 ∃Y ∀X attracts(X,Y)

9
 Everything is attracted by something.
 ∀Y ∃X attracts(X,Y)

10
 Something attracts everything.
 ∃X ∀Y attracts(X,Y)

Note that pairs 12, 34, and 56 are equivalent.
Here are some other members of the same family.
What do they mean?
 ∃X ∃Y ~attracts(X,Y)
 ∀X ∃Y ~attracts(X,Y)
 ∃Y ∀X~attracts(X,Y)
 ∀Y ∃X ~attracts(X,Y)
 ∃X ∀Y attracts(X,Y)

Something.
"Something" and similar words like "somebody", "sometime", and
"somewhere" are ambiguous.
In the sentence, "Everything attracts something", the word "something"
is ambiguous.
It could mean (1) something in particular, or (2) something or other.
For later reference, let us say that the former is the definite
and the latter the indefinite sense of the word "something".
(Logicians have no terms for this important distinction, so I've had to
introduce some.)
If we read "Everything attracts something" in its definite sense, it
asserts that everything attracts one and the same attractee; if we read
it in the indefinite sense, it asserts that everything attracts at least
one attractee, not necessarily the same one.
These are clearly not equivalent and must be translated differently, but
how?
From Tip 53 we learned the rule that order matters
for quantifiers of different types, and that the first is regarded as
quantifying the subject of the sentence.
We will use that rule here, but it is not enough.
We must add a new rule: that when an existential quantifier comes first,
or applies to the subject of a sentence, it is to be understood in the
definite sense, and when it does not come first, or applies to an object
of the verb, then it is to be understood in the indefinite sense.
With this understanding, then our sentence in its definite sense would
be translated, ∃Y ∀X attracts(X,Y).
This says that something in particular is attracted by everything, which
is the same as saying that everything attracts some particular thing.
In its indefinite sense, it would be,
∀X ∃X attracts(X,Y).
This says that everything attracts something or other, which is the
same as saying that something or other is attracted by everything.

Another example to clarify the rule.
"Everyone is offended by something".
In its definite sense it means that somewhere there is a universally
offensive object.
We might be wise to find it and destroy it immediately.
But in its indefinite sense it means that everyone is offended by
something or other, not necessarily by the same thing, perhaps each
by a different thing.
What offends you might not offend me, and might even be precious to
me.
The policy of destroying or prohibiting what offends you might be
one of the things that offends me.
So no obvious policy for eliminating offense suggests itself.
On the contrary, if my offended sensibilities count exactly as much
as yours, and vice versa, then we might have to live with a little
offense as a natural sideeffect of differing from one another in
our sensibilities.
The sentence in its definite sense (the former), should be
translated, ∃X ∀Y offends(X,Y) (something in
particular offends everyone).
In its indefinite sense (the latter) it should be translated,
∀Y ∃X offends(X,Y) (everyone is offended by
something or other).

Although I've said "something in particular, some one thing" for
simplicity and clarity, the definite sense of "something" could
signify a plurality of particular things.
So ∃Y ∀X attracts(X,Y) could mean that some
particular set of things attracts everything.
Similarly, the indefinite sense of "something" could signify a
plurality of things.
So ∀X ∃X attracts(X,Y) could mean that
everything attracts some plural number of things, but not
necessarily the same plurality or set of things.

For other examples, see Tip 53.iv.
For a related ambiguity in the word "some", see
Tip 20, above.

Temporal relations.
The word "sometimes" can refer to the existence of some times at which
a predicate is true of certain objects.
"Sometimes John loves Mary" would be translated
∃X (time(X) & loves(john,mary,X))  there is an X
such that X is a time and John loves Mary at that time.
"Once upon a time John loved Mary" would be translated the same way.
"John will always love Mary," would be translated
∀X (time(X) ⇒ loves(john,mary,X)).
"John never loved Mary":
∀X (time(X) ⇒ ~loves(john,mary,X)).

Spatial relations.
The word "somewhere" can refer to places where a predicate is true of
certain objects.
"Somewhere a wild boar is enjoying sunlight" would be translated
∃X ∃Y ∃Z (place(X) & boar(Y) & sunlight(Z) & enjoying(X,Y,Z))
 there is a place X, a boar Y, and some sunlight
Z such that that boar enjoys that sunlight at that place.

Numerical expressions.
We can go beyond the crude quantities of all, some,
and none to express more precisely how many things have a
certain property.
We can express the natural numbers as adjectives ("three blind mice"),
if not as nouns ("one, two, three").

Zero.
We can already express zero: zero things are human, nothing is
human, all things are not human:
∀X ~human(X), or ~∃X human(X).

Introducing identity.
To express natural numbers in predicate notation, we must introduce
a predicate to express identity, that is, to do the work that
the "=" symbol does in arithmetic.

Adjectival natural numbers.
Exactly one thing is human:
∃X ∀Y (human(X) & (human(Y) ⇒ (X = Y))):
there is at least one thing, X, such that X is
human, and if there is anything else, Y, that is human,
then it is the same as the first.
Exactly two things are human:
∃X ∃Y ∀Z (((human(X) & human(Y) & X != Y & (human(Z) ⇒ ((Z = X)  (Z = Y)))):
there is at least one thing, X, that is human, and another
thing, Y, that is human, and X is not the same as
Y, and if there is another thing, Z, that is
human, then it is either the same as X or the same as
Y.
And so on for three, four, five...things.

At least.
At least one thing is human: ∃X human(X).
This is the unadorned existential quantifier.
At least two things are human:
∃X ∃Y ((human(X) & human(Y)) & (X != Y)).
There is a human X, and a human Y, and X
is not the same thing as Y.

At most.
At most one thing is human:
∀X ∀Y (human(X) ⇒ (human(Y) ⇒ (X = Y))).
For all X and for all Y, if X is human,
then if Y is human too, then X is the same thing
as Y.
By avoiding the existential quantifier in this expression, we are
noncommittal on the question whether there are any humans.
At most two things are human:
∀X ∀Y ∀Z (((human(X) & human(Y)) & (X != Y)) => (human(Z) => ((Z = X)  (Z = Y))))

"Only (name)" and "All but (name)" expressions.
Only Socrates is human:
human(socrates) & ∀X ((X != socrates) ⇒ ~human(X)).
Socrates is human, and for all things X, if X is
not Socrates, then X is not human.
Only Socrates and Plato are human:
human(socrates) & human(plato) & ∀X (((X != socrates) & (X != plato)) ⇒ ~human(X)).
All but Socrates are human:
~human(socrates) & ∀X ((X != socrates) ⇒ human(X)).
Socrates is not human, and for all things X, if X
is not Socrates, then X is human.
All but Socrates and Plato are human:
~human(socrates) & ~human(plato) & ∀X (((X != socrates) & (X != plato)) ⇒ human(X)).