Semantics
Sai Gon University
tvanh@sgu.edu.vn

Unit: Logic
Lesson 12: ABOUT LOGIC
Logic is a word that means many things to different
people. Many everyday uses of the words logic and
logical could be replaced by expressions such as
reasonable behaviour and reasonable. You may say,
for instance, 'Sue acted quite logically in locking
her door', meaning that Sue had good, well
thought- out reasons for doing what she did. We
shall use the words logic and logical in a narrower
sense, familiar to semanticists. We give a partial
definition of our sense of logic below.

LOGIC deals with meanings in a language system, not
with actual behaviour of any sort. Logic deals most
centrally with PROPOSITIONS. The terms 'logic' and
'logical' do not apply directly to UTTERANCES (which
are instances of behaviour).
MODUS PONENS is a rule stating that if a proposition P
entails a proposition Q, and P is true, then Q is true. Put
in the form of a diagram, Modus Ponens looks like this:
• MODUS PONENS is a rule stating that if a proposition P
entails a proposition Q, and P is true, then Q is true. Put
in the form of a diagram, Modus Ponens looks like this:
P1Q
P
Q

Logic deals with meanings in a language system
(i.e. with propositions, etc.), not with actual
behaviour, although logical calculations are an
ingredient of any rational behaviour. A system for
describing logical thinking contains a notation for
representing propositions unambiguously and
rules of inference defining how propositions go
together to make up valid arguments.
Because logic deals with such very basic aspects
of thought and reasoning, it can sometimes seem
as if it is 'stating the obvious'.

The thing to remember is that one is not, in the end,
interested in individual particular examples of correct logical
argument (for, taken individually, such examples are usually
very obvious and trivial), but rather in describing the whole
system of logical inference, i.e. one is trying to build up a
comprehensive account of all logical reasoning, from which
the facts about the individual examples will follow
automatically. One only looks at individual examples in
order to check that the descriptive system that one is
building does indeed match the facts.
Logic, with its emphasis on absolute precision, has a
fascination for students who enjoy a mental discipline. Thus,
in addition to its contribution to our understanding of the
'Laws of Thought', it can be good fun.

Lesson 13: A NOTATION FOR SIMPLE
PROPOSITIONS
Logic provides a notation for unambiguously
representing the essentials of propositions.
Logic has in fact been extremely selective in the
parts of language it has dealt with; but the parts
it has dealt with it has treated in great depth.

Every SIMPLE proposition is representable by a
single PREDICATOR, drawn from the predicates
in the language, and a number of ARGUMENTS,
drawn from the names in the language. This
implies, among other things, that no formula for
a simple proposition can have TWO (or more)
predicators, and it cannot have anything which
is neither a predicate nor a name.

Ex: j LOVE m is a well-formed formula for a
simple proposition j m is not a well-formed
formula, because it contains no predicator j
IDOLIZE ADORE m is not a well-formed
formula for a simple proposition, because it
contains two predicators j and h LOVE m is not
a well-formed formula for a simple
proposition because it contains something
('and') which is neither a predicator nor a
name

We have presented a logical notation for simple
propositions. A well-formed formula for a simple
proposition contains a single predicator, drawn from
the predicates in the language, and a number of
arguments, drawn from the names in the language.
The notation we have given contains no elements
corresponding to articles such as a and the, certain
prepositions, and certain instances of the verb be,
as these make no contribution to the truth
conditions of the sentences containing them. We
have also, for convenience only, omitted any
representation of tense in our logical formulae.

The introduction of a notation for propositions
(to be refined in subsequent units) fills a gap
left empty since Unit 2, where we introduced
a way of representing sentences and
utterances, but not propositions. We now
have (the beginnings of) a way of representing
items at all three levels:
Utterance
Sentence
Proposition
'Jesus wept' Jesus wept
j WEEP

Lesson 14: CONNECTIVES: AND AND OR
The English words and and or correspond
(roughly) to logical connectives. Connectives
provide a way of joining simple propositions to
form complex propositions. A logical analysis
must state exactly how joining propositions by
means of a connective affects the truth of the
complex propositions so formed. We start
with the connective corresponding to and,
firstly introducing a notation for complex
propositions formed with this connective.

Any number of individual well formed formulae can
be placed in a sequence with the symbol & between
each adjacent pair in the sequence: the result is a
complex well formed formula.
Ex: Take the three simple formulae:
c COME g
(Caesar came to Gaul)
c SEE g
(Caesar saw Gaul)
c CONQUER g
(Caesar conquered
Gaul) From these, a single complex formula can be
formed:
(c COME g) & (c SEE g) & (c CONQUER g)

Commutativity of conjunction:
p&q
(premiss)
q&p
(conclusion)
Any number of wellformed formulae can be placed in
a sequence with the symbol V between each adjacent
pair in the sequence: the result is a complex
wellformed formula.
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.

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.

MORE CONNECTIVES
Unit 14: introduced connectives of conjunction
and disjunction. In this unit you will meet three
more connectives: implication 1, equivalence 6
and negation ~.
Rule p 6 q is equivalent to (p 1 q) & (q 1 p)

The logical negation operator ~ corresponds
fairly closely with English not or n't in meaning,
and can be defined both by truth table and by
rules of inference. The logical connectives 1
(conditional) and 6 (biconditional) cannot be
defined by truth table in any way which closely
reflects the meanings of English if . . . then and
if and only if. However, rules of inference can
be given for them which fairly accurately
reflect valid inferences in English involving
if . . . then and if and only if.

Now that you are familiar with these connectives,
the conjunction and disjunction connectives of
the previous unit, and the negation operator, you
have met all the formal apparatus that together
forms the system known as 'propositional logic',
or 'propositional calculus'. This branch of Logic
deals with the ways in which propositions can be
connected (and negated) and the effect which
these operations (of connection and negation)
have in terms of truth and falsehood. This
establishes a solid foundation for more advanced
work in logic.

GOOD LUCK!