1. Propositions

propositions

The proposition or statement has many concepts according to the study approach, even in a complete course of logic and philosophy it turns out to be a very extensive subject, but for this chapter of propositional logic, the study of propositions is a very simple and elementary subject.

Keep in mind that propositional logic is a completely informal course, since the study of logic in a formalized way defines its concepts with the theory of sets, they coexist with each other and it is neither possible nor logical to disassociate them (I tell you this from experience).

I have noticed that in the English language the word “declaration” is used more than the word “proposition” to refer to sentences that affirm or deny something. This mention is due to the fact that my native language is Spanish and the word “proposicion” is used more. I hope that the use of this word in my native language does not bring confusion since I will use it very often for this section. Without further ado, let’s start with the concept of proposition.

In mathematics and logic we call proposition and propositional variable that sentence that can be true or false, but not both at the same time.

While it is true that the examples we will shortly show are purely literal, propositional logic does not take into account the structure of the argument, but the values of truth that reside in them.

It is up to us to find out if the argument is true or false and that is the only thing that studies propositional logic, for this reason, propositions are symbolized with lowercase letters regardless of the type of argument studied, but this we will see later. Let’s look at the following propositional examples literally.

Proposition examples

  1. All dogs have four legs (True).
  2. A chicken is a dog (False).
  3. The chicken has four legs (False).
  4. If the dogs have four legs and the hen is a dog, then the hen has four legs (True). In the subtitle of the logical context, I explain the truth value of this strange incongruity.

In linguistics, these propositions are known as assertive sentences, however, the concept of proposition is completely different in linguistic theory, we can also present different approaches to the concept of proposition according to the area of study, we will begin with the etymological context.

Proposition examples

The proposition comes from Latin “propositio” and this is their substantive deformation “proposĭtus”, literally it can mean “action to put forth”. Apparently, the etymological context does not tell us much of its meaning.

Colloquial Context

In this context, a proposition can refer to an idea to express a situation that we had previously thought about, that is, decided, but with what objective?, we have two alternate cases:

  • To indicate a possibility or alternative of voluntary acceptance to a second person.
  • Or a proposal to ourselves, that is to say, to set ourselves a certain end.

Grammatical context

In grammar, a proposition is a sentence with subject and predicate, with the ability to achieve other similar or different meanings if we add or remove words (complements) without losing its particular meaning.

In fact, it can be related to other propositions to form a larger sentence, that is, compound sentences.

Example: “Most of my friends go to Cusco on a train that turns out to be super fast and they camp there to eat green noodles together with their respective mothers“. From this proposition, the following sentences with their own meaning can be derived:

  • Most of my friends go for the ride.
  • My friends go for a walk.
  • My friends are going camping there
  • A train that turns out to be super fast.
  • My friends will eat together with their mothers.
  • They will eat green noodles together with their mothers.

I think you already understood the idea.

Logical context

Since this type of proposition is the central theme of the current section, we say that a proposition is a sentence that can be true or false.

For this reason, a proposition must have a well formalized structure that takes into account the logic of reality as the logic of the abstract (like mathematics) even if they contradict each other.

For example: If we say that all even numbers are green, then the number 4 is green, in the context of reality it is illogical to think that numbers can be classified by colors, but in the abstract, the logic of the conclusion is true.

Philosophical context

In the philosophical context, a proposition is a value judgment that expresses something and is called a predicate applied to the subject of a sentence, in other words it affirms or denies something. Be careful, nothing has been said if the denial or affirmation is true or false.

In this case, the proposition is nothing more than a broadcast of information from a judgment by means of a verbal or written sentence. Finally, it is up to us to indicate whether it turns out to be true or false.

Context from first-order logic

In the logic of first order or logic of predicates not only the truth value of a proposition is taken into account, but also the subject and predicate of the assertive arguments, however, from this context also takes into account the open sentences, these they are called propositional functions and they are assigned a specific value to transform them into propositional variables, that is, logical propositions.

First order logic makes a deeper study of the logic of an argument, it is interested in how the argument is written and there are a series of concepts that include in a compulsory and necessary way the theory of sets.

Truth value of a sentence

In propositional logic the subject and predicate are not taken into account, at least not in a symbolic way (I will explain the reason later), although in order to study the truth value of a proposition, it is necessary to highlight both the subject and the predicate. A more formal study of the subject and predicate can be studied in a first-order logic course.

The subject is the variable and the predicate is the judgment of the variable, and the truth value of a sentence depends on what the predicate says about the variable. If the subject is a known variable, then the sentence is a proposition, if the subject is an unknown variable, then these types of sentences are called open sentences.

The Logic of Truth Value

The truth value of a proposition does not necessarily “speak” of the reality in which we live, that is, the common and current sense of our palpable environment, can also be abstract, even if such abstraction is illogical for those propositions that speak about reality.

examples

The truth value of a proposition does not necessarily “speak” of the reality in which we live, that is, the common and current sense of our palpable environment, can also be abstract, even if such abstraction is illogical for those propositions that speak about reality.

  • If the Spaniards are European and Alejandro Sanz is Spanish, then Alejandro Sanz is European.
  • If the even numbers are green, then the number 4 is green.
However, in mathematics the existence of subjects in propositions is very necessary since they can be represented by numerical or complex variables or values.

Open sentence

It is called an open sentence when the subject is an unknown variable for the predicate of a given statement, in other words, we do not know if it can be true or false.

The open sentence also called propositional functions or more correctly open formulas, represent uncertainty since we do not have data of the subject and we simply do not know if they are true or false.

The predicate only affirms or denies something of an incognito or indefinite subject. But when the subject admits a fixed value, the open sentence becomes a proposition where it can be false or true.

However, it is not necessary that the subjects of the open sentence assign them a given value to transform it into propositions, to this type of statements can be added some special phrases that have a special symbol called quantifiers, this we will study in the last sections of propositional logic.

examples

  • x < 7 it is an open sentence. (the variable x is an unknown)
  • She is very beautiful. (“She” is an unknown)
  • The group is very happy. (“Group” is an unknown)
In these examples, the variables x, She and Group are unknowns, which implies that the given sentences are open sentences.

Simple and compound proposition

Generally the propositions can be catalogued into simple and compound propositions, one containing the other, although they are also called particular and universal propositions respectively, but in the development of the course this distinction is very little useful and is merely reverential, little by little they will realize as we develop the course.

Simple and atomic proposition

Las proposiciones simples o atómicas son aquellas proposiciones que tienen un sujeto y predicado.

examples

  • Lima is the capital of Peru.
  • Propositional logic studies the truth values of propositions.
  • The fish live in the sea.

The words in red are the subjects and in green the predicates. Since we see that these propositions have at most one subject and predicate, then they are simple or atomic propositions.

Note: For formal logic, the predicate in the first example turns out to be “is the capital of” and the subjects “Lima” and “Peru”. The reason is very simple, formal logic generally studies variables, that is, open statements, since statement 1 can be written as “x is the capital of y” where “x” and “y” are the subjects of the statement.

But in the course of propositional logic we will only refer to the predicate concept from the point of view of the grammar of linguistic literature.

Compound or molecular proposition

Compound or molecular propositions are those propositions that have at least two subjects or two predicates.

A property of compound propositions is that they can be separated into atomic propositions.

examples

  • Ana and María are lawyers” are two propositions composed and can be separated into two atomic propositions: “Ana is a lawyer” and “María is a lawyer“.
  • My dog has ears and tail” can be separated into “My dog has ears” and “My dog has tail“.
  • Lucho is funny or sarcastic“, can be separated into “Lucho is funny” and “Lucho is sarcastic“.
  • If x is an even number, then x is an integer” can be separated into “x is an even number” and “x is an integer“.
The words in lilac are logical connectors, important for joining propositions. but before dealing with them, let’s see how logical propositions are represented symbolically.

Mathematical representation of propositions

The logic of propositions at the symbolic level is poor because there is no mathematical analysis of the structure of the arguments as does first-order logic, propositional logic is limited only to studying the properties of logical connectors.

Another counterpoint is that the symbolization of propositions is limiting because it makes no distinction between a simple and a compound proposition. Let’s see some examples of how we can symbolically represent the propositions. 

examples

  • pMy parrot is green.
  • qMy dog has four legs.
  • rMy parrot is green and my dog has four legs.
    In this case, we can write r = p and q.

Types of logical propositions

The logic of propositions at the symbolic level is poor because there is no mathematical analysis of the structure of the arguments as does first-order logic, propositional logic is limited only to studying the properties of logical connectors.

Another counterpoint is that the symbolization of propositions is limiting because it makes no distinction between a simple and a compound proposition. Let’s see some examples of how we can symbolically represent the propositions. 

examples

  • Those birds have wings and penguins do not fly (Conjunction).
  • Dogs have four legs or the cat has a tail (logical disjunction).
  • Either I’m cold or I’m hot (exclusive disjunction).
  • If the dogs have four legs, then it is quadruped (Conditional).
  • x is even if and only if x2n + 1. (Biconditional).
  • Cats are not fish (negation).
Only composite propositions admit logical connectives, the simple proposition does not admit any.

Relation proposition-truth value

If we have 4 propositions p, q, r and s where they can be true V and false F, we can find a relation “proposition-value of truth”, that is, a correspondence between propositions and their truth values as shown in the following diagram.

Diagrama de la correspondencia entre proposiciones y valores de verdadIf there is a set of propositions P = {p_1p_2p_3, … p_n} and a set of truth values V = {VF}, the relationship between P and V is:

f(p_m )={■(V,si p_n es verdadera@F,si p_n es falsa)┤

Where p_n can be true or false, but not both at the same time. This is called correspondence f from P to V.

Logical connective in the propositions

There are 6 logical connective and are the conjunction, inclusive disjunction, exclusive disjunction, conditional, biconditional and negation, the latter is not exactly a logical connective, is an operator that affects a single proposition, but we see each of them.

logical negation

Logical negation at the linguistic level changes the truth value of a proposition, it is also capable of changing an affirmation to a negation or vice versa.

The symbol of logical negation is “~” and if we have a proposition p, the denial of such a proposition would be ~ p.

examples

  • pHumans have two feet (True).
    ~ pHumans do not have two feet (False).
  • qThe Ubuntu operating system is not Microsoft (True).
    ~qThe Ubuntu operating system is Microsoft (False).

Logical conjunction

The logical conjunction is a logical connective linking two propositions, if the two statements are true, then the conjunction is true, otherwise, if at least one of them is false, then the conjunction is false.

The letter “y” is the logical conjunction with symbol “∧“, if we have two propositions p and q, the conjunction between these propositions is p ∧ q.

examples

  • p: The dogs have 2 ears.
    qThe dogs have 4 legs.
    p ∧ qThe dogs has 2 ears and 4 legs.
  • rEven numbers are natural numbers.
    sPrime numbers are natural numbers.
    r ∧ sThe even numbers and primes are natural numbers.

Logical disjunction

It is also called inclusive disjunction, usually represented by the letter “o” and symbolically written “∨“. If we have two propositions p and q, the proposition formed by the logical disjunction would be p ∨ q.

The truth value of the logical disjunction is false if the two statements that form them are false, but you turn out to be true if at least one proposition is true.

examples

  • p: The dogs have 2 ears.
    qThe dogs have 4 legs.
    p ∨ qThe dogs has 2 ears or 4 legs.
  • rEven numbers are natural numbers.
    sPrime numbers are natural numbers.
    r ∨ sThe even numbers or primes are natural numbers.

Exclusive disjunction

Exclusive disjunction is a logical connective represented by the symbol △ (notation of my native Spanish language) whose property is to include only those propositions where one and only of the two propositions must be true so that the exclusive disjunction is true, otherwise it will always be false.

If we have two propositions p and q, the proposition formed by the exclusive disjunction is represented as p △ q. Literally the symbol △ is the word “or” but twice as the statement “Either proposition A or proposition B”, let’s see the following examples.

examples

  • p: Rosa is out of the house.
    qRosa is inside the house.
    p △ qEither Rosa is out of the house or in the house.
  • rIt is day.
    sIt’s night.
    r △ sEither it is day or it is night.

Logical conditional

It is also called material conditional and is sometimes confused with logical implication when it really has different but similar signifiers.

The logical conditional works by taking a first statement as a condition and a second statement as a conclusion, but only focuses on the truth value but not on its argument.

It is symbolized by an arrow with sense to the right “→” such that for two propositions p and q called premise and conclusion respectively, form a new proposition of the form p → q and read “if p then q“.

examples

  • p: Today it rains.
    qThe floor is wet.
    p → qIf today it rains, then the floor is wet.
  • rToday the sun rises.
    sManuel will leave the house today.
    r → sIf today the sun rises, then Manuel will leave home.

Biconditional logic

The biconditional logic is a double conditional, that is, the premise and the conclusion can also be the conclusion and the premise respectively.

Biconditional is represented by the symbol “↔” such that for two propositions p and q, the proposition formed by the biconditional is p ↔ q and reads “p if and only if q“.

The proposition p is the premise of the conclusion q as well as q can be the premise of the conclusion p.

examples

  • p: The sun has come out.
    qIt is day.
    p ↔ qThe sun has risen if and only if it is daytime.
    q ↔ pIt is day if and only if the sun has risen.
  • r: 8 is even.
    s: 8 is a multiple of 2.
    r → s: 8 is even if and only if it is a multiple of 2.
    s ↔ r: 8 is a multiple of 2 if and only if it is even.

Equivalent propositions

Propositions can be written in different ways by altering only their logical connectives such that their truth value is the same.

For example, the proposition “Dogs has four legs” is equivalent to “It is not true that dogs do not have four legs”, the second proposition is a double negation of the first and they say exactly the same thing. Let’s see more examples.

examples

  • “Number 4 is an even number” is equivalent to “It is false that number 4 is not even”.
  • “If today it rains, then the floor gets wet” is equivalent to “If the floor has not been wet, then today it has not rained”.
  • “If I eat a lot of sweet, then it will come out tartar” is equivalent to “It is false that I eat a lot of candy or I will get tartar”.
With a little mental analysis, you will notice that these propositions are completely equivalent.

Semantics and syntax of a proposition

Semantic statements have an intuitive character, semantics represents the meaning of a statement, that is, it indicates the interpretation that we assign to the statements formed by symbols and well-defined characters of a linguistic language.

The statements that we have studied in greater proportion are the propositions, these acquire such a semantics that they can be determined as true and false.

We can study semantics from 3 angles, these are linguistic semantics, logical semantics and the semantics of cognitive sciences (psychology). Naturally this section has focused on logical semantics.

For example, suppose you couldn’t read, so you could never understand these paragraphs, all you’ll see is a set of strange, meaningless characters.

But if you read me so far is because you know what I’m saying, because for these lines to be intelligible must not only provide a standardized meaning but must also have a specific order and also standardized for understanding, however, there are other standards that you will not understand as that of a programmer, since we use the same characters to make programs and / or create web pages through codes like this image:

código html de una pagina web

This is a code snippet from the proposition chapter publication that you are starting to read right now. As you can see, we use the same characters with some more symbols to structure the publication. What your web browser such as Google Chrome, Mozilla, Opera or any other web browser does next is “render” it so that you can see it in your language, just as your language understands it.

We can say then that semantics is what you interpret according to the language with which you are familiar, in general, for communication to be fluid and shared among all, it must be a standardized semantics.

In propositional logic it is the same, it has a standard, although limited, but it serves to understand how propositions behave by means of logical connectives and grouping signs, in general, propositions are symbolized with lowercase letters such as “p”, “q” and “r” such that they can be determined as true “V” and false “F”, these values of truth are the only semantic values formalized in propositional logic, inasmuch as the semantics of the meaning of the arguments of the given propositions are taken intuitively but they are not part of the study of propositional logic but of the study of the logic of predicates or logic of first order.

propositional function

Sometimes open sentences are also called propositional functions when working on other branches of mathematics to move away from the grammatical linguistic context and towards more purely mathematical concepts.

A propositional function does not necessarily have to be assigned a specific value for it to be a proposition, it is possible to achieve such a transformation with the quantifiers commonly known as the words “For everything” and “There is some,” although there are other categories that transform propositional functions into propositions that we can study in a more advanced subject of logic called categorical propositions.

The propositions in the first-order logic

The first order logic or the logic of predicates is an extension of the propositional logic, it studies the structure of the arguments, generally of the open sentences since it takes into account the subject as a variable (although the predicate also), in this branch of the mathematical logic, these statements are usually called propositional functions and take into account the following:

  • Variable: May represent an indeterminate subject, however, may also represent an indeterminate predicate.
  • Constant: It represents a defined value of the variable of a propositional function, generally it is a specific value of the subject.
  • Functions: They are operators applied to the variables of either the subject or the predicate.
  • Logic connectors: are operators that link statements to form other statements.
  • Quantifiers (very related to categorical propositions): There are only two types, the universal and existential quantifier, are able to transform variables into constants without the need to give a specific value to the variables. An example is to take into account all even numbers no matter what it is, this argument is studied by the universal quantifier, in contrast, if a property meets at least one even number no matter what it is, this is studied by the existential quantifier.

One of the strong points of first-order logic is the ability to formalize the theory of demonstrations, with this, we can develop all mathematics to the present day.

To achieve this, the classification I have just indicated must be mathematically symbolized to develop a more solid argument that propositional logic is incapable of achieving, but for its strict formality of this branch, the logic of predicates must take very much into account the sombolization of predicates, punctuation signs, relational signs (the equal sign) and the list we just presented a moment ago.

In this way, first-order logic is able to differentiate an atomic proposition from a molecular one on a symbolic level, something that is not able to make propositional logic, the latter also called zero order logic or statement logic because it only works with statements or propositions.

references

  • Logic – Basic Mathematics | ninth edition – Ricardo Figueroa.
  • Notions of Logic – Basic Mathematics | ninth edition – Moisés Lázaro.
  • Introduction to Mathematical Logic – Mathematical Logic | Carlos Ivorra.
  • Propositional logic, proposition, open enunciation, semantics – wikipedia and wikiversity.