C HAPTER 3
T HE L OGIC OF S ENTENCES
Consider the following two arguments:
(a) If every event is causally determined, thman action is free. Every event is causally determined.
Therefore, no human action is free.
(b) If humans are aggressive by nature, then a strong government is needed to protect humansfrom
themselves. Humans are aggressive by nature. Ther, a strong government is needed to protect humans
The premises of both (a) and (b) are philosophically controversial, and most of us would admit that we do not
know whether they are true. Nevertheless, we can tell that they are valid arguments, that ifttheiran see thia
premises are true, then their conclusions must be true aswell. In fact, we could see this even if we did not really
understand what those sentences said. The reason for this is that validity is a formal property: it is a characteristic
of the form of an argument, as opposed to its content or subject-matter.
These two arguments have a commonlogical formorstructure, which we can represent in the following manner:
(c) If F, then G.
Notice that (c) does not count as an argument in the sense of our definition in section 1.1, above: an argument
consists of a set of sentences, and the expression ‘If F, then G’ is not a sentence. We can, however, obtain
argument from (c) by replacing the circle and squarein (c) with grammatically admissible English sentences. We
get (a) from this argument form by replacing theti the sentence ‘Every event is causally determined’ and
the square with the sentence ‘No human action is free’; we obtain argument (b) by replacing the circle with the
sentence ‘Humans are aggressive by nature’ and the squa re with ‘A strong government is needed to protect
humans from themselves’. While these arguments differ in content, they have the same logical structure.
Any argument having the form of (c) is a valid argument. What makes (c) a valid argument form is the fact
that it istruth-preserving: it isimpossible for any English argument having this form to have true premises and a false
conclusion. It is easy to construct:
instances of (c) having true premises and a true conclusion—for example, replace the circle with ‘Toronto
is west of Montreal’ and the square with ‘Toronto is west of Halifax’;
instances of (c) having false premises andonclusion—for example, replace the circle with ‘Montreal
is west of Toronto’ and the square with ‘Montreal is west of Halifax);
and instances of (c) having false premises afalse conclusion—for example, replace the circlewith
‘Montreal is west of Toronto’ and the square with ‘Montreal is west of Ottawa’.
But it is logically impossible to construct an instance of(c) which has true premises to a false conclusion. Hence,
we know that any argument having the form of (c ) is valid.
Sentences such as ‘It is not the case that Mars is theplanet closest to the Sun’, ‘If Winnipeg is the capital of
Manitoba, then it is east of Regina’, and ‘Galileo wasborn in the sixteenth century, and Newton was born in the
seventeenth century’ are called compound sentences. The constituent sentences are called component sentences.
Expressions such as ‘it is not the case that’, ‘and’, and ‘if ... , then’ are called sentential connectives.
There are, in fact, a great many such connectives iE nnglish. But a large class of important arguments depend
for their validity on the logic of a rather small numberof such connective expressions. The branch of logic that
studies such arguments is calledsentential logic, that is, the logic of sentences. We shall now consider a number of
argument forms that involve such connectives. Some of theesargument form are very simple, but, as we shall see,
a number of simple arguments can be combined to fashion a complex chain of reasoning.
3.1 Negations, Conjunctions, and Disjunctions
The negation of a sentence is a sentence formed by prefixing the phrase ‘it is not the case that’ to the original
sentence. Thus, for example, the negation of
(1) Socrates is a philosopher
(2) It is not the case that Socrates is a philosopher.
There are various ways of expressing the negation of agiven sentence in English. For example, each of the
following sentences says that it is not the case that Socrates is a philosopher:
Socrates is not a philosopher.
Socrates is a non-philosopher.
It is false that Socrates is a philosopher.
It isn’t true that Socrates is a philosopher.
It is by no means the case that Socrates is a philosopher.
We consider these sentences to bestylistic variants of (2) and take them as alternate ways of forming the negation
of (1) in English.
We often form the negation of a sentence by inserting ‘not’ or the like into the sentence to be negated, but
we sometimes use such prefixes as ‘un-’, ‘dis-’, ‘a-’, or ‘i’-to the same effect. For example, ‘The story was untrue’
expresses the negation of ‘The story was true’; similarly,‘The witnesses disagreed’ expresses the negation of ‘The
witnesses agreed’, ‘Alice is atypical’ expresses the negationof ‘Alice is typical’, and ‘It is impossible for Brian to
pass’ is a way of expresses the negation of ‘It is possible for Brian to pass’.
It would be a mistake, however, to suppose that every occurrence of the word ‘not’, or of one of these
prefixes, marks the negation of the sentence in which the ‘not’, or the prefix, occurs. The negation of a given
sentence is contradictory to it; that is, it denies exactly what the given sentence asserts (no less, no more). In other
words, the negation of a sentence is true if and only if the sentence that is negated is false. The use of ‘not’ or the
like does not always yield a sentence contradictory tothe original. For example, ‘Some people are at home’ and
‘Some people are not at home’ are not contradictories, s nice both may be true. Hence, the sentence ‘Some people
are not at home’ isnot the negation of ‘Some people are at home’.The negation of ‘Some people are at home’ is
‘It is not the case that some people are at home’, that is,
No people are at home
And the negation of ‘Some people are not at home’ is ‘Itis not the case that some people are not at home’, that
All people are at home.
For similar reasons, ‘Robin is unhappy’ does not express the negation of ‘Robin is happy’: both might be
false. The sentences ‘Robin is unhappy’ and ‘Robin is happy’ are contraries, not contradictories. The reason that
they are not contradictories is that happiness is a spectrum, so that Robin might be neither happy nor unhappy.
Something similar is true of belief. ‘Robin believes at there is life elsewhere in the universe’ and ‘Robin
disbelieves that there is life elsewhere in the univeeontraries rather than contradictories: she may withhold
belief, that is, neither believe nor disbelieve this claim.
The negation of ‘Robin is at home and it is raining’ is not ‘Robin is not at home and it is raining’, for both
of those sentences—that is, both ‘Robin is at home and iraining’ and ‘Robin is not at home and it is raining—
would be false if it is not raining. (Remember that a sentence and its negation cannot both be false.) Nor is the
negation of that sentence ‘Robin is not at home and it notis raining’, for again both of those sentences could be
false: suppose that Robin is not at home but it is raining. The negation of ‘Robin is at home and it is raining’ is
simply ‘It is not the case that both Robin is at home and itis raining’, which, as we shall see below, is equivalent
to ‘Either Robin is not at home or it is not raining’.
An important logical principle associated with negation islaw of double negation, which says that a sentence
is equivalent to the negation of its negation. Abbreviatingthe phrase ‘it is not the case that’ simply as ‘not-’, we
can represent the law of double negation schematically as follows:
Double negation (DN)
F ‰ Not-not-F
Principles of logical equivalence areinvertible, that is, each side implies the other; we will use the symbol‰‘’ in our
schematic representations of those principles to indicate this. According to the law of double negation, the
following pairs of sentences are equivalent:
The switch is on.
It is not true that the switch isn’t on.
It is not the case that it is raining.
It is not the case that it isn’t true that it isn’t raining.
A conjunction is a compound sentence consisting of two simpler sentences linked by the word ‘and’; for
(3) Jane Austen was an English novelist and David Hume was a Scottish philosopher.
The constituent sentences of a conjunction are calledconjuncts. A conjunction is symmetrical in the sense that the
order of the conjuncts is immaterial to the truth-value of the conjunction. In other words, (3) is equivalent to
David Hume was a Scottish philosopher and Jane Austen was an English novelist.
In English, words such as ‘although’, ‘however’, ‘butn‘yet’ do much the same work as ‘and’; for example,
the following sentences say the same thing:
Alfred is poor, but he is honest.
Alfred is poor, yet he is honest.
Although Alfred is poor, he is honest. 24
Alfred is poor; moreover, he is honest.
Alfred is poor; however, he is honest.
The phrase ‘both . . . and’ is a coordinate phrase thatmakes grouping clearer. A sentence such as ‘It is not
the case that the seal is broken and servicing has been preformed’ is ambiguous. It could be taken as the negation
of a conjunction: ‘It is not the case that both the seal is broken and servicing has been performed’. Or it could
be taken as a conjunction whose left conjunct is a ne gation: ‘The seal is not broken and servicing has been
A conjunction is true just in case both of its conjun cts are true; This gives us two absurdly simple but
perfectly valid forms of argument involving conjunctions:
ˆ BothF and G.
The following argument is an instance of adjunction:
The accused is guilty
There are witnesses
ˆ The accused is guilty and there are witnesses
Both F and G. Both F and G.
ˆ F ˆ G
The following argument is an instance of simplification:
The accused is guilty and there are witnesses.
ˆ The accused is guilty
Combining conjunction with negation, we get a slightlymore complicated form of argument, which also comes
in two forms:
Conjunctive syllogism (CS)
Not-bothF and G. Not-bothF and G.
ˆ Not-G ˆ Not-F
The following arguments are instances of conjunctive syllogism:
It is not the case that both Sam and Sue will attend the party.
Sue will attend the party.
ˆ Sam will not attend the party. 25
It is not the case that both Sam and Sue will attend the party.
Sam will attend the party.
ˆ Sue will not attend the party.
Though the following form of argument may appear similar to conjunctive syllogism, it is not a valid form
Invalid conjunctive argument
Not-bothF and G.
We can see that the foregoing argument form is invalid by means of the following counterexample:
It is not the case that both Montreal and Winnipeg are cities in Ontario.
Montreal is not a city in Ontario.
ˆ Winnipeg is a city in Ontario.
Since the premises are true and the conclusion false, onecan see that an argument having this form may not be
A disjunction is a compound sentence consisting of two simpler sentences linked by the word ‘or’; for
(4) The butler was framed or the maid was his accomplice.
The component sentences of a disjunction are calleddisjuncts. Like conjunction, disjunction is symmetrical in the
sense that the order of the disjuncts is immaterial to the truth-value of the disjunction.
The phrase ‘either ... or ...’ is interchange. he phrase ‘either . . . or’ is a coordinate phrase—like
‘both . . . and’—that makes grouping clearer. For example,
It is not the case that either the seal is broken or servicing has been performed.
is the negation of a disjunction, while
Either it is not the case that the seal is broken or servicing has been performed
is a disjunction whose left disjunct is a negation.
A disjunction is true just in case at least s disjuncts is true. This gives us the following basic
arguments involving disjunction:
Disjunctive addition (Add)
ˆ EitherF or G. ˆ EitheF orG . 26
Since a disjunction is true if at least one of its disjuncts istrue, if we are given that one of the disjuncts is true, we
may infer that the disjunction is true—regardless of the truth value of the other disjunct. Hence, any argumetn
having the form of disjunctive addition is valid: for example,
Eleanor will visit Australia this summer.
ˆ Either Eleanor will visit Australia this summer or she will visit Brazil next winter.
Disjunctive syllogism (DS)
Either F or G. Either F or G.
ˆ G ˆ F
Again, since a disjunction is true if at least one of its disjuncts is true, if we are given that one of the disjuncts is
false, we may infer that the other is true. Thus, arguments having the form of disjunctive syllogism are valid:
Either there are witnesses or the accused is innocent.
It is not the case that there are witnesses.
ˆ The accused is innocent.
It is sometimes claimed that the English word ‘or’ has two sensesinclusive sense and anexclusive sense. An
inclusive disjunction is true when either or both disjuncts are true; anexclusive disjunction is true when just one of
the disjuncts is true. There is no doubt that the word ‘or’ is often used in this inclusive sense. For example, if a
teacher says to her class, “You’ll do well if and only if you are smart or work hard”, she doesn’t mean that a
student who is smart and works hard will not do well. What is not so clear is whether ‘or’ is ever used in an
exclusive sense. No doubt, there are disjunctionshose disjuncts are incompatible so that both disjuncts cannot
be true (for example, ‘Either today is Mondayor today is Tuesday’), but the fact thadisjuncts are incompatible
does not show that the word ‘or’ is being used exclusively.
We need not settle this debate here. We shall, however, adopt the convention of always taking ‘or’ in the
inclusive sense. Thus, we shall always understand that a sentence of the form ‘EitherF or G’ is true if and only
if at least one disjunct is true. It follows that a sentence of this form is false just when both disjuncts are false.
We can, however, express the exclusive sense of disjunctionwhen we think it appropriate by explicitly adding the
phrase ‘but not both’. Thus, the sentence ‘We will go to the movies or to the beach, but not both’ says that we
will go to exactly one of those two places; thus it is false if both disjuncts are true.
Since we are reading ‘or’ in the inclusive sense, the following form of argument is invalid:
Invalid disjunctive argument
(F or G)
The following counterexample shows that this argument form is invalid (when ‘or’ is understood inclusively):
Either Ottawa is a city in Ontario or Toronto is a city in Ontario.
Ottawa is a city in Ontario.
ˆ Toronto is not a city in Ontario. 27
On our reading of ‘or’ both premises are true while the conclusion is false.
We often use ‘neither . . . nor’ to form the negation of a disjunction. Thus a sentence such as
Neither Abbot nor Cabot is guilty
says the same thing as
It is not the case that either Abbot or Cabot is guilty.
It turns out, however, that this last sentence is equivalent to the conjunction of a pair of negations,
Abbot is not guilty and Cabot is not guilty.
The equivalence of the last two sentences is a consequence of onDfeMorgan’s laws, which says that the negation
of a disjunction is equivalent to a conjunction of the negated disjuncts. Another of DeMorgan’s laws says that
the negation of a conjunction is equivalent to a disjunction of the negated conjuncts; in other words,
Not both Abbot and Cabot are guilty
is equivalent to
Either Abbot is not guilty or Cabot is not guilty.
This gives us two equivalences (and four valid argument forms):
DeMorgan’s laws (DeM)
Not-either F or G. ‰ Both not-F and not-G.
Not-both F and G. ‰ Either not-F or not-G.
According to DeMorgan’s laws,
It will neither snow nor rain
is equivalent to the sentence
It won’t rain and it won’t snow,
It won’t both rain and snow
is equivalent to
Either it won’t rain or it won’t snow.
It is worth noting that the negation of a disjunction—for example, ‘It will neither rain nor snow’ or ‘It won’t
either rain or snow’— implies the negation of conjunciton— ‘It won’t both rain and snow’— but not conversely: 28
The sentence ‘It won’t both rain and snow’ is true if it won’t rain (why?); but even though it won’t rain, it might
nevertheless snow, in which case the sentence ‘It won’t either rain or snow’ is false (why?).
A. For each of the following sentences, formulate a sentence that denies exactly what the sentence affirms (in
other words, a sentence that is opposite in truth value)without using the cumbersome phrase ‘it is not the case
1. Alfred is not a philosophy student.
2. Alice will go to Paris or she will get a job.
3. Betty and Charles both came to the party.
4. Neither Dave nor Emma are members of the group.
5. Ants never dance.
B. In the following examples, determine the truth-value of the compound sentences given the truth values of the
following component sentences:
(i) ‘Galileo was born before Descartes’ is true.
(ii) ‘Descartes was born in the sixteenth century’ is true.
(iii) ‘Newton was born before Shakespeare’ is false.
(iv) ‘Hume was born before Shakespeare’ is false.
(v) ‘Racine was a compatriot of Descartes’ is true.
(vi) ‘Racine was a compatriot of Galileo’ is false.
1. Galileo was not born before Descartes.
2. Galileo was born before Descartes and Descartes was born in the sixteenth century.
3. Galileo was born before Descartes or Descartes was not born in the sixteenth century.
4. Galileo was born before Descartes but Descartes was not born in the sixteenth century.
5. Either Galileo was not born before Descartes or Newton was not born before Shakespeare
6. Neither Newton nor Hume was born before Shakespeare.
7. Not both Newton and Hume were born before Shakespeare.
8. Racine was not a compatriot of both Descartes and Galileo.
9. Racine was a compatriot of neither Descartes nor Galileo.
10. Either Newton and Hume were both born before Shakespeare or Racine was a compatriot of Descartes. 29
11. Racine was a compatriot of Descartes but neither Newton and Hume were both born before Shakespeare.
C. Let us say that the result of replacing the circle ansquare in an argument form with grammatically admissible
sentences is an instance of that argument form. Thus, for example, the argument
Either Jones will win the election or Smith will win the election
Jones will not win the election.
ˆ Smith will win the election
is an instance of disjunction syllogism. Since we know that disjunctive syllogism is a valid form of argument, we
may conclude that the foregoing English argument is valid as well.
For each of the arguments below, abbreviate the Englh issentences occurring in the argument using the suggested
capital letters, indicating what sentence each letter stands for, and then set out the argument in schematic form
using those capital letters. Then say whether it is an instance of one of the argument forms or equivalences set
out in this section and whether it is valid. Question 1, below, is solved for illustration.
1. Sir Walter Scott either wrote Northanger Abbey or Ivanhoe. He didn’t write Norhanger Abbey. Therefore, he
If we let ‘N’ stand for the sentence ‘Sir Walter Scott wroe t Northanger Abbey’ and ‘I’ stand for ‘Sir Walter Scott
wrote Ivanhoe’, then we can represent argument 1 schematically as follows:
Either N or I.
This is an instance of disjunctive syllogism; hence, it is valid,
2. The won’t fail to arrive on time. So, they will arrive on time. (A)
3. This play was written by Shakespeare or by Bacon. Itwasn’t written by Shakespeare. So it must have been
written by Bacon. (S, B)
4. Toronto is larger than Montreal as well as Vancouver. So it is larger than Vancouver. (M, V)
5. The suspect wasn’t both at the victim’s house and in his lawyer’s office. He was at his lawyer’s office. So,
he wasn’t at the victim’s house. (V, L)
6. Either the contract is valid or Horatio is not liable.The contract is invalid. So, Horatio is not liable. (V, L)
7. It’s not the case that Brendan is both a physician and a lawyer. He isn’t a physician. So, he is a lawyer. (P,
8. Alice studies either maths or physics. She definitely studies maths. So she doesn’t study physics. (M, P)
9. Neither Bach nor Handel wrote that concerto. ThereforeB , ach didn’t write it and Handel didn’t write it. (B,
10. Bach didn’t write that concerto and Handel didn’t write it. Therefore, Bach didn’t write it. (B, H). 30
11 Not both John and Mary are telling the truth. So, John isn’t telling the truth and Mary isn’t. (J, M)
12. The company will not both remain profitable and fail to r iae its prices. So either it will not remain profitable
or it will raise its prices. (E, A)
13. It is not the case that some philosophers aren’t uitlitarians. So, some philosophers are utilitarians. (Choose
your own letters.)
D. Answer the following questions, justifying your answer in each case.
1. Paris has denied that she was dancing outside the Koala Klub at 3 in the morning and had her shoes off.
Assuming that Paris is telling the truth, does it follow that if Paris was dancing outside the Koala Klub at
3 in the morning, then she had her shoes on?
2. She: You love me but don’t want to marry me. Is that it?
She: Then either you don’t love me or you want to marry me.
Is she right? Explain.
3.2 Conditionals and Biconditionals
A conditional is a sentence composed of two component sentences joined by the sentential connective ‘if...,
(1) If Biscuit is a cat, then Biscuit is a carnivore.
The component sentences of a conditional have different names: the sentence introduced by the word ‘if’ is
called the antecedent of the conditional; the one that comes after ‘then’ is called the consequent. Thus, in (1), the
antecedent is ‘Biscuit is a cat’ and the consequent is ‘Biscuit is a carnivore’.
It will be useful to have a canonical form for a conditional, and we shall take the form
If F, then G.
as standard. It is important to recognize, however, that aconditional can be reformulated in a variety of different
but equivalent ways in English. Consider the sentence:
(2) If Alice is a physicist, then Alice is a scientist.
Since every physicist is a scientist, (2) is obviously true. The following rules show various ways in which (2) can
be reformulated without changing its meaning or truth value:
(i) A sentence of the form ‘IfF, then G’ may be written omitting the word ‘then’; hence, we may write (2)
If Alice is a physicist, Alice is a scientist.
(ii) The word order of a conditional can be invertedprovided that the same clause is still governed by the
phrase ‘if’; hence, we may write (2) as: 31
Alice is a scientist if Alice is a physicist.
(iii) The expression ‘if’ means the same thing as ‘provided that’, ‘assuming’, ‘in case’, and ‘given’. Thus (2)
may be written as:
Provided that Alice is a physicist, Alice is a scientist.
Assuming that Alice is a physicist, Alice is a scientist.
In case Alice is a physicist, Alice is a scientist.
Given that Alice is a physicist, Alice is a scientist.
And in light of rule (ii) above, the word order of these sentences can be inverted, as follows:
Alice is a scientist provided that Alice is a physicist,
Alice is a scientist assuming that Alice is a physicist.
Alice is a scientist in case Alice is a physicist.
Alice is a scientist given that Alice is a physicist.
(iv) A sentence of the form ‘IfF, then G’ is equivalent to one of the form ‘F only if G’; hence, we write
Alice is a physicist only if Alice is a scientist.
This, in turn, may be expressed as:
Only if Alice is a scientist is Alice a physicist .
In summary, a conditional can be rewritten a variety of equivalent ways, sometimes with the antecedent
written first and sometimes with the consequent written first. Each of the following forms are equivalent in
If F, then G.
If F, G.
G if F.
Assuming F, G.
G provided F.
F only if G.
When one encounters a conditional in an argument, it is agood idea to re-write it in our canonical, that is, in the
If F, then G.
Sometimes students find it odd to learn that a sentence of the form
F only if G
says the same thing as one of the form
If F, then G. 32
It may help to consider the matter in the following light. The antecedent of a conditional specifies a sufficient
condition for the consequent, that is, a condition such that if it obtains, then the condition specified by the
consequent obtains; and the consequent of a conditional specifies a necessary condition for the antecedent, that is,
a condition such that if it does not obtain, the condition specified by the antecedent does not obtain. Thus, for
example, in the sentence “If Biscuit is a cat, then Biscuitis a carnivore”, Biscuit’s being a cat is sufficient for her
to be a carnivore. On the other hand, Biscuit’s being a carnivore is not sufficient for her to be a cat: lots of
animals are carnivores. But Biscuit’s being a carnivore is a necessary condition for her to be a cat; that is, if she
is not a carnivore, she is not a cat.
Suppose, now, that some student is not doing very well ina certain course, and it occurs to that student that
it may be a waste of his time to write the final. He consults the instructor about his chances of passing. The
instructor says, “I don’t know whether you’ll pass, but Ican tell you this: You’ll pass the course only if you write
the final.” Is the instructor claiming that writing the final is a sufficient conditionfor passing—a condition such
that if the student does it, he will pass—or is she specifying a necessary condition—a condition such that if the
student doesn’t do it, he won’t pass. No doubt, the latt. What she is saying is that the student won’t pass without
writing the final. But in that case, what the teacher says may be paraphrased along the lines of ‘If you pass the
course, then you will have written the final’.
An important logical principle associated with conditionals is thaw of contraposition, which says that the result
of negating the antecedent and consequent of a conditio nal and switching them around is equivalent to the
If F, then G. ‰ If not-G, then not-F.
Thus, according to contraposition, the following two sentences are equivalent, and we may infer one from the
If the compound is an acid, the litmus turns red.
If the litmus does not turn red, then the compound is not an acid.
The phrase ‘unless’ means the same thing as the phrase ‘f i it is not the case that’. Accordingly, the sentence
The compound is not an acid unless the litmus turns red.
says the same thing as
The compound is not an acid if the litmus does not turn red.
In short, a conditional can be rewritten a variety of ofequivalent ways, sometimes with the antecedent written first
and sometimes with the consequent written first. For example, all of the statements below are equivalent to each
If the compound is an acid, then the litmus turns red. If F, then G
If the compound is an acid, the litmus turns red. If F, G
The litmus turns red if the compound is an acid. G if F
The litmus turns red provided that the compound is an acid. G provided F
The compound is an acid only if the litmus turns red. F only if G
Only if the litmus turns red is the compound an acid. Only if G F
The compound is not an acid unless the litmus turns red. Not-F unless G 33
Unless the litmus turns red, the compound is not an acid. Unless G not-F
We now turn to several valid argument forms involving the conditional:
Modus ponens (MP)
If F, then G.
MP tells us that if a sentence of the form ‘If F, then G’ is true and that antecedent of that conditional, that is,
F, is also true, then the consequent of the conditional, G, , must also be true. Thus, the following argument
is an instance of modus ponens:
If it is raining, then the game will be delayed.
It is raining.
ˆ The game will be delayed.
Modus tollens (MT)
If F, then G
MT tells us that if a conditional is true but its consequent is false, then its anteced