Introduction to Logic: Reasoning, Validity, and Formal Language
1. Logic and Its Object
Reasoning
Reasoning is the process that allows us to obtain new knowledge from existing knowledge. For example: “Today is a magnificent day,” “The grass is wet” -> “Today is a magnificent day and the grass is wet” -> therefore, new knowledge “Someone watered the grass.”
Relationship Between Reasoning and Conclusion
Reasoning is the relationship between propositions that leads to a conclusion.
Logic
The grass could be wet for another reason: “My brothers had a water balloon fight.” Logic deals with secure data. To guarantee the truth of the conclusion, we must ensure that the way we relate this data is appropriate—that is, logically correct. Logic is considered the philosophical discipline that studies the correctness or validity of reasoning.
Inference (Reasoning)
Any inference (reasoning) consists of:
- Premises: A set of statements that express the data from which we start. Example: The thief of my cheese is a cat or a mouse. / The footprints show it’s not a mouse.
- Conclusion: The final statement that expresses the new information derived from the premises. Example: The thief of my cheese is a cat.
- Deduction: Transferring from general premises to a less general conclusion. When this type of inference is correct, the conclusion is necessarily derived from the premises: if the premises are true, the conclusion cannot be false.
- Induction: Reaching a general conclusion from less general information given in the premises.
In deduction, the conclusion derives necessarily from the premises. In induction, we can only talk about a certain probability because the truth of the premises does not ensure the truth of the final conclusion.
The Validity of Reasoning
We are not talking about real reasoning, but correct or valid reasoning. Reasoning cannot be true or false, since it does not affirm or deny anything. The correctness of our reasoning is an important requirement for reaching true conclusions. However, it is not sufficient. To be sure of the truth of the conclusion, two conditions must be met: the correctness of the reasoning and the truth of the premises.
Logic deals exclusively with the validity of reasoning. The truth of the premises falls under other disciplines.
Differences Between Truth and Validity
The premises and the conclusion can be either true or false. Reasoning is not true or false, but valid or invalid. Reasoning is valid if the premises relate correctly to the conclusion. A premise is true if it corresponds to reality.
2. Informal Logic
Both informal and formal logic aim to study the validity of our reasoning. However, they approach it from different points of view.
- Formal logic focuses exclusively on determining whether arguments are well-constructed. It analyzes the relationships between the premises and the conclusion (the structure of the reasoning) and does not need to consider the meaning of the premises and conclusion.
- Informal logic deals with factors that have nothing to do with form. To determine the validity of reasoning, it focuses on issues external to the structure: whether the premises are adequate, whether the data justifies the conclusion, etc.—non-formal matters.
Fallacies
The study of fallacies (invalid reasoning) is a major subject in informal logic.
- Formal fallacies: Studied by formal logic, these are the consequence of violating some formal rule.
- Informal fallacies: Studied by informal logic, these do not depend on formal aspects, but on content-related issues, meaning, etc.
3. The Language of Formal Logic
Natural Language
The language we use daily is the result of changes over time. No natural language has been consciously created. They contain many ambiguous terms and syntactic rules with exceptions. This causes current languages to be full of inaccuracies and ambiguities.
Formal Language
The languages of logic and mathematics are artificial because they have been consciously designed to resolve the ambiguity and imprecision of natural language. These languages are also formal (which is what differentiates the language of logic from other artificial languages like Morse code) because they have a precise and rigorous formal system: both a vocabulary and rules for correctly forming phrases. The symbols in its vocabulary are meaningless (p, A, T, r…). Unlike the vocabulary of natural language, they say nothing about the world.
4. The Logic of Statements
- Atomic or simple statements: Cannot be broken down into other statements. They can be decomposed into subject and predicate, but contain no other statements.
- Molecular or complex statements: Can be decomposed into simple statements.
Symbols of Logic
Logical Symbols
- Variables: Lowercase letters (p, q, r, s, t) replace statements. They are called variables because the specific statements they replace can vary from one reasoning to another. Substituting different statements for p and q results in a different reasoning.
- Auxiliary Symbols: Brackets and parentheses are used to make complex statements easier to read and understand. Thanks to these, we can know the dominant relationship and interpret statements that could be understood in multiple ways. General rule: The main relationship is the one outside the parentheses, and if there are nested parentheses, the outermost one is the main one.
Logical Signs
These particles allow us to form molecular statements from simple statements:
- Negation (¬): Used to deny any statement. It corresponds to “not” in natural language.
Example: “It is not raining” = ¬p - Connectives: Used to join or connect simple statements to form molecular statements. They are equivalent to connectives in natural language. There are four types:
- Conjunction (&): “and”
- Disjunction (∨): “or”
- Conditional (→): “if…then”
- Biconditional (↔): “if and only if”