Introduction to Formal Logic - Critical Thinking

 

Lecture 10

May 20, 2007

 

I. Definitions

1. Formal Logic

Formal logic is the science which studies valid patterns of inference.

 

 

2. Propositional or Sentential Logic

Sentential logic is the simplest logic that is strong enough to capture a large portion of valid arguments.

 

Also known as SL.

 

3. Consistency

A group or set of statements is consistent if it is possible that all the statements could be true.

 

4. Inconsistency

A group of statements is inconsistent if it is not possible for all the statements to be true.

 

5. Validity

An argument is deductively valid if there exists no case such that:

 

a. All the premises are true.

b. The conclusion is not true.

 

6. Logical Truth

A statement is logically true if, and only if, it is not possible for it to be false.  ie. tautologies.

 

7. Logical Falsity

A statement is logically false if it is not possible for it to be true.  Thus is it logically impossible.

 

8. Logical Equivilence

Two statements are logically equivalent if it is not possible for them to have two different truth values.

 

9. Contingent

A statement that is neither logically true or logically false.

 

 

II. Other Considerations

 

1. Valid arguments may have false premises.

2. Valid arguments may have false conclusions.

3. Arguments may be invalid and have both true premises and true conclusions.

4. If the conclusion of an argument is logically true, the argument is valid regardless of the premises used.

5. A consistent set of premises MAY contain false statements.  For consistency you just need possibility that they are all true.

6. There are some arguments that even some of the best minds cannot determine if they are valid.

 

 

III. Preliminaries for Sentential Logic (SL)

Sentential logic considers arguments whose validity depends on the way complex sentences are composed out of simpler, declarative sentences using grammatical devices known as sentential connectives.

 

Remember when you standardize any argument, you should reword the sentences to use the same words when talking about the same concepts.

 

Sentential variables should use capital letters.

 

Some sentential connectives like "and" and "but" are importantly different for some purposes, but when used in arguments are considered the same.

 

IV. Translating English into SL

 

Our sentential logic or sentential language will include sentence letters (capitalized) and connectives like "&".

 

1. Translating Using &:  (AND/BUT)

"&" is used to join two statements together, both statements are then considered true.  The following words and phrases are typically translated using "&":

 

a. and

b. but

c. although

d. even though

e. however

f. in spite of

g. despite

h. yet

 

2. Translating Using v: (EITHER/OR)

"v" is used for when at least one of the two joined statements is true.  It is also used when exactly one of the two statements is true, this depends on the context of the situation.

 

3. Translating Using ¬: (NEGATION)

"¬"  in my web editor was called 'not sign'.

 

S: Bob is smart.

 

¬S (means Bob is not smart.)

 

B: All dogs bark.

 

¬B (not all dogs bark, at least one does not.)

 

Remember to use parentheses to show exactly what is being negated.

 

¬A & B does not equal ¬(A & B)

 

This should be easy for anyone who is a programmer.

 

 

4. Translating Using =>: (IF/THEN - IMPLICATION)

Symbolizes A implies B.

 

A => B, antecedent => consequent

 

Does not necessarily mean causation, only correlation.

 

One good use of the => symbol for for "if A, then B" relationships.

 

 

V. Necessary and Sufficient Conditions

The distinction between necessary and sufficient conditions is a very important one to make.  Failing to take into account their differences can mean you will make common errors in thinking.

 

1. Sufficient Conditions.

A sufficient condition A allows you to infer the thing B that it is a sufficient condition for.  If A, Then B.

 

eg. Striking a match in a room full of gasoline fumes is a sufficient cause for an explosion.

 

Sufficient conditions are not always necessary

 

2. Necessary Conditions

A necessary condition A is a condition that is required for a thing B to be true.  If not A, then not B.

 

Necessary conditions do not need to be sufficient conditions.

 

The lectures about scientific thinking and causation will look into these factors more.

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