Basic concepts of formal logic

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What do you need to know about formal logic?

Statement, Proposition, Judgment

Statement: any sentence in natural or formal language that can be true or false.
Example: “It is raining now”.
Proposition: It is the minimum unit of logical reasoning, an affirmation that has a truth value (T or F). It is what remains the same even if the language or the way of expressing it changes.

Example: “It is raining”, “Está lloviendo”, and “Il pleut”. They are three different statements, but they express the same proposition.

In formal logic we normally work with propositions (represented by letters p, q, r…).

Judgment: Mental act by which something is affirmed or denied; it is the psychological operation that gives rise to the proposition.

Premise and Argument

A premise is a proposition or statement that serves as a basis for establishing, justifying, or inferring a conclusion within an argument.

An argument is a structured set of propositions where one or more propositions, called premises, are offered as reasons to support another proposition, called the conclusion. The premises try to give reasons for accepting the conclusion.

Basic structure	Example
Premise 1	1. All mammals have a heart.
Premise 2	2. Dogs are mammals.
...
Conclusion	3. → Dogs have a heart.

Inference

Inference is the mental process or the logical act of deriving a conclusion from one or more premises. It is the step from the premises to the conclusion within an argument.

Types of Reasoning

These are the main types of inference or logical reasoning:

Type of Reasoning	Direction of the Inference	Key Property	Example
Deduction	From the general to the particular	-> If the premises are true, the conclusion is necessarily true (Valid arguments). Level of Certainty: Offers logical certainty.	All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
Induction	From the particular to the general	-> The premises only make the conclusion probable, but do not guarantee it (Strong arguments). Level of Certainty: Offers probable conclusions, not necessary ones. -> It is the type of reasoning used in science.	All the swans I have seen are white. Therefore, all swans are white. (until a black one was discovered)
Abduction	From effects to possible causes	Level of Certainty: Hypothetical (Best Explanation)	A hypothesis or the most probable explanation for a set of observations is formulated. It is the reasoning used for discovery.
Analogical	From the particular to the particular	Level of Certainty: Suggestive	One concludes that a case is similar to another in certain respects because they are similar in other known respects. The strength depends on the relevance of the similarity.

Truth and Validity

Truth

It is a property of propositions; a proposition is true if it corresponds with reality.

Validity

An argument is valid when if the premises were true, then the conclusion would necessarily also be true. It does not matter whether the premises are in fact true or false: what matters is the logical form.

Validity depends solely on the structure of the argument, not on its content.

If the argument has a form such that it is impossible for all the premises to be true and the conclusion false at the same time, then it is valid.
If there exists at least one case where the premises can be true and the conclusion false, then it is invalid.

Examples of valid arguments

Examples: Valid syllogism

Example	Structure
Premise 1: All dogs are mammals.	All A are B.
Premise 2: Rocky is a dog.	C is A.
Conclusion: Rocky is a mammal.	Therefore, C is B.

In this case, there is NO way for the premises to be true and the conclusion false → valid.

Examples: Modus Ponens

Example	Structure
Premise 1: If I study, I pass.	If P then Q.
Premise 2: I study.	P.
Conclusion: I pass.	Therefore Q.

Absolutely valid form.

Examples of invalid arguments

Fallacy of affirming the consequent

Example	Structure
Premise 1: If I study, I pass.	If P then Q.
Premise 2: I passed.	Q.
Conclusion: I studied.	Therefore P.

Here there exists a case where P is false and Q is true (one passed by cheating). That is enough to make it invalid.

Badly structured syllogism

Example	Structure
Premise 1: All cats are animals.	All A are B.
Premise 2: All dogs are animals.	All C are B.
Conclusion: All dogs are cats.	All C are A.

The premises can be true and the conclusion false → invalid.

How to detect validity

Always ask yourself:

Is there any possible situation where the premises are true and the conclusion false?

If none exists → the argument is valid.
If even a single one exists → the argument is invalid.

This criterion is the heart of formal logic.

Validity = impeccable structure.
Truth = correct content.
An argument can be:
- valid and with false premises,
- invalid and with true premises.

Only valid arguments can guarantee true conclusions when the premises are also true.

Form vs. Content

Form (Logical Structure):
- It refers to the structure or mold of the argument, the way in which the terms and propositions are related.
- The validity of an argument depends solely on its form, not on the meaning of the specific words.
- Example of form: If P, then Q. P. Therefore, Q. (This is a valid form called Modus Ponens).
Content (Logical Matter):
- It refers to the meaning or the real facts that the propositions deal with (the values of P and Q).
- Content is what determines whether a proposition is true or false.

In summary: Formal logic is concerned with the validity of the form, while truth is concerned with the content of the propositions.

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Concepts of Informal Logic

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Last updated: 2025-12-10

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