This is the first topic of the propositional calculus course, in this opportunity, we will study what are the statements and their properties. This concept has different definitions in different areas of study and even in linguistics, it is far from what we understand by a logical statement in mathematics.
Although this concept in propositional calculus (informal) is studied from different angles, the concept turns out to be the same. Now let’s study this concept in more depth.
What is a statement in mathematics?
We understand by statement both in logic and in mathematics whose statement can be true or false, but not both at once.
In propositional calculus, the only thing that matters is the truth values of a statement (Also called proposition). The arguments we will use in this section will only serve as an example to understand this concept.
Under this limitation they are usually represented by lowercase letters and are called propositional variables, then we will see these points later. Here are some examples of statements:
Examples
The following sentences are statements:
- The earth is not flat (true).
- Dogs have a beak instead of a muzzle (false).
- Notebooks are good for writing (true).
- \( 2+2=5 \) (false).
- \( 4^3 = 64 \) (true).
The truth value of a statement (the only thing that matters)
Note that grammatically written arguments regarding the following examples do not express or represent anything for mathematics unless these arguments for these statements are correctly defined, so our interpretation is subjective for mathematics.
In propositional calculus (but not everything in mathematical logic) it is only interested in the truth values of a statement than in the structure of the argument itself. We will clarify this point right after the following example.
Examples
The following examples explain which sentences are or are not statements:
- Every dog has two ears (true).
- A beetle is a donkey (False).
- I’m excited! (not a statement).
- That person is a woman (not a statement).
Explanation of each statement.
- The first example is based on our visual experience, we can verify that every dog has two ears resulting in a true statement.
- We know from our experience that a beetle is not a donkey, the proposition here is false.
- An exclamation does not indicate that something is true or false, which implies that the sentence is not a statement.
- This sentence indicates that a person can be a woman, but also not be, that is, we cannot say that it is true or false, so it is not a statement. These types of sentences are called propositional functions or predicates because they have unknown variables and are widely used in first-order logic.
Mathematics does not know or recognize if this sentence “Every dog has two ears” is true or false, it is not defined in its vocabulary “dog” and “ears”, in fact, each word of the sentence is not defined by mathematics unless a previous definition is made and the properties of each part of the sentence and even the order of each word of the sentence.
The propositional calculus is only limited to extracting the truth values regardless of the arguments since they are nothing more than simple subjective interpretations for mathematics.
There is a branch of mathematical logic that can study the structure of statements but it is a subject that is beyond the scope of this course.
Logical connectives
If we want to study the types of sentences correctly, we must first understand how these statements are formed, the first most important symbols in all mathematical logic, specifically in propositional logic are the logical connectives, for this reason, we are going to list right now in the following list:
| Conectivo | Palabra | Símbolo |
|---|---|---|
| Negation | not (and all its variations) | \( \sim \) |
| Conjunction | and (and all its variations) | \( \wedge \) |
| Disjunction (Inclusive) | or (and all its variations) | \( \vee \) |
| Exclusive disjunction | either … or (and all its variations) | \( \bigtriangleup \) |
| Conditional | If … then (and all its variations) | \( \rightarrow \) |
| Biconditional | …. if and only if (and all its variations) | \( \leftrightarrow \) |
Colloquially it is used “or” in an exclusive sense in the English language, however, here in mathematics it is usual to use it in an inclusive sense, and in the case of an exclusive disjunction we will use “either … or”,
In my native language, Spanish is used inversely, that is, we use “or” in an inclusive sense, and in this way we avoid explanations, and for the exclusive case, “or … or” is used, these would be some differences between Spanish and English.
Let’s see some examples:
Examples
The following sentences are statements made up of logical connectives:
- Denial of a statement.
\( p \): Dogs have tails.
\( \sim p \): Dogs do not have a tail. - Conjunction of a statement.
\( p \wedge q \): Dogs and cats have tails. - Disjunction of a declaration.
\( p \vee q \): Spiders or ants have legs. - Exclusive disjunction of a statement.
\( p \bigtriangleup q \): A light bulb is either on or it is off. - Conditional logic of a statement.
\( p \rightarrow q \): if the beetle is an insect, then it is not human. - Logical biconditional of a proposition.
\( p \leftrightarrow q \): It’s cold if and only if the temperature drops.
These logical connectives help us summarize the different types of declarative sentences that we present below.
Types of declarative sentences
There are two types of declarative sentences in the area of mathematical logic and they are the open sentence and the statements, the latter can be divided into two types or classes of statements and they are the simple and compound statements. Let’s see each one of them.
Open sentence
Open sentences also called propositional functions are those statements that do not have data when something is true or false, that is, they affirm or deny without knowing whether it is true or false.
Although there are other special words (with their own mathematical symbol) that, when added to open sentences, are transformed into statements, we will see this later in a topic called quantifiers.
Examples
The following examples are open sentences:
- She has a blue polo shirt.
- \( x>4 \).
- The house has two doors.
- \( y^2 + z^2 <5 \).
The word “She” that happens to be the subject of the sentence is a variable, also with the letters “\( x \)”, \( y \), \( z \) and “home” are variable. Since we do not know about its values, it implies that the previous sentences are open sentences.
Simple or atomic statement
It is that statement that is not formed by any logical connective.
Examples
- The sheet of paper is flat
- \( 5 > 3 \)
Compound or molecular statement
It is that statement that is formed by at least one logical connective.
Examples
- Los humanos no tiene cola y ni patas.
- 3 y 6 son divisibles por 3.
Relationship of a statement and its truth value
Let 4 statements \(p \), \(q \), \(r \) and \(s \) represent the set of statements \( \mathrm {P} \) where they can be true \( V \) or false \( F \) that represents the set of truth values \( \mathrm{V} \), then there is a relationship between statement and truth value as shown in the following diagram:
Based on this diagram, we can say that there are a certain number of statement \( \mathrm{P} = \{ p_{1}, \ p_{2}, \cdots p_{i}, \cdots p_{n} \} \) and sets of truth values \( \mathrm{V} = \{ V, \ F \} \), symbolically it is represented like this:
\[ f( p_{i} ) = \left \{ \begin{array}{ l } V, \ \text{if} \ p_{i} \ \text{is true} \\ F, \ \text{if} \ p_{i} \ \text{is false} \end{array} \right. \]
Where \(i \) represents positive integers. This is nothing more than a condition that the set of statements must fulfill according to the property \(f \), that is, that of being true or false.
Mathematical representation of propositions
The statements can be represented mathematically, but we must identify the parts of a statement, among them we have, the propositional variables, the logical connectives, and the grouping signs, let’s see each one of them:
Propositional variables
They are those statements represented by lowercase letters, generally by the letters \(p \), \(q \), \(r \), however, it can be represented with any lowercase letter as long as it is indicated that it is a propositional variable.
Propositional variables can help create more complex statements, the only drawback is that these variables do not identify if it turns out to be a compound or simple statement, we just don’t know, although in propositional logic it is not something that really matters, it is only used as a mention theoretical.
Returning to the logical connectives
The logical connectives give us more information about a statement, in fact, it enriches it since it connects the information of the propositional variables between them, expanding its meaning and also its structure.
We already indicated a table of truth values above, based on that, a new complex statement can be created.
Examples
Let the statements be \( p \) y \( q \):
- \( p \vee q \)
- \( q \wedge p \)
- \( p \rightarrow q \)
- \( \sim q \)
- \( p \leftrightarrow q \)
- \( p \bigtriangleup q \)
Grouping symbols
The grouping symbols help us to connect statements formed in turn by propositional variables and logical connectives, without these, a proposition would have a different meaning, up to be sometimes true and false at the same time if it is operated incorrectly, and it is what it wants to avoid.
The most used are the parentheses “()”, the brackets “[]” and the curly braces “{}”, these symbols help us avoid falling into ambiguities and maintain the meaning as well as the truth value of these declarations formed by the grouping symbols.
Examples
Let the statement \( p \vee q \rightarrow r \), using the symbols, it would be:
- \( ( p \vee q ) \rightarrow r \)
- \( p \vee ( q \rightarrow r ) \)
These two statements have not only two different meanings, but also different truth values, but we will see this later when we study the truth table of a statement.
Molecular schemes
Molecular schemes in propositional logic in the English language do not exist, but in Spanish, they do (my native language), it is really nothing more than just a compound statement.
However, a compound statement can also refer to a declarative grammar sentence as well as a logical formula such as “\( p \vee ( q \rightarrow r ) \)”.
To differentiate between a sentence and a formula, we will call these formulas molecular schemes, very common in the Spanish language.
Having clarified this point, we will say that a molecular scheme is the mathematical representation of the statements formed by the propositional variables, logical connectives and sometimes the grouping symbols.
Examples
Let 3 propositional variables \(p \), \(q \), \(r \), some examples of molecular schemes are:
- \( ( p \vee q ) \rightarrow r \)
- \( r \wedge [ \sim q \vee p ] \)
- \( \sim q \leftrightarrow { p \bigtriangleup ( r \rightarrow \sim q ) } \)
Equivalent statements
They are those statements where they can be altered such that they do not have the same logical connectives or have logical connectives and propositional variables ordered differently but with the same information and therefore, with the same truth value.
Examples
- Let the statement be “dogs bark”, an equivalent is:
- It is not true that dogs do not bark.
These two statements are equivalent and are symbolically represented like this:
\[ p \equiv \sim ( \sim p ) \]
In this case, the variable \(p \) is “dogs bark”.
- Let the statement be “cats and dogs are four-legged animals”, an equivalent is:
- Dogs and cats are four-legged animals.
Symbolically it is represented like this:
\[ p \wedge q \equiv q \wedge p \]
Here the variable \( p \) is “dogs are four-legged animals” and \( q \) is “cats are four-legged animals”, that is, the equivalent statements speak or express the same thing and therefore, their respective truth values as well.
Thus we end the first section, the next session is dedicated to the study of logical negation.