Classnotes
08/23/01
08/28/01 08/30/01
09/04/01
Syllabus handout. Click here for details.
Logic is a study concerned with the evaluation of reasoning. In reasoning, we attempt to go beyond what we already know or believe to establish some further proposition that is justified by what we already know. Reasoning is a mental activity and thus can be a private activity. When we make our reasoning public, we do so in the form of arguments.
An informal definition of ARGUMENT:
An ARGUMENT =def a public manifestation or communication of reasoning.
Origin of the study of logic:
Aristotle (384-324 bce) developed a system of rules for scientific reasoning
(the Syllogistic).
Euclid (300 bce) wrote the Elements of Geometry in which theorems
(solutions to specific geometrical problems) are proved by logical deduction
from basic axioms or postulates. Euclid employed no specific
logical principles.
In the nineteenth century, mathematicians developed approaches to logic
that more adequately characterized the principles of mathematical reasoning.
George Boole and Gottlob Frege (1848-1925).
A formal definition:
An ARGUMENT =def a sequence of sentences one of which (the conclusion) is taken to be supported by the remaining sentences (the premises).
Some comments on this definition:
(i) A sentence is a linguistic item, an expression used to communicate a piece of information. Thus, we use sentences to communicate our arguments.
(ii) We actually have three definitions – ‘argument’, ‘conclusion’, and ‘premises’.
(iii) What makes a sequence of sentences an argument is the intention of the communicator.
(iv) Synonyms for ‘be supported by’ are ‘be justified by’, ‘follow from’,
‘be a consequence of’. These identify the same logical relationship
between premises and conclusion.
Logical properties and relations:
Logic is concerned with general truth properties of and truth relations between sentences.
An argument is DEDUCTIVELY VALID if and only if it is not possible for the premises to be true and the conclusion at the same time to be false.
This states a truth relation between those sentences that are the premises and that which is the conclusion.
A set of sentences is LOGICALLY CONSISTENT if and only if it is possible for all of the members of the set to be true (at the same time.)
This states a truth relation that holds among these sentences.
The members of a pair (or more generally any set) of sentences are LOGICALLY EQUIVALENT if and only if it is not possible for one of the sentences to be true while the other is (any of the others are) false.
Again, this states a truth relation that holds among these sentences.
A sentence is LOGICALLY TRUE if and only if it is not possible for the sentence to be false.
A sentence is LOGICALLY FALSE if and only if it is not possible for the sentence to be true.
A sentence is LOGICALLY INDETERMINATE if and only if it is neither logically true nor logically false.
Each of these states a general truth property of a sentence.
The six main logical properties and relations:
These six properties and relations defined above give us the subject matter of logic. Logic is concerned with evaluating sentences, sets of sentences and arguments for these properties.
Note that each definition depends on the concepts of truth (and falsity)
and possibility. This notion of possibility is very vague. Successful attempts
at logical evaluation depend upon giving a precise content to the notion
of logical possibility. That will be our business.
Logicians evaluate sentences and sets of sentences for the main logical
properties and relations by developing formal systems, which act as models
of reasoning. A model abstracts from every feature of what is modeled that
is irrelevant to what is being examined. So a formal system is intended
to represent only the logical features of communication.
Formal systems:
A formal system consists necessarily in two components, and a third may be added.
(1) A formal language consisting of
(b) grammar (a set of rules for well-formed expressions.)
(3) A semantic scheme (a protocol for interpreting or assigning meanings to the expressions of the language.)
Note: a scheme of interpretation is not an interpretation, but only
a way of providing one. An interpreted formal system is sometimes called
a formal theory. One view of scientific theories is that they are
interpreted formal systems.
Sentential logic:
The first logical system we will develop is the sentential logic. Sentential logicians recognize the importance of sentences in the evaluation of the logical properties and relations. They also notice some significant differences and relationships between natural language sentences.
Take these sentences:
(1) Rover is a dog.
(2) Rover is a dog and he likes cookies.
(3) Either Rover is a dog or he is a wombat.
(1) differs from (2) and (3) in that it is simple whereas they are compound. (2) and (3) are made up from other sentences by the help of connective expressions (we’ll call them ‘sentential connectives’.) Also, the conditions under which a sentence like (2) will be true are different from the conditions under which (3) is true.
SENTENTIAL CONNECTIVE: An expression used to make compound sentences from other sentences.
A sentential connective is used TRUTH-FUNCTIONALLY if and only if the compound sentence it generates has a truth-value that is dependent on the truth-values of its components.
Sentential logic recognizes two truth-values – TRUE and FALSE. It is
called a bivalent (or classical) logic.
SL-the language of sentential logic:
Sentential logic employs a language that represents sentences and truth-functional connectives as logically significant. It recognizes five significant truth functions. They are:
DISJUNCTION (symbol Ú): the truth-functional connective that takes two sentences and forms a compound that is false when both components are false, and true otherwise.
CONDITIONAL (symbol É): the truth-functional connective that takes two sentences and forms a compound that is false when the antecedent (the sentence prior to the connective) is true and the consequent (the sentence after the connective) is false, and true otherwise.
BICONDITIONAL (symbol º): the truth-functional connective that takes two sentences and forms a compound that is true whenever the components share truth-values, and false when their truth-values are distinct.
NEGATION (symbol ~): the truth-functional connective that takes ONE sentence and forms a compound whose truth-value is the opposite of the truth-value of the component sentence.
For a summary of these functions, see the truth tables on pages 67-8.
SL employs roman capital letters to represent the logical function of individual (simple) sentences.
Parentheses:
SL employs parentheses for punctuating complex expressions.
Their use is largely for disambiguating expressions.
Common English expressions modeled by the SL connectives:
Conjunction: 'and', 'but', 'however', 'although', 'additionally', 'also', 'even though'.
Disjunction: 'or', 'either...or...', 'unless'.
Conditional: 'if...then...', 'so long as', 'on the condition that', 'only if' (but beware order!)
Biconditional: 'if and only if', 'just in case', 'is a necessary and sufficient condition for'.
Negation: 'not', 'it's not the case that', 'it's not true that', 'isn't'