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ยฉ philosophy.org.za
Critical Reasoning 14 - The Logic
of Relations
Most of what counts for Science or indeed Humani-
ties cannot be expressed solely in terms of the
predicate calculus we have learned so far. This is
because these disciplines proceed by discovering
relations among things. A relation then, is a connec-
tion that one (or more) things or phenomena have
to others. In the case that they involve two objects,
they can be expressed by, so called, binary predi-
cates (with two subjects) that represent the relation
that subsists between them. Thus:
John loves Sarah
Even though โSarahโ is the grammatical object of
the sentence above, both โJohnโ and โSarahโ are
regarded as the subjects of the binary predicate โโฆ
lovesโฆโ. Of course some relations like love can
relate to themselves, such as:
John loves himself
Quite naturally speaking, there are several triadic or even tetradic predicates that relate three or for
subjects respectively. E.g.
George is between Cape Town and Port Elizabeth. and
Jack traded his cow to the peddler for some magic beans.
Examples of polyadic predicates involving many subjects do exist but tend to result from longer lists
of subjects rather than naturally spoken relations; although sentences such as the following are the
exception:
Jean and Murray played Diane and Ben at bridge over a bowl of popcorn.
As Copi (p. 117) observes, relations enter into arguments in various ways, such as:
Al is older than Bill. Bill is older than Charlie. Therefore, Al is older than Charlie. This is simply a relational argument in which other relational predicates could have featured, such as
โโฆis taller than..,โ โโฆis wiser than..,โ โโฆ is heavier thanโฆโ etc. Consider the following, slightly more
complex argument involving quantification:
Helen likes David. Whoever likes David likes Tom.
Gottfried Wilhelm von Leibniz, (1646 - 1716 )
More Often Remembered for his Independent
Invention of Calculus, was also a Philosopher and
Logician of Note. Seen here in a Portrait housed at
the Regional Library of Hannover.
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Helen likes only good-looking men. Therefore, Tom is a good-looking man. The following more complex relational argument employed by Copi (l.c.) meanwhile involves
multiple quantification.
All horses are animals. Therefore, the head of a horse is the head of an animal. Fortunately, no novel methods of proof, other than those already encountered in Critical Reasoning
11, are required to prove the validity of such arguments. However we must pay particular attention
to the way we symbolise relational propositions.
Symbolising Relations
So far, we have used only one-place predicates of the form ๐น๐ฅ to represent โ๐ฅ is fastโ. Now we can
expand this notation to represent predicates of the form:
๐ฅ is taller than ๐ฆ as: ๐๐ฅ๐ฆ and
๐ฅ is between ๐ฆ and ๐ง as: ๐ต๐ฅ๐ฆ๐ง.
Although other publications follow the convention of symbolising binary or dyadic as โ๐ฅ๐๐ฆโ and so
on, we shall stick with Copiโs notation, if only for consistency.
Just as we regarded different substitution instances of the same predicate โ๐น๐ฅโ, as representing
different propositions, so we may regard different substitution instances of the same polyadic
predicate e.g. โ๐ต๐ฅ๐ฆ๐งโ, as representing different propositions. Note that the order of substitution is
very important here. If the predicate โ๐ฅ taught ๐ฆโ is substituted in terms of โ๐ฅโ for โSocratesโ and
โ๐ฆโ for โPlatoโ, the proposition:
๐๐ ๐ is true,
whereas the proposition:
๐๐๐ is false.
According to the meaning of a proposition, we may be required to use either existential or universal
quantification, either singly or in combination, when symbolising it. Either way we should pay
especial heed to which variables are free and which are bound (see Critical Reasoning 11 p.6) as well
as the scope of our quantifiers. Do they extend across the whole expression or only part of it? The
following examples were compiled from lecture notes as well as Copi (p. 119 ff.) Where the meaning
of an expression may not be immediately apparent, a paraphrasal is suggested.
1. Something is taller than everything.
(โ๐ฅ)(โ๐ฆ)๐๐ฅ๐ฆ
Paraphrasal: There exists an ๐ฅ, such that for all ๐ฆ, ๐ฅ is taller than ๐ฆ.
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2. George is between Cape Town and something or other.
(โ๐ฅ)๐๐๐ฅ
3. There is something between Cape Town and Port Elizabeth.
(โ๐ฅ)๐ต๐ฅ๐๐
4. There is something, such that that George is between that thing and Port Elizabeth.
(โ๐ฅ)๐๐ฅ๐
5. Everything attracts everything.
(โ๐ฅ)( โ๐ฆ)๐ด๐ฅ๐ฆ
Paraphrasal: For any ๐ฅ and for any ๐ฆ, ๐ฅ attracts ๐ฆ.
6. Everything is attracted by everything.
(โ๐ฆ)(โ๐ฅ)๐ด๐ฅ๐ฆ
Paraphrasal: For any ๐ฆ and for any ๐ฅ, ๐ฅ attracts ๐ฆ. (The passive voice of 5, above)
7. Something attracts something.
(โ๐ฅ)(โ๐ฆ)๐ด๐ฅ๐ฆ
8. Something attracts nothing.
(โ๐ฅ)(โ๐ฆ)~๐ด๐ฅ๐ฆ
9. Alice is attracted by something.
(โ๐ฅ)๐ด๐ฅ๐
10. Nothing attracts Bert.
(โ๐ฅ)~๐ด๐ฅ๐
11. Something is attracted by itself.
(โ๐ฅ)๐ด๐ฅ๐ฅ
12. Nothing attracts itself.
(โ๐ฅ)~๐ด๐ฅ๐ฅ
Exercise: Now that we have seen how some common relations are symbolised, you may wish to try
your hand at symbolising some of the more challenging sentences provided by Copi, section 5.1 part
II (p. 128). The first ten are reproduced below.
1. Dead men tell no tales.
2. A lawyer who pleads his own case has a fool for a client.
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3. A dead lion is more dangerous than a live dog.
4. Uneasy lies the head that wears the crown.
5. If a boy tells only lies, none of them will be believed.
6. Anyone who consults a psychiatrist ought to have his head examined.
7. No one ever learns anything unless he teaches it to himself.
8. Delilah wore a ring on every finger and had a finger in every pie.
9. Any man who hates children and dogs cannot be all bad.
10. Anyone who accomplishes anything will be envied by everyone.
Solutions: The following solutions, some of which are not the same as Copiโs, were provided by
Professor Peter Suber in the form of a hand-out. The full set of 45 translations is available here. They
are reproduced below with additional paraphrasals. Note that the format adopted by Suber is known
as โprenex normal formโ. Accordingly, โall the quantifiers in a formula are stacked at the left end,
none is negated, and the scope of each extends to the end of the whole formula.โ (l.c.) This is not
the only preferred form. Indeed many logicians do not follow a consistent format, preferring to
preserve the form of natural language wherever possible.
1. (โ๐ฅ)(โ๐ฆ)[(๐ท๐ฅ โข ๐๐ฅ โข ๐๐ฆ) โ ~๐๐ฅ๐ฆ]
Paraphrasal: For all ๐ฅ and for all ๐ฆ, such that if ๐ฅ is dead and ๐ฅ is a man and ๐ฆ is a tale,
then ๐ฅ does not tell ๐ฆ.
2. (โ๐ฅ)(โ๐ฆ)[(๐ฟ๐ฅ โข ๐๐ฅ๐ฅ) โ (๐ถ๐ฆ๐ฅ โข ๐น๐ฆ)]
Paraphrasal: For all ๐ฅ and for all ๐ฆ, such that if ๐ฅ is a lawyer and ๐ฅ pleads ๐ฅโs case (i.e.
his own), then ๐ฆ has ๐ฅ as a client and ๐ฆ is a fool.
Note that the meaning of the proposition is preserved in this formulation, however we
only appreciate the wit when we realise that the lawyer and client are the same person.
This can be made explicit as follows: (โ๐ฅ)[(๐ฟ๐ฅ โข ๐๐ฅ๐ฅ) โ (๐ถ๐ฅ๐ฅ โข ๐น๐ฅ)]
3. (โ๐ฅ)(โ๐ฆ)[(~๐ด๐ฅ โข ๐ฟ๐ฅ โข ๐ด๐ฆ โข ๐ท๐ฆ) โ ๐ท๐ฅ๐ฆ]
Paraphrasal: For all ๐ฅ and for all ๐ฆ, such that if ๐ฅ is not alive and ๐ฅ is a lion and ๐ฆ is alive
and ๐ฆ is a dog, then ๐ฅ is more dangerous than ๐ฆ.
4. (โ๐ฅ)(โ๐ฆ)[(๐ป๐ฅ โข ๐ถ๐ฆ โข ๐๐ฅ๐ฆ) โ ๐๐ฅ]
Paraphrasal: For all ๐ฅ and there exists a ๐ฆ, such that if ๐ฅ is a head and ๐ฆ is a crown and ๐ฅ
wears ๐ฆ, then ๐ฅ lies uneasy.
5. (โ๐ฅ)(โ๐ฆ)(โ๐ง)(โ๐ข){[๐ต๐ฅ โข (๐๐ฅ๐ฆ โ ๐ฟ๐ฆ)] โ (๐๐ฅ๐ฆ โ ~๐ต๐ข๐ง)}
Paraphrasal: For all ๐ฅ, ๐ฆ, ๐ง and ๐ข, if ๐ฅ is a Boy and if ๐ฅ tells ๐ฆ such that ๐ฆ is a lie, then if ๐ฅ
tells ๐ฆ then ๐ข will not believe ๐ง.
6. (โ๐ฅ)(โ๐ฆ)[(๐๐ฅ โข ๐๐ฆ โข ๐ถ๐ฅ๐ฆ) โ ๐๐ฅ]
Paraphrasal: For all ๐ฅ and there exists some ๐ฆ, then if ๐ฅ is a person and ๐ฆ is a
psychiatrist and ๐ฅ consults ๐ฆ then ๐ฅ ought to have his head examined.
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7. (โ๐ฅ)(โ๐ฆ)[(๐๐ฅ โ ~๐ฟ๐ฅ๐ฆ) โ ~๐๐ฅ๐ฆ๐ฅ]
Paraphrasal: For all ๐ฅ and there exists some ๐ฆ, such that if ๐ฅ is a person and ๐ฅ did not
learn ๐ฆ, then ๐ฅ did not teach ๐ฆ to ๐ฅ (himself).
8. (โ๐ฅ)(โ๐ฆ)[(๐น๐ฅ๐ โข ๐ ๐ฆ) โ ๐๐ฆ๐ฅ] โข (โ๐ฅ)(โ๐ฆ)[(๐น๐ฅ๐ โข ๐๐ฆ) โ ๐ผ๐ฅ๐ฆ]
Paraphrasal: For all ๐ฅ and there exists some ๐ฆ, such that ๐ฅ is a finger of Delilahโs and ๐ฆ is
a ring then ๐ฆ is on ๐ฅ, and there exists some ๐ฅ and for all ๐ฆ, such that ๐ฅ is a finger of
Delilahโs and ๐ฆ is a pie, then ๐ฅ is in ๐ฆ.
Note that this is a conjunction between two quantified statements. Unlike the prenex
normal form, none of the quantifiersโ scope extends across the whole formula.
9. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ โข ๐ถ๐ฆ โข ๐ท๐ง โข ๐ป๐ฅ๐ฆ โข ๐ป๐ฅ๐ง) โ ~๐ต๐ฅ]
Note: Copi treats โ๐ฅ is all badโ as a predicate so there is no need to quantify badness.
Paraphrasal: For all ๐ฅ, ๐ฆ and ๐ง such that if ๐ฅ is a man, ๐ฆ is a child and ๐ง is a dog and ๐ฅ
hates ๐ฆ and ๐ฅ hates ๐ง, then ๐ฅ is not all bad.
10. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ โข ๐๐ง โข ๐ด๐ฅ๐ฆ) โ ๐ธ๐ง๐ฅ]
Paraphrasal: For all ๐ฅ and there exists some ๐ฆ and for all ๐ง such that if ๐ฅ is a person and
๐ง is a person and ๐ฅ accomplishes ๐ฆ, then ๐ง envies ๐ฅ.
Note that Copiโs sentence allows that any and every thing that is accomplished will
result in the accomplisher being envied by all persons, including themselves.
Arguments Involving Relations
As mentioned, no new techniques are required in dealing with arguments involving relations. The
four quantification inferences learned in Critical Reasoning 11 now take on the following forms:
Universal Instantiation (UI) Universal Generalisation (UG)
(โ๐ฅ)๐น๐ฅ ๐น๐ฅ โด ๐น๐ฅ โด (โ๐ฅ)๐น๐ฅ Existential Instantiation (EI) Existential Generalisation (EG)
(โ๐ฅ)๐น๐ฅ ๐น๐ฆ ๐น๐ฅ โด (โ๐ฆ)๐น๐ฆ โฎ ๐ โด ๐ Unlike the previous quantification rules, here we do not require that ๐ and ๐ be different variables.
Instead we can simply instantiate with respect to the same variable in a premise. Of course the other
restrictions apply, such as the correct nesting of assumptions. Similarly, where we have two
premises (โ๐ฅ)๐น๐ฅ and (โ๐ฅ)~๐น๐ฅ we begin our instantiation of one by simply dropping the quantifier,
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then we must instantiate the other in terms another letter if it is within the scope of the first
instantiation. (Copi, p. 131)
Consider, by way of example, the following proofs of validity from Copi (Ex II p. 133-4) again
provided by Professor Suber (available here.)
2. All circles are figures. Therefore all who draw circles draw figures.
1. (โ๐ฅ)[๐ถ๐ฅ โ ๐น๐ฅ] /โด (โ๐ฅ)(โ๐ฆ)[(๐ถ๐ฅ โข ๐ท๐ฆ๐ฅ) โ (๐น๐ฅ โข ๐ท๐ฆ๐ฅ)] 2. ๐ถ๐ฅ โข ๐ท๐ฆ๐ฅ 3. ๐ถ๐ฅ 2 Simp. 4. ๐ท๐ฆ๐ฅ 2 Simp. 5. ๐ถ๐ฅ โ ๐น๐ฅ 1 UI 6. ๐น๐ฅ 5,3 M.P. 7. ๐น๐ฅ โข ๐ท๐ฆ๐ฅ 6,4 Conj. 8. (๐ถ๐ฅ โข ๐ท๐ฆ๐ฅ) โ (๐น๐ฅ โข ๐ท๐ฆ๐ฅ) 2-7 CP 9. (โ๐ฆ)[(๐ถ๐ฅ โข ๐ท๐ฆ๐ฅ) โ (๐น๐ฅ โข ๐ท๐ฆ๐ฅ)] 8 UG 10. (โ๐ฅ)(โ๐ฆ)[(๐ถ๐ฅ โข ๐ท๐ฆ๐ฅ) โ (๐น๐ฅ โข ๐ท๐ฆ๐ฅ)] 9 UG
5. Whoever belongs to the Country Club is wealthier than any member of the Elks Lodge. Not
everyone who belongs to the Country Club is wealthier than anyone who does not belong.
Therefore, not everyone belongs to either to the Country Club or the Elks Lodge.
Paraphrasal: For all ๐ฅ and for all ๐ฆ such that if ๐ฅ belongs to the Country Club and ๐ฆ belongs
to the Elks Lodge then ๐ฅ is wealthier than ๐ฆ. There exists an ๐ฅ and there exists a ๐ฆ such that
๐ฅ belongs to the Country Club and ๐ฆ does not belong to the Country Club such that ๐ฅ is not
wealthier than ๐ฆ. (I.e. At least one member of the Country Club is not wealthier than at least
one non-member.) Therefore, there exists an ๐ฅ that is neither a member of the Country Club
nor a member of the Elks Lodge.
1. (โ๐ฅ)(โ๐ฆ)[(๐ถ๐ฅ โข ๐ธ๐ฆ) โ ๐๐ฅ๐ฆ] 2. (โ๐ฅ)(โ๐ฆ)(๐ถ๐ฅ โข ~๐ถ๐ฆ โข ~๐๐ฅ๐ฆ) /โด (โ๐ฅ)(~๐ถ๐ฅ โข ~๐ธ๐ฅ) 3. (โ๐ฆ)(๐ถ๐ฅ โข ~๐ถ๐ฆ โข ~๐๐ฅ๐ฆ) 4. ๐ถ๐ฅ โข ~๐ถ๐ฆ โข ~๐๐ฅ๐ฆ 5. (โ๐ฆ)[(๐ถ๐ฅ โข ๐ธ๐ฆ) โ ๐๐ฅ๐ฆ] 1 UI 6. (๐ถ๐ฅ โข ๐ธ๐ฆ) โ ๐๐ฅ๐ฆ 5 UI 7. ๐ถ๐ฅ 4 Simp. 8. ~๐ถ๐ฆ 4 Simp. 9. ~๐๐ฅ๐ฆ 4 Simp. 10. ~(๐ถ๐ฅ โข ๐ธ๐ฆ) 6,9 M.T. 11. ~๐ถ๐ฅ v ~๐ธ๐ฆ 10 De M. 12. ~๐ธ๐ฆ 11,7 D.S. 13. ~๐ถ๐ฆ โข ~๐ธ๐ฆ 8,12 Conj. 14. (โ๐ฅ)(~๐ถ๐ฅ โข ~๐ธ๐ฅ) 13 EG 15. (โ๐ฅ)(~๐ถ๐ฅ โข ~๐ธ๐ฅ) 3,4-14 EI 16. (โ๐ฅ)(~๐ถ๐ฅ โข ~๐ธ๐ฅ) 2,3-15 EI
7. Everything on my desk is a masterpiece. Anyone who writes a masterpiece is a genius.
Someone very obscure wrote some of the novels on my desk. Therefore, some very obscure
person is a genius.
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1. (โ๐ฅ)(๐ท๐ฅ โ ๐๐ฅ) 2. (โ๐ฅ)(โ๐ฆ)[(๐๐ฅ โข ๐๐ฆ โข ๐๐ฅ๐ฆ) โ ๐บ๐ฅ] 3. (โ๐ฅ)(โ๐ฆ)(๐๐ฅ โข ๐๐ฅ โข ๐๐ฆ โข ๐ท๐ฆ โข ๐๐ฅ๐ฆ) /โด (โ๐ฅ)(๐๐ฅ โข ๐๐ฅ โข ๐บ๐ฅ) 4. (โ๐ฆ)(๐๐ฅ โข ๐๐ฅ โข ๐๐ฆ โข ๐ท๐ฆ โข ๐๐ฅ๐ฆ) 5. ๐๐ฅ โข ๐๐ฅ โข ๐๐ฆ โข ๐ท๐ฆ โข ๐๐ฅ๐ฆ 6. ๐ท๐ฆ โ ๐๐ฆ 1 UI 7. ๐ท๐ฆ 5 Simp. 8. ๐๐ฆ 6,7 M.P. 9. ๐๐ฅ 5 Simp. 10. ๐๐ฅ๐ฆ 5 Simp. 11. ๐๐ฅ โข ๐๐ฆ โข ๐๐ฅ๐ฆ 9,8,10 Conj. 12. (โ๐ฆ)[(๐๐ฅ โข ๐๐ฆ โข ๐๐ฅ๐ฆ) โ ๐บ๐ฅ] 2 UI 13. (๐๐ฅ โข ๐๐ฆ โข ๐๐ฅ๐ฆ) โ ๐บ๐ฅ 12 UI 14. ๐บ๐ฅ 13,11 M.P. 15. ๐๐ฅ 5 Simp. 16. ๐๐ฅ โข ๐๐ฅ โข ๐บ๐ฅ 9,15,14 Conj. 17. (โ๐ฅ)(๐๐ฅ โข ๐๐ฅ โข ๐บ๐ฅ) 16 EG 18. (โ๐ฅ)(๐๐ฅ โข ๐๐ฅ โข ๐บ๐ฅ) 4,5-17 EI 19. (โ๐ฅ)(๐๐ฅ โข ๐๐ฅ โข ๐บ๐ฅ) 3,4-18 EI
Jon Ross has provided more proofs to this and further exercises in Copiโs text. The interested reader
can find the relevant introduction and links here or direct links to the remaining chapters as follows :
chapter 5, chapter 7, and Chapter 8.
Some Attributes of Binary or Dyadic Relations
Relations themselves have several interesting attributes, some of which may enter into proofs. For
now we shall confine ourselves to binary or dyadic relations, however there is no upper limit on the
number of attributes of polyadic relations.
1. Symmetry: A relation is symmetrical if the following is true: (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ๐ ๐ฆ๐ฅ)
Paraphrasal: If ๐ฅ has a relation ๐ to ๐ฆ, then ๐ฆ has relation ๐ to ๐ฅ.
E.g. โ๐ฅ is next to ๐ฆโ, or โ๐ฅ is married to ๐ฆโ, or โ๐ is the twin of ๐โ etc.
On the other hand, a relation is asymmetrical if the following is true (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ)
Paraphrasal: If ๐ฅ has a relation ๐ to ๐ฆ, then ๐ฆ does not have a relation ๐ to ๐ฅ.
E.g. โ๐ฅ is older than ๐ฆโ, or โ๐ฅ is North of ๐ฆโ etc.
Meanwhile, a relation is non-symmetrical if it is neither symmetrical nor asymmetrical.
E.g. โ๐ฅ is a sister of ๐ฆโ, or โ๐ฅ loves ๐ฆโ etc.
2. Transitivity: A relation is transitive if the following is true: (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ๐ ๐ฅ๐ง]
Paraphrasal: If ๐ฅ has a relation ๐ to ๐ฆ and ๐ฆ has the same relation ๐ to ๐ง, then ๐ฅ has the same
relation ๐ to ๐ง.
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E.g. โ๐ฅ is North of ๐ฆโ, or โ๐ฅ is larger than ๐ฆโ, or โ๐1 predates ๐2โ etc.
On the other hand, a relation is intransitive if: (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ~๐ ๐ฅ๐ง]
Paraphrasal: If ๐ฅ has a relation ๐ to ๐ฆ and ๐ฆ has the same relation ๐ to ๐ง, then ๐ฅ cannot have the
same relation ๐ to ๐ง.
E.g. โ๐ฅ is the father of ๐ฆโ, or โ๐ฅ weighs exactly twice as much as ๐ฆโ etc.
Meanwhile, a relation is non-transitive if it is neither transitive nor intransitive.
E.g. โ๐ฅ loves ๐ฆโ, or โ๐ฅ has a different mass to ๐ฆ"
3. Reflexivity: A relation ๐ ๐ฅ๐ฆ is reflexive if: (โ๐ฅ)[(โ๐ฆ)(๐ ๐ฅ๐ฆ v ๐ ๐ฆ๐ฅ) โ ๐ ๐ฅ๐ฅ]
Paraphrasal: ๐ฅ has a relation ๐ to itself if there is some ๐ฆ such that either ๐ ๐ฅ๐ฆ or ๐ ๐ฆ๐ฅ.
E.g. โ๐ฅ is the same age as ๐ฆโ, or โ๐ฅ is a contemporary of ๐ฆโ, or โ๐ฅ has the same eye colour as ๐ฆโ etc.
As a special case, a relation ๐ ๐ฅ๐ฆ is totally reflexive if the following is true that: ๐ ๐ฅ๐ฅ
E.g. โ๐ฅ is identical to itselfโ
Meanwhile, a relation ๐ ๐ฅ๐ฆ is irreflexive if it is true that: (โ๐ฅ)~๐ ๐ฅ๐ฅ
Paraphrasal: ๐ฅ cannot have a relation ๐ to itself.
E.g. โ๐ฅ is North of ๐ฆ", or โ๐ฅ is married to ๐ฆโ, or โ๐ฅ > ๐ฆ for Real numbersโ
Finally, a relation is non-reflexive if it is neither totally reflexive, reflexive nor irreflexive.
E.g. โ๐ฅ loves ๐ฆโ, or โ๐ฅ hates ๐ฆโ etc.
Note: Most binary or dyadic relations have more than one property or attribute. E.g. the relation:
โโฆ weighs more thatโฆโ is at the same time asymmetrical, transitive and irreflexive, while that of โโฆ
weighs the same asโฆโ is symmetrical, transitive and reflexive. (Copi, p. 136) See if you can identify
the attributes in the following relations.
1. ๐ฅ is uncle of ๐ฆ. 2. ๐ฅ = ๐ฆ 3. ๐ฅ is as big as ๐ฆ. 4. ๐ฅ is a sister of ๐ฆ.
Solutions:
1. โ๐ฅ is uncle of ๐ฆโ is at the same time:
(a) asymmetrical: (โ๐ฅ)(โ๐ฆ)(๐๐ฅ๐ฆ โ ~๐๐ฆ๐ฅ) (b) intransitive: (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ~๐๐ฅ๐ง] and (c) irreflexive: (โ๐ฅ)~๐๐ฅ๐ฅ
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2. โ๐ฅ = ๐ฆโ is at the same time:
(a) symmetrical: (โ๐ฅ)(โ๐ฆ)[(๐ฅ = ๐ฆ) โ (๐ฆ = ๐ฅ)] (b) transitive: (โ๐ฅ)(โ๐ฆ)(โ๐ง){[(๐ฅ = ๐ฆ) โข (๐ฆ = ๐ง)] โ (๐ฅ = ๐ง)} and (c) totally reflexive: (โ๐ฅ)(๐ฅ = ๐ฅ)
3. โ๐ฅ is as big as ๐ฆโ is at the same time:
(a) symmetrical: (โ๐ฅ)(โ๐ฆ)(๐ต๐ฅ๐ฆ โ ๐ต๐ฆ๐ฅ) (b) transitive: (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ต๐ฅ๐ฆ โข ๐ต๐ฆ๐ง) โ ๐ต๐ฅ๐ง] and (c) reflexive: (โ๐ฅ)[(โ๐ฆ)(๐ต๐ฅ๐ฆ v ๐ต๐ฆ๐ฅ) โ ๐ต๐ฅ๐ฅ]
4. โ๐ฅ is a sister of ๐ฆโ
(a) non-symmetrical: neither symmetrical nor asymmetrical - no symmetrical relation (b) transitive: (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ๐๐ฅ๐ง] (c) irreflexive: (โ๐ฅ)~๐๐ฅ๐ฅ
Copi (p. 136) points out furthermore, that some attributes entail the presence of others, such as that
all asymmetrical relations must also be irreflexive. This he derives as follows:
1. (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) Definition of asymmetry 2. (โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) 1 UI 3. ๐ ๐ฅ๐ฅ โ ~๐ ๐ฅ๐ฅ 2 UI 4. ~๐ ๐ฅ๐ฅ v ~๐ ๐ฅ๐ฅ 3 Impl. 5. ~๐ ๐ฅ๐ฅ 4 Taut. 6. (โ๐ฅ)~๐ ๐ฅ๐ฅ 5 UG
This last line, as can be seen, is the definition of irreflexivity and so the proof is complete.
Note: When using models to prove invalidity, such as those introduced in Critical Reasoning 11, we
must be cautious when assessing arguments that involve relational attributes. We cannot, for
example, consistently begin with a model ๐ containing only one object when a relation of
irreflexivity, (โ๐ฅ)~๐ ๐ฅ๐ฅ obtains. There must be at least two non-identical objects. Similarly with the
relation of transitivity, (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ๐ ๐ฅ๐ง] a potentially consistent model must
contain at least three non-identical objects.
Enthymemes
One problem with relational attributes is precisely that they are assumed and therefore unstated. In
Critical Reasoning 01 we called such arguments enthymemes. Where premises are supressed or
โunderstoodโ we shall have to take them into account when considering validity. (Copi p. 136-137)
Consider the argument on page 1 above: Al is older than Bill. Bill is older than Charlie. Therefore, Al is older than Charlie.
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Technically it is invalid as it stands, but we account it valid just the same because the relationship of
โโฆ being older thanโฆโ has the obvious, but unstated, attribute of transitivity. Sometimes however
the missing or auxiliary premise may be about factual linguistic assumptions. Consider:
All dogs are tame. Therefore, Fido is tame. Clearly we are assuming that Fido is a dog (๐ท๐) and so we have:
1. (โ๐ฅ)(๐ท๐ฅ โ ๐๐ฅ) /โด ๐๐ 2. ๐ท๐ (auxiliary premise) 3. ๐ท๐ โ ๐๐ 1 UI 4. ๐๐ 3,2 M.P. Even within the same relatively simple argument we might make multiple assumptions about
relations and matters of fact, as in the following example by Copi (p. 138-139).
Any horse can outrun any dog. Some greyhounds can outrun any rabbit. Therefore, any horse can outrun any rabbit. Here we are making the assumption that the relation โโฆ can outrunโฆโ is a transitive one and the
fact that all greyhounds are dogs. This makes the argument only a little longer to state but more
lengthy to prove. Here is Copiโs proof of the same:
1. (โ๐ฅ)[๐ป๐ฅ โ (โ๐ฆ)(๐ท๐ฆ โ ๐๐ฅ๐ฆ)] 2. (โ๐ฆ)[๐บ๐ฆ โข (โ๐ง)(๐ ๐ง โ ๐๐ฆ๐ง)] /โด (โ๐ฅ)[๐ป๐ฅ โ (โ๐ง)(๐ ๐ง โ ๐๐ฅ๐ง)] 3. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ๐๐ฅ๐ง] (auxiliary premise of transitivity) 4. (โ๐ฆ)(๐บ๐ฆ โ ๐ท๐ฆ) (auxiliary premise that all greyhounds are dogs) 5. ๐ป๐ฅ 6. ๐ ๐ง 7. ๐บ๐ฆ โข (โ๐ง)(๐ ๐ง โ ๐๐ฆ๐ง) 8. ๐บ๐ฆ 7 Simp. 9. ๐บ๐ฆ โ ๐ท๐ฆ 4 UI 10. ๐ท๐ฆ 9,8 M.P. 11. ๐ป๐ฅ โ (โ๐ฆ)(๐ท๐ฆ โ ๐๐ฅ๐ฆ) 1 UI 12. (โ๐ฆ)(๐ท๐ฆ โ ๐๐ฅ๐ฆ) 11,5 M.P. 13. ๐ท๐ฆ โ ๐๐ฅ๐ฆ 12 UI 14. ๐๐ฅ๐ฆ 13,10 M.P. 15 (โ๐ง)(๐ ๐ง โ ๐๐ฆ๐ง) 7 Simp. 16. ๐ ๐ง โ ๐๐ฆ๐ง 15 UI 17. ๐๐ฆ๐ง 16,6 M.P. 18. ๐๐ฅ๐ฆ โข ๐๐ฆ๐ง 14,17 Conj. 19. (โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ๐๐ฅ๐ง] 3 UI 20. (โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ๐๐ฅ๐ง] 19 UI 21. (๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ๐๐ฅ๐ง 20 UI 22. ๐๐ฅ๐ง 21,18 M.P. 23. ๐๐ฅ๐ง 2,7-22 EI 24. ๐ ๐ง โ ๐๐ฅ๐ง 6-23 C.P. 25. (โ๐ง)(๐ ๐ง โ ๐๐ฅ๐ง) 24 UG 26. ๐ป๐ฅ โ (โ๐ง)(๐ ๐ง โ ๐๐ฅ๐ง) 5-25 C.P. continued over pageโฆ
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27. (โ๐ฅ)[๐ป๐ฅ โ (โ๐ง)(๐ ๐ง โ ๐๐ฅ๐ง)] 26 UG Exercise: If you have a copy of Copiโs textbook at hand you may wish to try and prove a selection of
some of the enthymemes from the exercise at the end section 5.3 on pages 139-140. Otherwise,
here are the first two. The solutions below were provided by John Ross.
1. A Cadillac is more expensive than any low-priced car. Therefore, no Cadillac is a low-priced
car. (๐ถ๐ฅ: ๐ฅ is a Cadillac. ๐ฟ๐ฅ: ๐ฅ is a low-priced car. ๐๐ฅ๐ฆ: ๐ฅ is more expensive than ๐ฆ.)
1. (โ๐ฅ)[๐ถ๐ฅ โ (โ๐ฆ)(๐ฟ๐ฆ โ ๐๐ฅ๐ฆ)] /โด (โ๐ฅ)(๐ถ๐ฅ โ ~๐ฟ๐ฅ) 2. (โ๐ฅ)~๐๐ฅ๐ฅ (auxiliary premise of irreflexivity) 3. ๐ถ๐ฅ 4. ๐ถ๐ฅ โ (โ๐ฆ)(๐ฟ๐ฆ โ ๐๐ฅ๐ฆ) 1 UI 5. (โ๐ฆ)(๐ฟ๐ฆ โ ๐๐ฅ๐ฆ) 4,3 M.P. 6. ๐ฟ๐ฅ โ ๐๐ฅ๐ฅ 5 UI 7. ~๐๐ฅ๐ฅ 2 UI 8. ~๐ฟ๐ฅ 6,7 M.T. 9. ๐ถ๐ฅ โ ~๐ฟ๐ฅ 3-8 C.P 10. (โ๐ฅ)(๐ถ๐ฅ โ ~๐ฟ๐ฅ) 9 UG
2. Alice is Bettyโs mother. Betty is Charleneโs mother. Therefore, if Charlene loves only her
mother, then she does not love Alice. (๐: Alice. ๐: Betty. ๐: Charlene. ๐๐ฅ๐ฆ: ๐ฅ is mother of ๐ฆ.
๐ฟ๐ฅ๐ฆ: ๐ฅ loves ๐ฆ.
1. ๐๐๐ 2. ๐๐๐ /โด (โ๐ฅ)(๐ฟ๐๐ฅ โ ๐๐ฅ๐) โ ~๐ฟ๐๐ 3. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ~๐๐ฅ๐ง] (auxiliary premise of intransitivity) 4. (โ๐ฅ)(๐ฟ๐๐ฅ โ ๐๐ฅ๐) 5. (โ๐ฆ)(โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ~๐๐ฅ๐ง] 3 UI 6. (โ๐ง)[(๐๐ฅ๐ฆ โข ๐๐ฆ๐ง) โ ~๐๐ฅ๐ง] 5 UI 7. (๐๐๐ โข ๐๐๐) โ ~๐๐๐ 6 UI 8. ๐ฟ๐๐ฅ โ ๐๐ฅ๐ 4 UI 9. ~๐๐๐ โ ~๐ฟ๐๐ 8 Trans. 10. (๐๐๐ โข ๐๐๐) โ ~๐ฟ๐๐ 7,9 H.S. 11. ๐๐๐ โข ๐๐๐ 1,2 Conj. 12. ~๐ฟ๐๐ 10, 11 M.P. 12. (โ๐ฅ)(๐ฟ๐๐ฅ โ ๐๐ฅ๐) โ ~๐ฟ๐๐ 4-12 C.P.
Identity
The special relation of identity, symbolised by the โ=โ sign merits special treatment, both because of
its importance and ubiquity. According to the Identity of Indiscernibles or Law of Identity1, two
entities are identical if and only if they have all their properties in common; thus:
(โ๐ฅ)(โ๐ฆ)[(๐ฅ = ๐ฆ) โก (โ๐น)(๐น๐ฅ โก ๐น๐ฆ)]
1 The term โLeibnizโ lawโ is confusing because it may also refer to the โproduct ruleโ or the โgeneralized product ruleโ in calculus.
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This formulation of identity is much stronger that of our informal use of the term when, for example
referring to โidenticalโ twins. Even if two twins, say Bobby and Joe, are truly indiscernible even by
their parents, one would have the property of having one name, the other another. One may have
the property of standing on the left, the other on the right. One may have the property of having
been born minutes before the other, and so on. Therefore, they are not logically identical.
We should also pause here to point out that quantifying over properties as we have done here is
regarded as illegitimate by a significant proportion of philosopher-logicians for reasons that will take
us too far from the present topic. Suffice it to say that you may regard the Identity of Indiscernibles
and what follows as given, until such time as you have developed your own reasoned position on the
matter.
Rules of Identity (Id.)
ฮฆ๐ ฮฆ๐ ๐ = ๐ ๐ ๐ = ๐ ~(ฮฆ๐) โด ๐ = ๐ฃ โด ๐ = ๐ โด ฮฆ๐ โด ~(ฮฝ = ฮผ) Consider the following instances of use of the rules of identity, left to right:
1. Dollar Brand is a musician. Dollar Brand is Abdullah Ibrahim. Therefore, Abdullah Ibrahim is a
musician.
1. ๐๐ 2. ๐ = ๐ /โด ๐๐ 3. ๐๐ 1,2 Id.
2. Dollar Brand is talented. Phil is not talented. Therefore, Dollar Brand is not Phil.
1. ๐๐ 2. ~๐๐ /โด ๐ โ ๐ 3. ๐ โ ๐ 1,2 Id.
3. Dollar Brand is Abdullah Ibrahim. Therefore, Abdullah Ibrahim is Dollar Brand.
๐ = ๐ /โด ๐ = ๐ (Identity is symmetrical.)
4. If Dollar Brand is identical to himself, then Phil is South African. Therefore, Phil is South
African.
1. (๐ = ๐) โ ๐๐ /โด ๐๐ 2. ๐ = ๐ 1. Id. (Identity is totally reflexive.) 3. ๐๐ 1,2 M.P. Note that in the above examples identity is not used on only part of line. Also observe the shorthand
๐ โ ๐ for ~(ฮฝ = ฮผ).
Examples of sentences involving the identity predicate: ๐๐ฅ: ๐ฅ is a person; ๐= Jane; ๐๐ฅ๐ฆ: ๐ฅ is taller than ๐ฆ.
1. Jane is taller than anyone else.
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(โ๐ฅ)[(๐๐ฅ โข ๐ฅ โ ๐) โ ๐๐๐ฅ]
2. There is someone who is taller than any other person. (โ๐ฅ){๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ]}
3. Jane is the only intelligent person. (โ๐ฅ)[(๐๐ฅ โข ๐ผ๐ฅ) โ ๐ฅ = ๐] or (โ๐ฅ)[๐๐ฅ โข ๐ฅ โ ๐) โ ~๐ผ๐ฅ]
4. There is only one intelligent person. (โ๐ฅ){(๐๐ฅ โข ๐ผ๐ฅ) โข (โ๐ฆ)[(๐๐ฆ โข ๐ผ๐ฆ) โ ๐ฆ = ๐ฅ]}
Note that the left hand side of the โข with the largest scope ensures that there is at least one
such person, while the expression to the right ensures that there is at most one.
5. There are exactly two intelligent persons. (โ๐ฅ)(โ๐ฆ){(๐๐ฅ โข ๐ผ๐ฅ โข ๐๐ฆ โข ๐ผ๐ฆ โข ๐ฅ โ ๐ฆ) โข (โ๐ง)[(๐๐ง โข ๐ผ๐ง) โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)]}
Note again, how in similar fashion, the left hand side of the โข with the largest scope insures
that there are at least two such persons, while the expression to the right ensures that there
are at most two. There is no upper limit on how logic can โcountโ in this way, except to
observe that the expressions very quickly become increasingly cumbersome.
6. Al is on the team and can outrun anyone else on it. (๐ = Al; ๐๐ฅ: ๐ฅ is on the team. ๐๐ฅ๐ฆ: ๐ฅ can
outrun ๐ฆ) If, in this example by Copi, (p. 143) we were to write: ๐๐ โข (โ๐ฅ)(๐๐ฅ โ ๐๐๐ฅ), it
would entail that Al can outrun himself, whereas we know that being able to outrun is an
irreflexive relation. What we have to capture is the meaning of the word โelseโ, hence:
๐๐ โข (โ๐ฅ)[(๐๐ฅ โข ๐ฅ โ ๐) โ ๐๐๐ฅ] Arguments Involving Identity Consider the following examples of proofs involving identity before trying your own hand at nos. 5,
6, 9 and 10 from the exercise at the end of section 5.4 of Copi (p. 150):
1. Bill is Smith. Smith is taller than Jones. Therefore, Bill is taller than someone or other. (Note
that the โisโ in the first premise is one of identity not predication.)
1. ๐ = ๐ 2. ๐๐ ๐ /โด (โ๐ฅ)๐๐๐ฅ 3. ๐๐๐ 1,2 Id. 4. (โ๐ฅ)๐๐๐ฅ 3 E.G.
2. 1. (โ๐ฅ)(โ๐ฆ)(๐ฅ = ๐ฆ โ ๐ ๐ฅ๐ฆ) 2. โผ ๐ ๐๐ 3. ๐ = ๐ฅ /โด ๐ โ ๐ continued over pageโฆ
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4. (โ๐ฆ)(๐ = ๐ฆ โ ๐ ๐๐ฆ) 1 UI 5. ๐ = ๐ โ ๐ ๐๐ 4 UI 6. ~๐ ๐๐ 2,3 Id. 7. ๐ โ ๐ 5,6 M.T.
3. 1. ๐น๐ 2. (โ๐ฅ)(๐บ๐ฅ โข ๐ = ๐ฅ) /โด (โ๐ฅ)(๐น๐ฅ โข ๐บ๐ฅ) 3. ๐บ๐ฅ โข ๐ = ๐ฅ 4. ๐ = ๐ฅ 3 Simp. 5. ๐น๐ฅ 1,4 Id. 6. ๐บ๐ฅ 3 Simp. 7. ๐น๐ฅ โข ๐บ๐ฅ 5,6 Conj. 8. (โ๐ฅ)(๐น๐ฅ โข ๐บ๐ฅ) 7 E.G. 9. (โ๐ฅ)(๐น๐ฅ โข ๐บ๐ฅ) 2,3-8 EI
4. Only Jane is intelligent. Someone who is intelligent is tall. Therefore Jane is tall. 1. (โ๐ฅ)(๐ผ๐ฅ โ ๐ฅ = ๐) 2. (โ๐ฅ)(๐ผ๐ฅ โข ๐๐ฅ) /โด ๐๐ 3. ๐ผ๐ฅ โข ๐๐ฅ 4. ๐ผ๐ฅ โ ๐ฅ = ๐ 1 UI 5. ๐ผ๐ฅ 3 Simp. 6. ๐ฅ = ๐ 4,5 M.P. 7. ๐๐ฅ 3 Simp. 8. ๐๐ 7,6 Id. 9. ๐๐ Selected Solutions to Copi (p. 150)
5. All entrants will win. There will be, at most, one winner. There is at least one entrant.
Therefore, there is exactly one entrant. (๐ธ๐ฅ: ๐ฅ is an entrant; ๐๐ฅ: ๐ฅ will win)
1. (โ๐ฅ)(๐ธ๐ฅ โ ๐๐ฅ) 2. (โ๐ฅ)(โ๐ฆ)[(๐๐ฅ โข ๐๐ฆ) โ ๐ฆ = ๐ฅ] (note the standard translation for โat most oneโ)
3. (โ๐ฅ)๐ธ๐ฅ /โด (โ๐ฅ)[๐ธ๐ฅ โข (โ๐ฆ)(๐ธ๐ฆ โ ๐ฆ = ๐ฅ)] (and for โthere is exactly oneโ) 4. ๐ธ๐ฅ 5. ๐ธ๐ฅ โ ๐๐ฅ 1 UI 6. ๐๐ฅ 5,4 M.P. 7. (โ๐ฆ)[(๐๐ฅ โข ๐๐ฆ) โ ๐ฆ = ๐ฅ] 2 UI 8. (๐๐ฅ โข ๐๐ฆ) โ ๐ฆ = ๐ฅ 7 UI 9. ๐๐ฅ โ (๐๐ฆ โ ๐ฆ = ๐ฅ) 7 Exp. 10. ๐๐ฆ โ ๐ฆ = ๐ฅ 9,6 M.P. 11. ๐ธ๐ฆ โ ๐๐ฆ 1 UI (having used UI on 5, we must now use ๐ฆ) 12. ๐ธ๐ฆ โ ๐ฆ = ๐ฅ 11,10 H.S. 13. (โ๐ฆ)(๐ธ๐ฆ โ ๐ฆ = ๐ฅ) 12 UG 14. ๐ธ๐ฅ โข (โ๐ฆ)(๐ธ๐ฆ โ ๐ฆ = ๐ฅ) 4,13 Conj. 15. (โ๐ฅ)[๐ธ๐ฅ โข (โ๐ฆ)(๐ธ๐ฆ โ ๐ฆ = ๐ฅ)] 14 EG 16. (โ๐ฅ)[๐ธ๐ฅ โข (โ๐ฆ)(๐ธ๐ฆ โ ๐ฆ = ๐ฅ)] 3,4-15 EI
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9. There is exactly one penny in my right hand. There is exactly one penny in my left hand.
Nothing is in both my hands. Therefore there are exactly two pennies in my hands. (๐๐ฅ: ๐ฅ is
a penny; ๐ ๐ฅ: ๐ฅ is in my right hand; ๐ฟ๐ฅ: ๐ฅ is in my left hand.)
1. (โ๐ฅ){๐๐ฅ โข ๐ ๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ ๐ฆ) โ ๐ฆ = ๐ฅ]} 2. (โ๐ฅ){๐๐ฅ โข ๐ฟ๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฟ๐ฆ) โ ๐ฆ = ๐ฅ]} 3. (โ๐ฅ)(~๐ ๐ฅ v ~๐ฟ๐ฅ) /โด (โ๐ฅ)(โ๐ฆ)[๐๐ฅ โข ๐๐ฆ โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ฆ v ๐ฟ๐ฆ) โข ๐ฅ โ ๐ฆ โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)}] 4. ๐๐ข โข ๐ ๐ข โข (โ๐ฆ)[(๐๐ฆ โข ๐ ๐ฆ) โ ๐ฆ = ๐ข] 5. ๐๐ฅ โข ๐ฟ๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฟ๐ฆ) โ ๐ฆ = ๐ฅ] 6. ๐๐ข 4 Simp. 7. ๐๐ฅ 5 Simp. 8. ๐ ๐ข 4 Simp. 9. ๐ฟ๐ฅ 5 Simp. 10. ๐ ๐ข v ๐ฟ๐ข 8 Add. 11. ๐ฟ๐ฅ v ๐ ๐ฅ 9 Add. 12. ~๐ ๐ข v ~๐ฟ๐ข 3 UI 13. ๐ ๐ข โ ~๐ฟ๐ข 12 Impl. 14. ~๐ฟ๐ข 13, 8 M.P. 15. ๐ฅ โ ๐ข 9,14 Id. 16. ๐๐ฅ โข ๐๐ข 7,6 Conj. 17. ๐๐ฅ โข ๐๐ข โข (๐ฟ๐ฅ v ๐ ๐ฅ) 16,11 Conj. 18. ๐๐ฅ โข ๐๐ข โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ข v ๐ฟ๐ข) 17,10 Conj. 19. ๐๐ฅ โข ๐๐ข โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ข v ๐ฟ๐ข) โข ๐ฅ โ ๐ข 18,15 Conj. 20. ๐๐ง โข (๐ฟ๐ง v ๐ ๐ง) 21.(โ๐ฆ)[(๐๐ฆ โข ๐ ๐ฆ) โ ๐ฆ = ๐ข] 4 Simp. 22.(โ๐ฆ)[(๐๐ฆ โข ๐ฟ๐ฆ) โ ๐ฆ = ๐ฅ] 5 Simp. 23. (๐๐ง โข ๐ ๐ง) โ ๐ง = ๐ข 21 UI 24. (๐๐ง โข ๐ฟ๐ง) โ ๐ง = ๐ฅ 22 UI 25. (๐๐ง โข ๐ฟ๐ง) v (๐๐ง โข ๐ ๐ง) 20 Dist. 26. [(๐๐ง โข ๐ฟ๐ง) โ ๐ง = ๐ฅ] โข [(๐๐ง โข ๐ ๐ง) โ ๐ง = ๐ข] 24,23 Conj. 27. ๐ง = ๐ฅ v ๐ง = ๐ข 26,25 C.D. 28. [๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ข) 20-27 C.P. 29. (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ข)]} 28 UG 30. ๐๐ฅ โข ๐๐ข โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ข v ๐ฟ๐ข) โข ๐ฅ โ ๐ข โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ข)} 19,29 Conj. 31. (โ๐ฆ)[๐๐ฅ โข ๐๐ฆ โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ฆ v ๐ฟ๐ฆ) โข ๐ฅ โ ๐ฆ โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)}] 30 EG 32. (โ๐ฅ)(โ๐ฆ)[๐๐ฅ โข ๐๐ฆ โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ฆ v ๐ฟ๐ฆ) โข ๐ฅ โ ๐ฆ โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)}] 31 EG 33. (โ๐ฅ)(โ๐ฆ)[๐๐ฅ โข ๐๐ฆ โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ฆ v ๐ฟ๐ฆ) โข ๐ฅ โ ๐ฆ โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)}] 2,5-32 EI 34. (โ๐ฅ)(โ๐ฆ)[๐๐ฅ โข ๐๐ฆ โข (๐ฟ๐ฅ v ๐ ๐ฅ) โข (๐ ๐ฆ v ๐ฟ๐ฆ) โข ๐ฅ โ ๐ฆ โข (โ๐ง){[๐๐ง โข (๐ฟ๐ง v ๐ ๐ง)] โ (๐ง = ๐ฅ v ๐ง = ๐ฆ)}] 1,4-33 EI
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Russellโs Theory of Descriptions
What is the meaning of the word โtheโ? Despite
โtheโ being the most commonly used word in the
English language, very few of us (who havenโt done
a course in Linguistics or the Philosophy of Lan-
guage) are likely to know. True, we know how to
use it and pace Wittgenstein, for whom meaning is
use, it was not until Bertrand Russellโs 1905, now
classic paper โOn Denotingโ (available here) pub-
lished in the journal โMindโ that most Philosophers
were in agreement about the logical structure and
semantics of expressions involving denoting
phrases.
Consider the following sorts of expressions:
1. phrases that fail to denote, such as โthe
current King of Franceโ;
2. phrases which do denote one object, such
as โthe President of the United statesโ
which refers unambiguously to one known
individual; or โthe tallest spyโ which refers unambiguously to one unknown individual;
3. phrases which denote ambiguously, such as โan aardvarkโ. (Wikipedia: Theory of
descriptions)
Russellโs theory of descriptions is regarded as a paradigm of Analytic Philosophy because it
apparently solved three problems simultaneously, namely:
โข the problem of empty names;
โข how words attach to things, i.e. reference, and
โข truths (or falsehoods) that fail to refer.
In On Denoting (1905) Russell considers the sentence: โThe current King of Face is baldโ. Clearly the
sentence has a meaning that we understand but it fails to refer to anybody. So how it possible for it
to be either true or false? Russell analyses the sentence into the following components:
1. There is an ๐ฅ such that ๐ฅ is the current King of France.
2. For every ๐ฅ and ๐ฆ such that if ๐ฅ and ๐ฆ are current Kings of France then ๐ฅ is ๐ฆ (i.e. there is at
most one current King of France).
3. Anything that is a current King of France is bald.
Thus, any definite description of the form โthe ๐น is ๐ตโ can be expressed as:
(โ๐ฅ)[๐น๐ฅ โข (โ๐ฆ)(๐น๐ฆ โ ๐ฅ = ๐ฆ) โข ๐ต๐ฅ]
According to Russell, this immediately solves two problems. Firstly, the sentence: โThe current King
of Face is baldโ is straightforwardly seen to be false because there are no entities of which the
Bertrand Russell in 1916
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predicate โโฆ is the current King of Franceโ is true. In fact, all instances of โ๐ฅ has the property ๐นโ are
false for all ๐ฅ, without ๐ฅ having to refer to any mysterious non-entity. Secondly, this solves Gottlob
Fregeโs problem of informative identities; the identity of the โmorning starโ and the โevening starโ
being just such a case. Both have different meanings but the same referent i.e. โThe planet Venusโ.
Instead of treating a sentence like, โThe morning star rises in the morningโ as having a subject-
predicate form, Russell analyses it as โthere is one unique thing such that it is the morning star and it
rises in the morningโ which is not semantically the same as โthere is one unique thing such that it is
the evening star and it rises in the eveningโ, even though both expressions ultimately have the same
referent. (Wikipedia: Theory of descriptions)
Although few linguists would say that Russell has โsolvedโ the question of reference, his theory of
descriptions has recast such questions in terms of descriptions with which we may become familiar,
rather than entities with which we are acquainted. Thus, I may learn about the existence of a special
point, the centre of gravity of the Solar System, by its definite descriptions. Similarly, I might know
about the not existence of other entities by their definite descriptions.
As far as indefinite descriptions are concerned, we know that the sentence โSome dog is annoyingโ
is true, not because we are acquainted with the subject-predicate โ๐ด๐โ, but because of the
existentially quantified conjunction of two descriptions:
1. There is an ๐ฅ such that ๐ฅ is a dog, and
2. ๐ฅ is annoying.
In symbols: (โ๐ฅ)(๐ท๐ฅ โข ๐ด๐ฅ) (Wikipedia: Theory of descriptions)
Paraphrasal: There exists an ๐ฅ such that ๐ฅ is a dog and ๐ฅ is annoying.
Arguments Involving Definite Descriptions
Some of the exercises at the end of Copiโs section 5.4 involve definite descriptions, introduced by
the article โtheโ. Try first to symbolise, and then prove nos. 1-4 (p. 149-150) Again, the solutions
below were provided by John Ross. E.g.
1. The architect who designed Tappan Hall designs only office buildings. Therefore, Tappan Hall
is an office building. (๐ด๐ฅ: ๐ฅ is an architect. ๐ก: Tappan Hall. ๐ท๐ฅ๐ฆ: ๐ฅ designed ๐ฆ. ๐๐ฅ: ๐ฅ is an
office building.)
1. (โ๐ฅ){๐ด๐ฅ โข ๐ท๐ฅ๐ก โข (โ๐ฆ)[(๐ด๐ฆ โข ๐ท๐ฆ๐ก) โ ๐ฆ = ๐ฅ] โข (โ๐ง)(๐ท๐ฅ๐ง โ ๐๐ง)} /โด ๐๐ก 2. ๐ด๐ฅ โข ๐ท๐ฅ๐ก โข (โ๐ฆ)[(๐ด๐ฆ โข ๐ท๐ฆ๐ก) โ ๐ฆ = ๐ฅ] โข (โ๐ง)(๐ท๐ฅ๐ง โ ๐๐ง) 3. (โ๐ง)(๐ท๐ฅ๐ง โ ๐๐ง) 2 Simp. 4. ๐ท๐ฅ๐ก โ ๐๐ก 3 UI 5. ๐ท๐ฅ๐ก 2 Simp. 6. ๐๐ก 4,5 M.P. 7. ๐๐ก 1,2-6 EI
3. The smallest state is in New England. All states in New England are primarily industrial.
Therefore, the smallest state is primarily industrial. (๐๐ฅ: ๐ฅ is a state. ๐๐ฅ: ๐ฅ is in New England.
๐ผ๐ฅ: ๐ฅ is primarily industrial. ๐๐ฅ๐ฆ: ๐ฅ is smaller than ๐ฆ.
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1. (โ๐ฅ){๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐๐ฅ} 2. (โ๐ฅ)[(๐๐ฅ โข ๐๐ฅ) โ ๐ผ๐ฅ] /โด (โ๐ฅ){๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐ผ๐ฅ} 3. ๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐๐ฅ 4. ๐๐ฅ โข ๐๐ฅ 3 Simp. 5. (๐๐ฅ โข ๐๐ฅ) โ ๐ผ๐ฅ 2 UI 6. ๐ผ๐ฅ 5,4 M.P. 7. ๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] 3.Simp. 8. ๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐ผ๐ฅ 7,6 Conj. 9. (โ๐ฅ){๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐ผ๐ฅ} 8 EG 10. (โ๐ฅ){๐๐ฅ โข (โ๐ฆ)[(๐๐ฆ โข ๐ฆ โ ๐ฅ) โ ๐๐ฅ๐ฆ] โข ๐ผ๐ฅ} 1,3-9 EI Predicate Variables and Properties of Properties So far we have been employing a first order calculus in which the Roman letters โ๐นโ, โ๐บโ, โ๐ปโ etc.
(but not the Greek letters โ๐โ and โ๐โ) have been treated as constants. Therefore we have treated
an expression like โ(โ๐ฅ)๐น๐ฅโ as a proposition, capable of being true or false, since it contains no free
variables. On a second order calculus a predicate letter like โ๐นโ is treated as a variable. Therefore,
on a second calculus, the expression โ(โ๐ฅ)๐น๐ฅโ is considered to be a propositional function,
incapable of being true or false because it is incomplete, containing the free variable, โ๐นโ. To
become a proposition, the expression would have to be written:
(โ๐น)(โ๐ฅ)๐น๐ฅ or (โ๐น)(โ๐ฅ)๐น๐ฅ.
Some standard translations of second order propositions (Copi, 151-152):
1. Socrates has all properties.
(โ๐น)๐น๐
2. Plato has some attribute.
(โ๐น)๐น๐
3. Everything has every attribute.
(โ๐ฅ)(โ๐น)๐น๐ฅ
4. Everything has some property or other.
(โ๐ฅ)(โ๐น)๐น๐ฅ
5. Something has every property.
(โ๐ฅ)(โ๐น)๐น๐ฅ
6. Every property belongs to something or other.
(โ๐น)(โ๐ฅ)๐น๐ฅ
7. Some property belongs to everything.
(โ๐น)(โ๐ฅ)๐น๐ฅ
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8. Something has some property or other.
(โ๐ฅ)(โ๐น)๐น๐ฅ
Although such expressions are meaningful to us, it is a highly contentious question as to whether
they could refer to anything in the external world outside our own mental constructs. There would
almost certainly be no properties in a universe devoid of beings to think about them. Like Hume on
causality, (Classic Text 04) we should be ontologically sceptical. However for example, we speak
almost naturally about classes of particles having certain properties in common, or even being
defined by their properties. Given that we do, a second order calculus must admit the possibility of
properties having properties. Indeed it has to admit of the possibility of sentences such as the
following (with second order predicates in bold):
1. All useful attributes are desirable.
(โ๐น)(๐ผ๐น โ ๐ซ๐น)
2. Some desirable attributes are not useful.
(โ๐น)(๐ซ๐น โข ~๐ผ๐น)
3. Nothing which possesses every rare attribute has any ordinary attribute.
(โ๐ฅ)[(โ๐น)(๐น๐น โ ๐น๐ฅ) โ (โ๐บ)(๐ถ๐บ โ ~๐บ๐ฅ)]
4. Nothing has all attributes.
~(โ๐ฅ)(โ๐น)๐น๐ฅ or (โ๐ฅ)(โ๐น)~๐น๐ฅ
5. Some attribute belongs to nothing.
(โ๐น)(โ๐ฅ)~๐น๐ฅ
6. Napoleon had all the attributes of a great general.
(โ๐น)[(โ๐ฅ)(๐บ๐ฅ โ ๐น๐ฅ) โ ๐น๐]
Second order expressions involving the identity predicate can also be quantified. As Copi (p. 152)
shows beginning with a purely symbolic definition of the identity, i.e.
(๐ฅ = ๐ฆ) = ๐๐ (โ๐น)(๐น๐ฅ โก ๐น๐ฆ)
from which it follows that,
(โ๐ฅ)(โ๐ฆ)[(๐ฅ = ๐ฆ) โก (โ๐น)(๐น๐ฅ โก ๐น๐ฆ)]
which will be recognised as Leibnizโs law. Consider the following two examples.
7. Any two things have some common attribute.
(โ๐ฅ)(โ๐ฆ)[๐ฅ โ ๐ฆ โ (โ๐น)(๐น๐ฅ โข ๐น๐ฆ)]
8. Jones and Smith share all their good qualities but have no bad quantities in common.
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(โ๐น)[๐ฎ๐น โ (๐น๐ โก ๐น๐)] โข (โ๐น){๐ฉ๐น โ [(๐น๐ โ ~๐น๐) โข (๐น๐ โ ๐น๐ )]}
Finally, we can use the method of conditional proof learned in Critical Reasoning 09 to prove the
following tautologies involving attributes or properties:
1. If circles are ellipses then circles have all the properties of ellipses.
(โ๐ฅ)(๐ถ๐ฅ โ ๐ธ๐ฅ) โ (โ๐น)[(โ๐ฆ)(๐ธ๐ฆ โ ๐น๐ฆ) โ (โ๐ง)(๐ถ๐ง โ ๐น๐ง)]
1. (โ๐ฅ)(๐ถ๐ฅ โ ๐ธ๐ฅ) 2. (โ๐ฆ)(๐ธ๐ฆ โ ๐น๐ฆ) 3. ๐ถ๐ง โ ๐ธ๐ง 1 UI 4. ๐ธ๐ง โ ๐น๐ง 2 UI 5. ๐ถ๐ง โ ๐น๐ง 3,4 H.S. 6. (โ๐ง)(๐ถ๐ง โ ๐น๐ง) 5 UG 7. (โ๐ฆ)(๐ธ๐ฆ โ ๐น๐ฆ) โ (โ๐ง)(๐ถ๐ง โ ๐น๐ง) 2-6 C.P. 8. (โ๐น)[(โ๐ฆ)(๐ธ๐ฆ โ ๐น๐ฆ) โ (โ๐ง)(๐ถ๐ง โ ๐น๐ง)] 7 UG 9. (โ๐ฅ)(๐ถ๐ฅ โ ๐ธ๐ฅ) โ (โ๐น)[(โ๐ฆ)(๐ธ๐ฆ โ ๐น๐ฆ) โ (โ๐ง)(๐ถ๐ง โ ๐น๐ง)] 1-8 C.P.
2. All asymmetrical binary relations are irreflexive.
Recall that for binary or dyadic relationships the relationship is asymmetrical if (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) is true and irreflexive if (โ๐ฅ)~๐ ๐ฅ๐ฅ
is true. If we connect these as a conditional statement and then universally quantify over both we shall arrive at the desired translation: (โ๐ )[(โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) โ (โ๐ฅ)~๐ ๐ฅ๐ฅ]
1. (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) 2. (โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) 1 UI 3. ๐ ๐ฅ๐ฅ โ ~๐ ๐ฅ๐ฅ 2UI 4. ๐ ๐ฅ๐ฅ v ~๐ ๐ฅ๐ฅ 3 Impl. 5. ๐ ๐ฅ๐ฅ 4 Taut. 6. (โ๐ฅ)~๐ ๐ฅ๐ฅ 5 UG (legitimate use of UG because ๐ฅ is bound in 1) 7. (โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) โ (โ๐ฅ)~๐ ๐ฅ๐ฅ 1-6 C.P. 8. (โ๐ )[(โ๐ฅ)(โ๐ฆ)(๐ ๐ฅ๐ฆ โ ~๐ ๐ฆ๐ฅ) โ (โ๐ฅ)~๐ ๐ฅ๐ฅ] 7 UG Task Because most of the exercises at the end of Copi 5.5 (p. 155-156) have been used as examples in the
text above, we ask you to attempt only two more by way a task:
I. Symbolise the following propositions:
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10. Everyone has some unusual attribute or other. It would be an unusual person who had no unusual attributes. (๐๐ฅ: ๐ฅ is a person; ๐๐ฅ: is an unusual individual; ๐ผ๐น: ๐น is an unusual attribute)
II. Prove the following:
7. All intransitive binary relations are irreflexive. Feedback These solutions were again provided by John Ross.
10. (โ๐ฅ)[๐๐ฅ โ (โ๐น)(๐ผ๐น โข ๐น๐ฅ)] and
(โ๐ฆ){[๐๐ฆ โข (โ๐บ)(๐ผ๐บ โ ~๐บ๐ฆ)] โ ๐๐ฆ} or (โ๐ฆ){[๐๐ฆ โข ~(โ๐บ)(๐ผ๐บ โข ๐บ๐ฆ)] โ ๐๐ฆ} 7. 1. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ~๐ ๐ฅ๐ง] (definition of intransitivity) 2. (โ๐ฅ)(โ๐ฆ)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ฅ) โ ~๐ ๐ฅ๐ฅ] 1 UI 3. (โ๐ฅ)[(๐ ๐ฅ๐ฅ โข ๐ ๐ฅ๐ฅ) โ ~๐ ๐ฅ๐ฅ] 2 UI 4. (๐ ๐ฅ๐ฅ โข ๐ ๐ฅ๐ฅ) โ ~๐ ๐ฅ๐ฅ 3 UI
5. ๐ ๐ฅ๐ฅ โ ~๐ ๐ฅ๐ฅ 4 Taut. 6. ~๐ ๐ฅ๐ฅ v ~๐ ๐ฅ๐ฅ 5 Impl. 7. ~๐ ๐ฅ๐ฅ 6 Taut. 8. (โ๐ฅ)~๐ ๐ฅ๐ฅ 7 UG 9. (โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ~๐ ๐ฅ๐ง] โ (โ๐ฅ)~๐ ๐ฅ๐ฅ 1-8 C.P. 10. (โ๐ ){(โ๐ฅ)(โ๐ฆ)(โ๐ง)[(๐ ๐ฅ๐ฆ โข ๐ ๐ฆ๐ง) โ ~๐ ๐ฅ๐ง] โ (โ๐ฅ)~๐ ๐ฅ๐ฅ} 9. UG
Reference
COPI, I.M. (1979) Symbolic Logic, 5th Edition. Macmillan: New York
SUBER, P. (1997) Answers to Copiโs Translation and Derivation Exercises: Polyadic Predicate Logic.
http://legacy.earlham.edu/~peters/courses/log/answers4.htm
Ross, J. (2010) Unfortunately this link has expired or has been moved elsewhere.
Acknowledgement: Unless otherwise acknowledged in the text, the bulk of explanations and
examples above have been compiled from lecture notes on Symbolic Logic presented by the
University of Cape Town by Professor Ian Bunting. The section on definite descriptions and the
meaning of โtheโ were reconstructed from notes on the Philosophy of Language by Professor David
Brooks at the same institution.
The next Critical Reasoning study unit concerns hypothesis testing.