A maior rede de estudos do Brasil

Grátis
367 pág.

## Pré-visualização | Página 50 de 50

```we have said, is not exactly equivalent to the logical
connective &, whose meaning is deﬁned solely in terms of truth. It is
interesting to ask about the meaning of the related English word but.
Practice (1) Make a quick, intuitive judgement here. Do the English
words and and but mean the same thing? Yes / No
(2) Now we will work out a truth table for but. Remember that we
are only concerned with the TRUTH in given situations of the
sentences we cite.
(a) In a situation where Henry died and Terry resigned are both
true, is the sentence Henry died but Terry resigned true or false? T / F
(b) Where Henry died is true, but Terry resigned is false, is
Henry died but Terry resigned true or false? T / F
(c) Where Henry died is false, but Terry resigned is true, is
Henry died but Terry resigned true or false? T / F
(d) Where Henry died and Terry resigned are both false, is
Henry died but Terry resigned true or false? T / F
(3) On the basis of your answers to questions (a)–(d) above, ﬁll in the values
(T or F, as appropriate) in the right-hand column of the truth table for
but below.
p q p but q
T T
T F
F T
F F
(4) Does the truth table that you have ﬁlled out diﬀer in any way
from the truth table given for the logical connective &? Yes / No
(5) Considered only from the point of view of their eﬀect on the
truth of complex sentences containing them, do and and but
diﬀer in meaning? Yes / No
Feedback (1) No (2) (a) T (b) F (c) F (d) F (3) T, F, F, F (reading down the column)
(4) No (5) No
Comment The conclusion reached about and and but need not be perplexing, so long as
one bears in mind that truth-conditional meaning, of the kind described in
UNIT 14 Connectives: and and or
173
PART FOUR Logic
174
truth tables, may be only a part of the meaning, in a wider sense, of a word.
From the point of view of truth alone, and and but make the same
contribution to meaning; but they diﬀer in other aspects of meaning. In
particular, the word but is preferred when the speaker wishes to indicate
some kind of contrast between the two propositions involved. If the second
proposition, but not the ﬁrst, gives an unexpected piece of information, for
instance, the use of but, rather than and, is appropriate. The logical
connective & captures the truth-conditional aspects of the meanings of both
and and but.
A truth table can also be given for V, corresponding approximately
to English or (or either . . . or). We get you to construct one in practice
below.
Practice (1) In a situation where Henry died and Terry resigned are both true, is
(Either) Henry died or Terry resigned true or false? T / F
(2) Where Henry died is true, but Terry resigned is false, is (Either)
Henry died or Terry resigned true or false? T / F
(3) Where Henry died is false, but Terry resigned is true, is (Either)
Henry died or Terry resigned true or false? T / F
(4) Where Henry died and Terry resigned are both false, is (Either)
Henry died or Terry resigned true or false? T / F
(5) Using your answers to (1)–(4) above as a basis, ﬁll in the values (T or F,
as appropriate) in the right-hand column of the table below.
p q p V q
T T
T F
F T
F F
Feedback (1) T (2) T (3) T (4) F (5) T, T, T, F (reading down the column)
Comment The values T and F which appear in truth tables are the same values as those
assigned to simple propositions in relation to the situations in the world
which they describe (recall the previous unit). In the case of simple
propositions, the values T and F ‘come from’ the world. In the case of
complex propositions with connectives such as & and V, the combinations of
the values of the component simple propositions are ‘looked up’ in the
appropriate truth table, and the value of the whole complex proposition
(either T or F) is arrived at. Thus in the case of complex propositions, their
truth values ‘come from’ the truth values of their constituent simple
propositions.
UNIT 14 Connectives: and and or
175
Example Metaphorically, the truth value of a complex proposition is like the trunk of
a tree whose roots reach down into the world. Truth values ﬂow from the
world upwards through the roots, being aﬀected in various ways where
the roots connect with each other, and eventually arriving at the trunk of
the tree.
The following practices will give you additional experience in working
with complex propositions. Try to work through them completely before
checking the answers in the feedback sections.
((j BEHIND e) & (r SMILE)) V (e STAND) T
V
((j BEHIND e) & (r SMILE)) T
&
(j BEHIND e) T r SMILE T e STAND F
Practice Using the world shown in the picture above as the source of truth values for
simple propositions, and calculating the values of complex propositions from
truth tables, ﬁll in the values T and F, as appropriate, in the boxes in the
diagrams below.
PART FOUR Logic
176
(1) ((e STAND) & (s PULL j)) V (e BEHIND j)
V
(e STAND) & (s PULLj)
&
e STAND s PULL j e BEHIND j
↑ ↑ ↑
(2) (In this example, the boxes are on the left-hand side of their
propositions, simply to make the diagram neater.)
(j BETWEEN r e) & ((e SIT) V (j STAND))
&
(e SIT) V (j STAND)
V
j BETWEEN r e e SIT j STAND
↑ ↑ ↑
Feedback F T
F T
(1) F T F (2) T T T
Practice Use the map overleaf as a ‘world’ to determine the truth values of the
formulae (some simple, some complex) listed. We have used unabbreviated
logical names, for convenience. Assume that the predicates have their usual
English meanings. Boldface is not used in this practice.
(1) mumbai IN India T / F
(2) karachi IN iran T / F
(3) tibet BORDER afghanistan T / F
(4) (tashkent IN iran) & (kabul IN tibet) T / F
(5) (karachi IN pakistan) V (iran BORDER india) T / F
(6) (karachi SOUTH tashkent) & ((tashkent EAST tehran) &
(nepal BORDER tibet)) T / F
(7) ((chennai IN india) V (tehran IN tibet)) V (mumbai IN iran) T / F
(8) ((chennai IN iran) V (tibet WEST iran)) & (kabul IN afghanistan) T / F
Feedback (1) T (2) F (3) T (4) F (5) T (6) T (7) T (8) F
Comment Truth values of complex propositions are derived from the truth values of
their constituent simple propositions. This is an example of what has been
called the compositionality of meaning.
Definition The thesis of COMPOSITIONALITY of meaning is that the meaning of any
expression is a function of the meanings of the parts of which it is composed.
Comment The truth tables we have given for & and V are in fact functions of the kind
mentioned in this deﬁnition.
Summary The logical connectives & (corresponding to English and and but) and
V (roughly English or) are used to form complex propositional formulae
by connecting simple propositional formulae. Rules of inference can be
given involving these connectives, and they can be deﬁned by means of
truth tables.
Unit 14 Study Guide and Exercises
Directions After you have read Unit 14 you should be able to tackle the following
questions to test your understanding of the main ideas raised in the unit.
UNIT 14 Connectives: and and or
177
I R A N
PA
KI
ST
A
N
I N D I A
Tehran
Karachi
Mumbai
Chennai
N E P A L
Kabul
Tashkent
UZBEKISTAN
A
FG
HA
NI
ST
AN
TIBET
PART FOUR Logic
178
1 You should understand these terms and concepts from this unit:
logical conjunction (with &)
logical disjunction (with V)
rules of inference
commutativity of conjunction
commutativity of disjunction
compositionality of meaning
truth tables
2 Translate the following into logical notation using either & or V (or both as
required).
a Fred arrived and Mary left
b Either Fred is a Republican or Mary is a Democrat
c Max and Erma are happy
d Bill visited Jim and Mary
e Max saw Mary or Fred
f Jim briefed Jane and he saw Mary or he met Fred
(disambiguate by means of diﬀerent bracketing)
g Either Mary briefed Fred and she saw Jane or she saw David
(is this ambiguous?)
h Mary briefed Fred and either she saw Jane or she saw David
(is this ambiguous?)
3 What can be connected with the logical connectives```