Formal and Informal Logic: A Deep Dive
Abstract Philosophy: Formal and Informal Logic
Logic and Its Object
Definition: Logic is the knowledge or science that aims at formal reasoning (studying the correctness or validity of reasoning).
Reasoning: The process that allows us to obtain new knowledge from existing knowledge.
Reasoning or Inference
These are processes by which we obtain information from known data. Any inference consists of:
- Premises: A set of statements that express the data from which we start.
- Conclusion: A final statement that expresses the new information derived from the premises.
Types of Reasoning:
- Deduction: Moving from general premises to a less general conclusion. When such an inference is correct, the conclusion is necessarily derived from the premises: it is impossible for the conclusion to be false if the premises are true.
- Induction: Reaching a general conclusion from premises that provide less general information. We can only speak of a probability that the conclusion is true because the truth of the premises does not assure that the final conclusion is true.
Formal Logic
Types of Formal Logic:
- Propositional Logic: Studies the formal validity of arguments, taking into account only the truth value of each statement. It considers statements as a whole and does not internally analyze the subject and predicate. This involves some limitations, especially in determining the validity of those arguments which cannot be determined without analyzing the statements that compose it.
- Predicate Logic: Analyzes the internal structure of statements, because the propositions are considered in which a property (predicate) is attributed to the subject.
- Class Logic: Believes that statements are propositions that express ties between individuals and classes. Predicates are analyzed as properties belonging to individuals who share the same class or series.
- Logic of Relations: Incorporates elements into the language, symbols, and rules that are needed to express relationships.
The Language of Logic
Natural Language: What we use in our daily conversations.
The language of logic is artificial; it has been consciously designed to resolve the ambiguity and imprecision of natural language. It is also a formal language, so everything is accurately defined and rigorous.
Elements of Logical Language
- Vocabulary:
- Letters represent the statements, names, or predicates of reasoning.
- Symbols are used to represent relationships between terms and statements.
- Rules of Formation: Establish which combinations of symbols are well-formed sentences (formulas of logical language).
- Rules of Transformation: Show how we can transform a well-formed formula into another well-formed formula.
All these symbols and rules are well-defined; this property of language is called precision.
Formal Systems of Logic
- Consistency: No contradiction exists within the system because, from the transformation rules, it is not possible to deduce a formula and its opposite.
- Completeness: All correct formulas are deductible from the defined transformation rules.
- Decidability: The system has a mechanical procedure that allows us to decide whether a formula or reasoning is correct or not.
Limitations of Logic
Statement: A linguistic expression that can be either true or false.
- Simple or Atomic Statements: Cannot be composed of other statements.
- Complex or Molecular Statements: Can be decomposed into simple statements.
Logical Symbols
- Non-Logical:
- Variables: Lowercase letters.
- Auxiliaries: Parentheses and brackets.
- Logical: Allow us to form complex statements from simple statements.
Negation (¬): Serves to deny any statement.
Connectives: Serve to unite or connect simple statements. There are four types:
- Conjunction: ^ (and)
- Disjunction: V (or)
- Conditional: -> (then)
- Biconditional: <-> (if and only if)
Inference Rules
These can be presented in two ways:
- Inference Outline: A schematic representation of reasoning. A-> B / A = B
- Logical Law: Representation of a statement. It is a conditional that is always true; the law is a logical tautology. [(A-> B) ^ A] -> B