- What are simple and compound propositions?
- Simple propositions
- Compound propositions
- Other types of propositions
- Truth value of a proposition
- Proposition and prayer
We explain what simple and compound propositions are, the characteristics of each one and their differences with a sentence.
What are simple and compound propositions?
In logic Y math, propositions are sentences or statements that can be given a true or false value, as the case may be, and that express a logical relationship of some kind between a subject (S) and a predicate (P). Propositions are related to each other through judgments, and are the basis of the deductive and inductive system of formal logic.
Now, a first classification of propositions offers two fundamental types of proposition, taking into account their internal structure:
- Simple propositions. Or atomic propositions, they have a simple formulation devoid of negations and links (conjunctions or disjunctions), so they constitute a single logical term.
- Compound propositions. Or molecular propositions, they have two terms joined by a nexus, or they use negations within their formulation, resulting in more complex structures.
To understand it better, we will see each case separately below.
Simple propositions
A simple proposition is one in which there are no logical operators. In other words, those whose formulation is precisely simple, linear, without links or negations, but rather expresses a content in a simple way.
For example: "The world is round", "Women are human beings", "A triangle has three sides" or "3 x 4 = 12".
Compound propositions
On the contrary, compound propositions are those that contain some type of logical operators, such as negations, conjunctions, disjunctions, conditionals, etc. They generally have more than one term, that is, they are formed by two simple propositions between which there is some type of conditioning logical link.
For example: “Today is not Monday” (~ p), “She is a lawyer and comes from Ireland” (pˆq), “I was late because there was a lot of traffic” (p → q), “I will eat omelette or I will leave without lunch” (pˇq).
Other types of propositions
According to Aristotelian logic, there are the following types of propositions:
- Affirmative universals. All S is P (where S is universal and P is particular). For example: “All humans they must breathe ”.
- Negative universals. No S is P (where S is universal and P is universal). "No human lives under Water”.
- Affirmative individuals. Some S is P (where S is particular and P is particular). "Some humans live in Egypt."
- Negative individuals. Some S is not P (where S is particular and P is universal). "Some humans don't live in Egypt."
Truth value of a proposition
The truth value or value of truth of a proposition is a value that indicates to what extent it is true (V) or false (F), sometimes represented as 1 and 0.
Knowing this data we can know when a proposition is a contradiction (true and false at the same time), and it allows us to transfer its statement to other logical-formal systems, such as algebra or to binary code.
To determine the truth value of a proposition, we must first express it in symbolic language, formulate it logically, and introduce the values of true and false in each of its terms, to form what is known as a "truth table", in which the possibilities of the truth value of the proposition are expressed.
This can be summarized as follows:
| p what | pˆq | pˇq | p → q | p↔q | pΔq |
| V V | V | V | V | V | F |
| T F | F | V | F | F | V |
| F V | F | V | V | F | V |
| F F | F | F | V | V | F |
The symbols used above mean:
- ˆ (and): conjunction.
- ˇ (o): disjunction.
- → (If… then): conditional.
- ↔ (If and only if): biconditional
- Δ (or ... or): exclusive disjunction
Thus, for example, the proposition "If and only if I win the lottery, then I will buy a house" would be expressed symbolically as: p ("I win the lottery") ↔ q ("I will buy a house"), since in case if he didn't win the lottery, he couldn't buy it. Your true values would be:
- True. In case you win the lottery and buy the house (p = V q = V), or if you don't win the lottery and don't buy the house (p = F q = F).
- Fake. In the remaining cases, that is, he did not win the lottery but still bought the house (p = F q = V), or he won the lottery and did not buy anything (p = V q = F).
Proposition and prayer
The central difference between a sentence and a proposition is that the first can have several of the second, that is, the propositions are part of a sentence.
This is because the sentence is a unit of greater and complete meaning, which has by itself all the meaning it requires, while a proposition is a unit of lesser, incomplete meaning, which requires the rest to be able to express its meaning completely. .
For example, the sentence "I want to go to the movies, but I have no money" contains two propositions:
- p = I want to go to the movies
- ~ q = I don't have money