Introduction to Logic
Week Sixteen: 14 Jan., 2008
Subjects, Predicates and The Universal Quantifier
000. Office Hours this term: Tuesdays and Wednesdays 1-2, or by appointment; 5B-120.
00. Results from the Second Marked Homework:
Statistical Observations:
Average Score of Students who attended at least 8 out of 9 Autumn Term Lectures: 83.8
Average Number of Autumn Term Lectures attended by students who scored less than 50: 4
0. Reminder: Tutorial Assistance is readily available for anyone who feels the need for it. Please come to
office hours to make a plan.
1. The Limitations of Propositional Logic
-- Propositional Logic cannot represent the internal structure of atomic sentences. Each atomic
sentence is simply represented by a sentence letter.
-- But the validity of some arguments crucially depends on the internal structure of its premises
and conclusion. For example,
(1) All logic students are wise.
(2) Jeff is a logic student.
Therefore,
(3) Jeff is wise.
-- This is clearly a valid argument, but note what happens if we represent it as a sequent of
Propositional Logic:
Key:
P: All logic students are wise.
Q: Jeff is a logic student.
R: Jeff is wise.
P, Q: R
-- But this sequent is obviously invalid. We need some way of representing the internal structure
of these sentences.
2. Subject and Predicate
-- We do this by distinguishing between names and predicates. Take the sentence, ‘Jeff is
wise’. The subject of this sentence is the name ‘Jeff’ and to that subject we ascribe the predicate
‘…is wise’. Roughly speaking, names stand for individuals (things) and predicates stand for their
properties. When we say ‘Jeff is wise’, we are saying that the individual Jeff has the property of
being wise.
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-- We represent this in Predicate Logic by introducing some new notation. We use lower-case
letters from the start of the alphabet (a, b, c) as names for individuals (sometimes called constants,
as their reference is held constant). And we use the upper case letters F, G, H, etc. to represent
predicates.
-- A simple subject-predicate sentence in English is thus represented as follows:
Fa
-- So we can now represent the premise “Jeff is a logic student” as follows:
Key:
F: … is a logic student
a: Jeff
Fa
Notes:
(i) Note that in Predicate Logic the predicate letter is written first, followed immediately by the
constant – a reversal of the usual order of an English subject-predicate sentence.
(ii) Notice the occurrence of the ellipsis marks (…) in the specification of the meaning of a
predicate. This serves to indicate that the predicate only expresses a compete proposition (a
complete thought) when completed by filling in a name in place of the ellipsis mark.
3. First Translation Exercises using Predicate Logic
Part I: Translate the following sentences into well-formed-formulae (wffs) of the propositional (or
sentential) calculus. Recall that the Propositional Calculus is the Logical Language we studies last
term, in which upper case letters (P. Q, R. etc.) represent complete propositions. Be sure to
provide a key for your translation.
i) Jeff is wise.
ii) Jeff is a logic student.
iii) Tomassi’s textbook is blue.
iv) Tomassi’s textbook is thick.
Part II: Now provide translations of the same sentences into wffs of the Predicate Calculus.
Recall that in the Predicate Calculus, lower case letters (a, b, c, etc.) represent names, while upper
case letters (F, G, H, etc.) represent predicates. Be sure to provide a key for your translation.
4. Compound Formulae of Predicate Logic
In moving from the Propositional (or Sentential) Calculus to the Predicate Calculus, we are
learning a more powerful logical language. But we are not really leaving anything behind. For
all of the logical connectives of the Sentential Calculus also apply in the Predicate Calculus. The
formulae we have considered are formed by simple combination of a single Predicate symbol and
a single constant, but we can take these simple wffs of the Predicate Calculus and form compound
wffs using the connectives we learned last term. Examples:
Fa & Ga
~Fa
Fa & Ga means that a is both F and G; ~Fa means that a is not F. If we let a stand for Andrew,
and the predicates F and G stand for “is friendly” and “is generous”, then Fa & Ga represents the
English sentence , “Andrew is both friendly and generous.” Fa v Ga would mean that Andrew is
either friendly or generous, etc. NOTE: Notice here that the connectives have been used to
connect name-predicate complexes, not to to connect the predicates themselves. The following is
NOT a wff:
F&Ga
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5. Second Translation Exercise.
Translate the following sentences into wffs of the Predicate Calculus. Provide a key, and use
parentheses as needed to be clear about the scope of connectives.
v) Andrew is not friendly
vi) Andrew is friendly but not generous.
vii) Andrew is neither friendly nor generous.
viii) If Jeff is a logic student then Jeff is not wise.
ix) It is just not true that if Jeff is a logic student then Jeff is not wise.
6. The Logical Representation of Generality: Variables and Quantifiers
-- We are now in a position to represent two of the premises in our initial inference. We let the
constant a stand for Jeff, and we can introduce the predicate symbols F and G to represent the
predicates “is a logic student” and “is wise” respectively. So the first premise in that argument is
Fa and the conclusion is Ga.
-- But the sentence ‘All logic students are wise’ presents us with a new challenge. It is not a
singular sentence but a general one; it makes a claim not about one single named individuals but
about all logic students. We thus cannot represent this sentence as ‘Fa’ or ‘Gb’.
-- Again we need some new notation. A variable is a lower-case letter from the end of the alphabet
(x, y, z) which does not stand for any particular individual. Each simply means ‘thing’.
-- By itself a variable is not complete, it needs to be bound by a quantifier.
7. What is a Quantifier? A quantifier is a term which is used to answer a question of the form “How
many?” How many British players remain in the 2008 Australian Open Tennis Tournament? Yesterday
the answer was: “One”; today the answer is “None.” “One” and “None” are thus quantifiers, or
quantificational terms.
8. Exercise: Make a list of quantificational terms in English.
Hint: Think of different ways in which you might answer a question like the following: “How
many members of George Bush’s original cabinet had been directors of corporations in the US Oil
Industry?”
9. Quantifiers in Logic: Logical Languages do not have nearly as many quantifiers as a natural language
like English. As with connectives, the logician typically tries to do with a minimum number of strictly
definable basic quantifiers. More complex quantifiers can then be constructed out of these basic
quantifiers. In the logical system we shall be learning this term, we will make do with two basic
quantifiers, known as the universal quantifier and the existential quantifier. For today we can just focus
on the first of these two.
10. The Universal Quantifier: We will represent the Universal Quantifier with an inverted upper case A:
∀
You can think of the Universal Quantifier as meaning ‘every’ or ‘all’. Quantifiers range over a domain of
individuals (usually taken to be all things in the universe – but may be restricted).
-- The Universal Quantifier combines with a variable as follows: ∀x […]. The square brackets represent the
scope of the quantifier (also known as its matrix) and must be filled with something in order to form a wellformed formula. One might fill it with a simple combination of a predicate letter and a variable. One then
obtains:
∀x [Fx]
--Note that in a well-formed formula, the universal quantifier always appears with a variable and with a
completed matrix.
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11. Interpreting the Universal Quantifier: In thinking about what the universal quantifier means, it will
be useful to proceed in stages. First we need to learn simply to pronounce a universally quantified formula
in the language of logic. We can then think about how to translate such a formula into idiomatic English.
Here are some examples:
Formula: ∀x [Fx]
Pronounciation:
Alternate:
English equivalent:
For any x, x is F.
For every x, x is F.
Everything is F.
Formula: ∀x [~Fx]
Pronounciation:
Alternate:
English equivalent:
For any x, it is not the case that x is F.
For every x, it is not the case that x is F.
Nothing is F.
Suppose we let F stand for the predicate: … was created by God. In that case, the first formula above
would say that everything was created by God. The second formula would say that nothing was created by
God.
12. Be careful about the scope! As always, one must pay close attention to the scope of quantifiers and
the logical connectives with which they occur. Consider the following two logical formula.
∀x [~Fx]
∼∀x [Fx]
Notice that the scope of the negation differs between the two formula. How does this alter the meaning of
two formula? To find out complete the following exercise:
13. Exercise: Fill in the answers:
∀x [~Fx]
Pronounciation:
English Equivalent
∼∀x [Fx]
Pronounciation:
English Equivalent
Now provide a translation of each of the two sentences into English, letting F stand for the predicate “was
created by God.”
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14. More translation Exercises: Work together with a partner to translate the following formula into
idiomatic English sentences, using the following key:
{Domain of Discourse: Everything}
F: … was created by God
G: … is good.
∀x[Gx]
∀x[Fx v ~Fx]
∀x[Gx] v ~∀x[Gx]
∀x[Fx & Gx]
∀x[Fx] → ∀x[Gx]
~(∀x[Fx] → ∀x[Gx])
∀x[~Gx] → ~∀x[Fx]
15. Completing our original exercise: We are now in a position to translate the final premise of the
argument we were considering at the outset. We wanted to translate the sentence ‘All logic students are
wise’. To see how we can do this, consider that we can render this sentence, ‘For any thing, if it is a logic
student then it is wise’. Hence we get:
∀x [Fx → Gx]
-- We can now represent the original argument as follows:
1) ∀x
[Fx → Gx]
2) Fa
∴ 3) Ga
To represent the argument as a sequent in the predicate calculus we follow the same conventions we used
in the Propositional Calculus, separating premises by commas and marking the conclusion with a colon:
∀x
[Fx → Gx], Fa: Ga
16. Final Translation Exericse: Now let’s try translation in the opposite direction. Translate the
following English sentences into the Predicate Calculus
x) Every living thing consumes energy.
xi) If Tom is funny then everyone is happy.
xii) No bachelor in married.
Homework for the Next Session
Reading: Tomassi, P. Logic, Ch.5, §§I, III-IV.
Exercises: Tomassi, Exercises 5.1, 5.3
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Solutions to Christmas Puzzles:
(1) Three people check into a hotel. They pay $30 to the manager and go to their room. The
manager finds out that the room rate is $25 and gives $5 to the bellboy to return. On the way to the
room the bellboy reasons that $5 would be difficult to share among three people so he pockets $2
and gives $1 to each person. Now each person paid $10 and got back $1. So they paid $9 each,
totaling $27. The bellboy has $2, totaling $29. Where is the remaining dollar?
Solution: Each person paid $9, totaling $27. The manager has $25 and the bellboy $2.
The bellboy's $2 should be added to the manager's $25 or subtracted from the tenants'
$27, not added to the tenants' $27.
(2) Father Christmas tells his two elves to race their reindeer to a distant city to see who will get
all the presents. The one whose reindeer is slower will win. The elves, after wandering aimlessly
for days, ask a wise man for advise. After hearing the advice they jump on the reindeer and race as
fast as they can to the city. What does the wise man say?
Solution: The wise man tells them to switch camels.
(3) Three logicians, A, B, and C, are wearing hats, which they know are either black or white but
not all white. A can see the hats of B and C; B can see the hats of A and C; C is blind. Each is
asked in turn if they know the color of their own hat. The answers are: A: "No." B: "No." C:
"Yes." What color is C's hat and how does she know?
Solution: A must see at least one black hat, or she would know that her hat is black since
they are not all white. B also must see at least one black hat, and further, that hat had to
be on C, otherwise she would know that her hat was black (since she knows A saw at
least one black hat). So C knows that her hat is black, without even seeing the others'
hats.
(4) Two men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads
to Nowheresville. One of these people always answers the truth to any yes/no question which is
asked of him. The other always lies when asked any yes/no question. By asking one yes/no
question, can you determine the road to Someplaceorother?
Solution: The fact that there are two is a red herring - you only need one of either type.
You ask him the following question: "If I were to ask you if the left fork leads to
Someplaceorother, would you say 'yes'?". If the person asked is a truthteller, he will
answer "yes" if the left fork leads to Someplaceorother, and "no" otherwise. But so will
the liar. So, either way, go left if the answer is "yes", and right otherwise.
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