Truth tables (part two)

A complex truth table

We can use the above truth tables to create a more complex truth table. For example, “Socrates is a human and he’s not a vampire.” We can translate this into logical form as “A ∧ ¬B.” (We use capital letters because they stand for something specific.) In this case “A” stands for “Socrates is a human” and “B” stands for “Socrates is not a vampire.”

The truth table for this is the following:

A	B	¬B	A ∧ ¬B
T	T	F	F
T	F	T	T
F	T	F	F
F	F	T	F

We have “¬B” on the truth table because simpler statements must be resolved before we can find the truth values of more complicated statements. “¬B” is contained in “A ∧ ¬B.”

The truth value for “¬B” is the opposite of the truth vale for “B,” so we just write in the opposite values there.

The truth value for “A ∧ ¬B” will be true whenever both “A” and “¬B” are true. There’s only one place on the truth table where “A ∧ ¬B” is true.

Tautologies

A tautology is a statement that’s always true because of it’s logical form. For example, “there are life forms on other planets or there are no life forms on other planets.” That statement has the form “P ∨ ¬P.”

We can identify a tautology by looking at the truth table because all the possible truth values of a tautology are true. When looking at all the above truth tables, you will notice that none of the statements are tautologies because there’s always one possibility of each of the above statements to be false.

A truth table of a tautology:

P	¬P	P ∨ ¬P
T	F	T
F	T	T

“P ∨ ¬P” is true whenever at least on of those statements is true. That’s why it’s always true—Whenever P is false, ¬P is true and vise versa.

Self-contradictions

A self-contradiction is a statement that’s always false. For example, “there are life forms on other planets and there are no life forms on other planets.” That statement has the form “A ∧ ¬A.”

We can identify a self-contradiction on a truth table by seeing when a statement is always false. None of the above truth tables contain self-contradictions because none of those statements are always false.

A truth table of a self-contradiction:

P	¬P	P ∧ ¬P
T	F	F
F	T	F

“P ∧ ¬P” is false whenever either “P” or “¬P” is false. One of those simple statements is always false.

Consistent statements

Statements are logically consistent as long as they can all be true at the same time, and contradictory (or inconsistent) whenever they can’t be. “P” and “Q” are consistent because it’s possible they are both true, but “P” and “¬P” are inconsistent because it’s not possible that they’re both true.

Consider the following two statements:

¬P → Q

¬(P ∨ Q)

We can make a truth table for them:

P	Q	¬P	¬P → Q	P ∨ Q	¬(P ∨ Q)
T	T	F	T	T	F
T	F	F	T	T	F
F	T	T	T	T	F
F	F	T	F	F	T

“¬P” must be given truth values before we can find the truth values of “¬P → Q” because it is part of that more complex statement. We need to find the truth values for “P ∨ Q” before “¬(P ∨ Q)” because it’s also part of that more complex statement.

“¬P → Q” will only be false when “¬P” is true and “Q” is false. That only happens on the bottom row.

“¬(P ∨ Q)” will only be true when “P ∨ Q” is false. That only happens on the bottom row as well.

The truth table above shows that “¬P → Q” and “¬(P ∨ Q)” are contradictory statements because they’re never both true at the same time.

Equivalent statements

Equivalent statements always have the same truth values. For example, “all humans are mammals and all humans are animals” is logically equivalent to “all humans are animals and all humans are mammals.” “P ∧ Q” is logically equivalent to “Q ∧ P.”

Consider the following two statements:

¬(P ∧ Q)

¬P ∨ ¬Q

We can make a truth table for them:

P	Q	P ∧ Q	¬(P ∧ Q)	¬P	¬Q	¬P ∨ ¬Q
T	T	T	F	F	F	F
T	F	F	T	F	T	T
F	T	F	T	T	F	T
F	F	F	T	T	T	T

This truth table shows why they are equivalent—they always have the same truth values. Whenever “¬(P ∧ Q)” is false “¬P ∨ ¬Q” is also false, and they are both always true at the same time.

Valid arguments

An argument is logically valid whenever it’s impossible for the premises to be true and the conclusion false at the same time. They are logically invalid whenever it is possible for the premises to be true and the conclusion to be false at the same time.

Argument 1

Consider the following argument:

1. Socrates is a human and Socrates isn’t a vampire.
2. If Socrates is a human and Socrates isn’t a vampire, then Socrates is mortal.
3. Therefore, Socrates is mortal.

We can translate this argument into the following logical statements:

A ∧ ¬B

(A ∧ ¬B) → C

C

Each letter stands for a specific statement:

A: Socrates is a human.

B: Socrates is a vampire.

C: Socrates is mortal.

The truth table for this argument is the following:

A	B	C	¬B	A ∧ ¬B	(A ∧ ¬B) → C	C
T	T	T	F	F	T	T
T	T	F	F	F	T	F
T	F	T	T	T	T	T
T	F	F	T	T	F	F
F	T	T	F	F	T	T
F	T	F	F	F	T	F
F	F	T	T	F	T	T
F	F	F	T	F	T	F

There’s only one spot on the table where both premises are true, and the conclusion is also true. Therefore, this argument is logically valid.

Argument 2

Consider the following argument:

1. Socrates is an animal.
2. If Socrates is a mammal, then Socrates is an animal.
3. Therefore, Socrates is a mammal.

We can translate this argument into logical form:

A

B → A

B

The truth table for this argument is the following:

A	B	B → A	B
T	T	T	T
T	F	T	F
F	T	F	T
F	F	T	F

“B → A” is only false when “B” is true and “A” is false. That’s only on the second row from the bottom.

This truth table proves that the argument is invalid because there’s a row of true premises and a false conclusion.

Now, what about our first argument?, is it valid or invalid:

"they will kill me if I work

and they will kill me if I don´t work

either if I work or I don´t work

They will kill me."

1- firstly we will formalize it:

p= I work

q= they will kill me.

p → q

¬p → q

(p ∨ ¬p) → q

2- then we built the truth table:

p	q	¬p	p → q	¬p → q	p ∨ ¬p	(p ∨ ¬p) → q
T	T
T	F
F	T
F	F

3- Finally I figure out whether there are any   row where the premises are true and the the conclusion is also true

Truth tables (part one)

Remember: our target is being  able to distinguish between a valid or an invalid argument, for instance: 

they will kill me if I work

and they will kill me if I don´t work

either if I work or I don´t work

They will kill me.

 In a quick overview this argument  seems to be valid, but how could we be sure?. Truth tables can help us. Let´s see!

A statement can be true o false: P = T v F, in other words, each single statement has two truth values.

The truth value In a complex statement  is the result of the  combination of truth values that single logical statements  which compose it, can have.

In this example, two statements have four possible combinations of truth values, but  If the truth values ​​of three propositions combine eight outcomes are obtained. In general the total number of possible combinations is 2ⁿ,  considering "n" the number of singles statements contained in a complex statement.

A complex statement is composed by many single statements linked with different connectives, so we have to take them in consideration, and figure up the way that each connective link affects the final complex statement´s truth value.

There is a column (vertical area) under each statement, which contains every possible truth value. The column under “p” has “T, T, F, F” (true, true, false, false). The column under “q” is “T, F, T, F” (true, false, true, false). The column under “p ∧ q” contains “T, F, F, F” (true, false, false, false).

Every row (horizontal area) beneath the statements contains every combination of truth values. The first row of truth values states that “p,” “q,” and “p ∧ q” are all true. The second row states that “p” is true, “q” is false, and “p ∧ q” is false. The third states that “p” is false, “q” is true, and “p ∧ q” is false. The fourth states that “p,” “q” and “p ∧ q” are all false.

Negation: 

p	¬p
T	F
F	T

Generally  speaking, “p” is any possible statement and “¬p” means “it’s not the case that p.”

Each box on the top row contains a logical statement. (In this case “p” or “¬p.”) Each box below a statement tells us the possible truth values of that statement. “p” can be true or false, and “¬p” can be false or true.

Each row of boxes below the logical statements contains the possible combinations of truth values of the statements above. The first row down says “p” is true and “¬p” is false. Whenever “p” is true, “¬p” will be false. For example, “p” can stand for “rocks exist.” In that case the statement is true, and “¬p” is false because it stands for “it’s not the case that rocks exist.”

The final row says “p” is false and “¬p” is true. Whenever “p” is false, “¬p” will be true. For example, “p” could stand for “1+1=3,” which is false. In that case “¬p” is true because it means “it’s not the case that 1+1=3.”

Truth tables provide every possible combination of truth values that logical statements can have. The only two truth values needed here are true and false, so there are only two rows beneath the logical statements.

Conjunction

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Lower case letters represent “predicate constants.” These lower case letters stand for any possible statement, such as “rocks exist” or “if rocks exist, then bananas are pink.”

The first row contains various statements (“p,” “q,” and “p ∧ q”). “p ∧ q” roughly translates to mean “both p and q.” For example, “p” can mean “rocks exist” and “q” can mean “bananas exist.” In that case “p ∧ q” means “rocks and bananas exist.”

You can notice that the result is "true" just  in the case that both single statements have the same  truth value. 

Disjunction

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

“p ∨ q” generally  translates as “either p or q.” For example, “p” can be “dogs are mammals” and “q” can be “dogs are reptiles.” In that case “p or q” will be “dogs are mammals or dogs are reptiles.

The truth table indicates that every “p ∨ q” statement is true unless both “p” and “q” are false, which is shown on the final row down. For example “p” can be “dogs are reptiles” and “q” can be “dogs are lizards.” In that case “p ∨ q” stands for “either dogs are reptiles or they’re lizards.” That statement is false.

Conditional

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

“p → q” generally translates as “if p, then q.” For example, “if humans are mammals, then humans are animals.”

The truth table indicates that “p → q” is true unless “p” is true and “q” is false. “p” can be “the President of the USA is a human” and “p” can be “the President of the USA is a reptile.” In that case “p → q” will mean “if the President of the USA is a human, then the President of the USA is a reptile.” That statement is false.

We can also consider a true statement where “p” is false and “q” is false. For example, “p” can be “the President of the USA is a lizard” and “q” can be “the President of the USA is a reptile.” In that case “p → q” will be “if the President of the USA is a lizard, then the President of the USA is a reptile.” That statement is true.

Finally, let’s consider a conditional statement where “p” is false and “q” is true. “p” can stand for “the President of the USA is a lizard” and “q” can stand for “the President of the USA is an animal.” In that case the statement is “if the President of the USA is a lizard, then the President of the USA is an animal.” That statement is true.

Biconditional

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

“p ↔ q” translates as “p if and only if q.” For example, “p” can stand for “1+2=3” and “q” can stand for “2+1=3.” In that case “p ↔ q” stands for “1+2=3 if and only if 2+1=3.”

The table above makes it clear that “p ↔ q” is only true when “p” and “q” have the same truth values. They must both be true or false. If not, the statement is false.

Consider when “p” stands for “dogs are animals” and “q” stands for “dogs are reptiles.” In that case “p ↔ q” stands for “dogs are animals if and only if dogs are reptiles.” That statement is false.

La verdad como autenticidad: Platón

       ¿Te conformarías con algo que solo parece verdadero?

Platón, hace veinticinco siglos nos advertía que no siempre las cosas que parecen verdaderas lo son, realmente. Más aún, él defendía la existencia de una realidad auténtica de la  que nosotros, los seres que poblamos este mundo material, no somos más que una copia. 

Platón se asombraba de la caducidad y la inestabilidad que parece presidir nuestro mundo, para él esta inestabilidad era una señal de imperfección, él veía la perfección en lo eterno. Por eso diseñó un sistema filosófico en el cuál la realidad estaba dividida en dos dimensiones: un mundo de esencias inmutables que contienen los rasgos auténticos de las cosas y un mundo de seres materiales poblado por una infinidad de copias de las esencias. Al primer mundo lo denominó el "Mundo de las Ideas" contiene la auténtica realidad y a nuestro mundo, de copias más o menos perfectas,  lo llamó "Mundo Material". 

Todos los caballos, diría Platón, son una copia de la "Idea de caballo"

Desde la perspectiva de Platón en un ser humano (por poner un ejemplo), hay cosas que cambian y cosas que permanecen inmutables. Cambia nuestra estatura , nuestro peso, también cambiamos nuestro aspecto, tiñendonos el pelo por ejemplo. Pero hay una serie de características que permanecen siempre, y que definen lo que somos, por ejemplo, somos inteligentes, tenemos sentimientos morales, somos sociables... Si de verdad queremos saber qué es un ser humano, tenemos que ir a la esencia, a la Idea o modelo -que diría Platón- del cual cada uno de nosotros somos una copia. 

Pues bien, resulta que lo que hace que un ser humano sea humano, y no una silla o un caballo es algo que no se puede ver, pero si se puede descubrir si pensamos en ello, así que si queremos acceder a lo que define lo que las cosas son verdaderamente, tenemos que usar la razón, no podemos fiarnos de los sentidos. 

Lógica: transformar razonamientos en fórmulas, no es cosa de mágia!

  Formalizar un razonamiento no es cosa de magia, sino de lógica. Ten en cuenta que todo razonamiento no es más que la unión o suma o conjunción de una serie de premisas que nos conducen a una conclusión. 

Esquemáticamente la cosa sería tal que así:

Si premisa A + premisa B + premisa C + ...premisa N... entonces... Conclusión.
O en lenguaje de lógica proposicional: AÙBÙC...ÙN®Conclusión

Tendríamos que formalizar cada una de las premisas, y la conclusión, y luego expresar el razonamiento como una implicación, en la que las premisas serían el antecedente de la conclusión. 
Veamos un ejemplillo..., adaptación de un poema de Nicolás Guillén
"Me matan si no trabajo 
y si trabajo me matan
trabaje o no trabaje, siempre me matan"
Vamos a llamar:
"p" a trabajar
"q" a matarme 
El análisis del argumento sería el siguiente:
premisa A: "me matan si no trabajo", que formalizaría como:                     Øp®q
premisa B: "si trabajo me matan", cuya formalización sería:                               p®q

  Conclusión: "trabaje o no trabaje, siempre me matan", es decir: (p˅ Øp) ®q, descorazonador, ¿verdad?

      Ahora, uniéndolo todo en una implicación en la que las premisas son el antecedente de la conclusión, tenemos que:

   (Øp®q)Ù(p®q)®((p˅ Øp) ®q)

¡Voilà!

Este otro ejemplo es de San Agustín de Hipona, intenta formalizarlo tú:

"Si me engaño, existo, el que no existe, no puede engañarse, 

pero yo me engaño, por lo tanto, existo"

Atrévete con estos otros argumentos:

a)    Mi tío dice que es un hombre honrado pero no paga sus impuestos, el que no paga sus impuestos es un delincuente, por eso mi tío no es un hombre honrado.

b)   Si Doña Angelitas ama a Pedro, entonces no ama a Don Marcelino, pero si doña Angelitas ama a Marcelino, entonces no ama a Roberto, si doña Angelitas ama a Roberto, no puede amar a Pedro o a Don Marcelino, pero Doña Angelitas ama a Marcelino, luego no ama a Pedro ni a Roberto.

c) Si como demasiado, engordo, y si engordo no me sirve la ropa, si como demasiado no me sirve la ropa.

Violencia contra las mujeres-brujas: Hipatia de Alejandría

Hipatia tuvo todo lo necesario para acabar masacrada; era mujer, inteligente, independiente, culta, atea e influyente, en una época en la que se imponía un fanatismo religioso judeocristiano para el cual la mujer debía ser sumisa, retraída, ignorante, obediente, dependiente, callada, sometida. 

La película "Agora", es un biopic de los últimos años de la vida de Hipatia de Alejandría que contiene una buena dosis de ficción. Como la película es larga y apenas tendremos tiempo para introducirla brevemente, es muy aconsejable que consultes los documentos que aparecen a continuación, para poder entender mejor el argumento del film.

• Aquí encontrarás información sobre la biblioteca de Alejandría, extraida de la obra "Cosmos" de Carl Sagan. Lee el artículo e intenta responder a la siguiente cuestión: ¿Cuáles fueron las verdaderas razones por las que la sabiduría y la ciencia almacenadas en la biblioteca nunca llegaron a florecer y prosperar?. ¿Cúal debe ser la función de la ciencia?
• La revista "Muy Interesante" publicó una breve reseña de la vida de Hipatia, que puede ayudarte a comprender el carácter del personaje.
• Si quieres saber algo sobre el argumento de esta película, el contexto histórico y científico de la vida de Hipatia y algunos datos interesantes sobre la producción y el estreno de esta película ¡a Wikipedia!. Apunta los datos que aparecen sobre los enfrentamientos entre distintas religiones que se produjeron en la época. ¿De qué modo determinaron el trágico final de Hipatia?
• Y en este artículo, que trata con gran rigor la vida y la obra de Hipatia, encontrarás información para responder a estas cuestiones:

1- ¿Cuáles fueron las aportaciones de Hipatia a la ciencia y a las matemáticas?
2- ¿Qué enseñanzas filosóficas y morales impartía Hipatia?, en la película ¿crees que estas ideas morales influyen en su decisión de rechazar la propuesta de matrimonio que le hace Orestes?
3- Por último: ¿cuáles fueron las verdaderas razones de la muerte de Hipatia?

Solo nos queda prestar atención a las ideas científicas que aparecen en la película. Después de lo que hemos estudiado sobre el cambio de paradigma desde el geocentrismo de Ptolomeo al nuevo modelo heliocéntrico de Galileo, verás que en Hipatia se produce una evolución intelectual. ¿Qué ideas científicas modernas atribuye Amenabar a Hipatia?, ¿qué experimentos y razonamientos realiza nuestra protagonista para tratar de resolver el problema de "los errantes" (el movimiento retrógrado de los planetas)?


No puedo acabar esta entrada sobre la figura de Hipatia sin  rendir homenaje a Carl Sagan, cuya serie "Cosmos" me maravilló, cuando era chica. En este fragmento de su último capítulo oí hablar, por primera vez, de Hipatia y la biblioteca de Alejandría. 

Propositional logic: let´s formalize!

Logic deals with the formalization of natural language and reasoning methods. We will begin with an introduction to the logic of compound statements, which is a basis for many other logical systems and is also called propositional logic.

Propositional logic is a formal system in which the basic units are statements (or propositions)

The basic assumption is that each proposition represents a sentence that is either true or false (but not both)

Simple propositions are denoted by propositional variables (p, q, r, s……), statements can be combined via logical connectives into more complex , compound propositions.

The connectives used to form more complex propositions include:

·         negation (¬, ~,  . , read “not” or every natural language expression that means “not”, like: “It is not the case”, “It´s false that”, and so on)

·         Conjuntion: (˄, &,  read “and” or every natural language expression that means “and”, like “but”, althoug, also, as well as, and so on)

·         Disjuntion (˅, read “or” and every natural language expression that means “or”, like “either”)

·         Conditional (→,  ᴝ, read “if….then….” but it also can be used to translate “if,” “whenever”, “…provided that…”,  “…is sufficient for….”, “….is necessary for…” and so on.

·         Biconditional (↔, read “…if and only if…”, but also: “when and only when”, “is equivalent to “, “is equal to”, “is both necessary and sufficient for”, “just in case of..then..”

Parentheses are also used to mark the way the connectives linked the propositions, they are used when the connectives changed the ordinary logical order which is:

• Whenever a negation affects the whole conjunction or disjunction, conditional or bicontional , for example:

“It´s false that my brother can speak French and my sister too”: ¬(p˄q)

• Whenever the logical dominance (the usual capability that  the connectives have, to connect the propositions) changes  in the natural language formulation. So the order in “logical dominance” is: ↔, →, and ˄,˅ which have the same dominance.

Parentheses are often omitted to increase readability but one has to be careful to avoid  ambiguous expressións.

Here you can practice what you has just learn. you will find exercises at the bottom of the page.

On this page there are more ejercises, solutions are at the bottom. 

Anyway, if you are still in a mess, you can watch  this videos (in Spanish, ufff!!) that show you how solving certain  sort of  formalizacion exercises.