presentacion de filosofia en ingles

1. 2. Modern
Class
Logic

2. Symbolic Logic
01
Mathematical Logic
02
Table of contents
2.1. Propositional Calculus
2.2. Fundamental Operations
2.3. Truth Tables
2.4. Tautologies,Contradictions, and
Contingencies
1.1. Characteristics
1.2. Language and Metalanguage
1.3 Logical Connectors

3. Symbolic Logic
CHARACTERISTICS
01

4. Introduction
Symbolic logic, like syllogistics, is formal and focuses
on the validity of reasoning, not on the content of
terms or statements but on the formal structure of
relationships between terms and statements. It is an
improvement of syllogistics, as it creates more
rigorous and suitable means and tools to verify the
validity of arguments.

5. • Formalization:
• In the signs of language, we can distinguish three dimensions:
syntax, which is the relationship of a sign with other signs in the
same system
• semantics, which is the relationship of a sign with its meaning
• pragmatics, which is the relationship of a sign with what the human
being wants to communicate.
Characteristics of symbolic logic

6. • When we use a language only in its syntactic aspect, disregarding its
semantic and pragmatic dimensions, we obtain a formalized language.
• Examples of this type of language are found in mathematics, especially
in algebra. An equation, for example, is a formalized language because
it disregards all significant content and only adheres to certain rules.
Characteristics of symbolic logic

7. • Calculus:
• When we know how to use a sign, that is, the
syntactic rules it must adhere to, we can operate
with it.
• We define calculus as the operations we can
perform with signs, such as transforming
expressions through the application of exact and
explicit operating rules.

8. • Symbolization:
• Symbolic logic receives this name precisely
because it uses symbolic signs; the use of
symbology for the formalization process is
carried out in a consistent and complete
manner.

9. • Primary Source:
• The proposition is the figure of reality. The
proposition is a model of reality as we think of it. At
first glance, the proposition - as it is printed on
paper - does not seem to be a figure of the reality it
deals with. Nor does musical notation appear at first
glance to be a figure of music, nor does our
phonetic writing (the letters) appear to be a figure of
our spoken language. However, these symbols
show clearly that, in the ordinary sense of the word,
they are figures of what they represent. The figure
is a model of reality.
• Wittgenstein. Tractatus Logico-Philosophicus.

11. • Axiomatization: In this regard, M. Sacristán writes:
• The axiomatic system is, since the times of Greek
geometry, the typical form of presentation in calculus
or in formalized language.
• The characteristic of the axiomatic system as a
realization of the idea of calculation consists of having
a set of statements or formulas that are accepted
without proof, and from which all other statements in
the theory are obtained, which we call theorems.
• The formulas accepted without proof are called axioms
or postulates.
• The set of axioms, along with the definition of a
system statement or formula, and the set of rules for
obtaining theorems from the axioms (transformation
rules), constitute the primitive basis of the system.

12. Language consists of conventional signs (symbols) that are substitutes for
objects. In language, we can distinguish various planes or levels:
• At the first level, that of the plane of signs, we find ourselves at the basic level
of language.
• At the second level, we find ourselves with the metalanguage.
• For example, if we say the human being is a mammal, we find ourselves on
the plane of language because we use the expression human being as a
sign of the reality that it corresponds to.
1.2. Language and metalanguage

13. • But if we say human being as the subject of the judgment the human being is
a mammal, we have placed ourselves at a second level, at the level of
metalanguage.
• In this second case, human being is a sign that represents another sign.
• Signs form language, also called object-language, while signs of signs, from
the second level, form the metalanguage. In short, the language that speaks
of language is called metalanguage.
• Metalanguage consists of the language used to refer to other languages.

14. • In symbolic logic, atomic propositions are not affected by any connector and are
symbolized with the letters p, q, r...
• Logical connectors are operators that allow the formation of compound propositions from
these atomic propositions.
• Below is a table with the connectors, their respective symbols, schematics, and
meanings.
1.3. Logical Connectors

16. • Examples:
• p: Ethics is necessary; ¬p: Ethics is not necessary.
• p: Ethics is necessary; q: Ethics is important; p^q: Ethics is both necessary
and important.
• p v q: Ethics is necessary or important.
• p q: Ethics is neither necessary nor important.
• p ⊻ q: Either ethics is necessary or important, but not both.
• p → q: If ethics is necessary, then it is important.
• p ↔ q: Ethics is necessary only if it is important.

17. These expressions allow us to construct complex statements symbolized with logical
connectors and the letters p and q, which are the atomic propositions. These expressions
exhibit a consistent behavior and function as syntactic or logical particles.
To delve deeper into how we express everyday language in symbolic logic, follow this link to a video:
http://bit.ly/299gtMW

18. Mathematical logic
2.1 Propositional calculus
02

19. • In logic, calculus is a system of signs that are not interpreted, which
means that in calculus it is possible to perform operations without
knowing what the symbols mean. In this sense, calculus differs from
language, which is a system of interpreted signs.
• Propositional calculus consists of deriving one proposition from another,
generally more complex, based on one or more propositions that are not
broken down into their elements and are not interpreted.
Mathematical logic represents propositions using letters. In
propositional calculus, its truth value is sought, that is, whether it
is true or false. Thus, we are in a bivalent logic (two values are
admitted: V = true and F = false). There are also multivalent logics
that have several truth values. We symbolize the true or false
proposition with a single letter: p, q, r..., etc.

20. • ACTIVITIES
• Explain why we say that symbolic logic is formal.
• Provide two examples of language and two examples of metalanguage.
• Research who the main philosophers of symbolic logic are.
• Answer: What is the difference between mathematical calculus and propositional
calculus?
• Answer: What is the difference between bivalent logic and multivalent logic?
• Express the following propositions in symbolic form (p, q, r, and connectors):
• a. Kant's book is not for beginners.
• b. Either we win the game or we go down to the B league.
• c. The engineer accepts the contract only if he is paid the advance.
• d. If the day is gloomy, then it is gray.
• e. The provided space is good only if they put up a sign.

21. o The fundamental logical operation is the one through which we delimit the scope of a
term as well as its meaning, using the aforementioned logical connectors:
o Negation: It is the logical operation that, based on a given proposition, forms a new
proposition that denies the initial one. Its symbol is "¬p". The law that defines negation
is: if p is true, ¬p is false; and vice versa, if p is false, ¬p is true. We determine the truth
table of negation as follows:
2.2. Fundamental Operations

22. • Conjunction: It is the operation of joining two propositions. In everyday language, we
express it with the letter "and" (Y), whose logical symbol is ^.
• This operation is only true when the propositions it forms are both true, and it is false
when one of them is false. Its truth table is as follows:

23. • Disjunction: It is expressed in everyday
language using the word OR.
• It consists of the logical operation that forms a
complex proposition by joining two propositions.
• The symbol for disjunction is v, derived from the
Latin conjunction "vel," which means 'or.'
• The law of disjunction states: it is false only when
the propositions that compose it

24. • Implication: We express it in our everyday language
through words like "If... then..." and similar constructions,
for example, "if it's sunny, then we will go to the
countryside."
• The symbol used to represent it is "→". So, "p→q" is read
as "p implies q."
• In implication, it is very important to keep in mind the
order in which the propositions appear.
• The one preceding the sign is called the antecedent, and
the one following is called the precedent.
• The truth table for implication is as follows:

25. o Equivalence: It means that two or more propositions are interchangeable with each
other.
o The symbol used to represent equivalence is "↔".
o Thus, "p↔q" is read as "p is equivalent to q".
o The truth table for equivalence is as follows:

26. o Contradiction: Two propositions are logically contradictory when they cannot both be
true or both be false.
o For example: Either it's raining or it's cold, and it's raining if and only if it's cold because if
one is true, the other is false, and vice versa.

27. o Truth tables are a method of testing propositional calculus, based on a conception in
which the goal of calculation is to determine the necessary and sufficient conditions for
the truth of a proposition or statement.
o Each row of the table that makes the statement in question true determines a sufficient
condition for its truth. In each row, the values assigned to each propositional variable
determine a necessary condition for the row.
o Let's take an example. Suppose we want to create the truth table for the following
statement:
o "If you don't pass the test, you will be in serious trouble.“
o First, we identify the atomic propositions and assign them a variable:
o p: you pass the test; q: you will be in serious trouble.
2.3. Truth Tables

28. o "Then we can formalize as (-p)→q, and finally, we construct its table:"

29. o Ludwig Wittgenstein was one of the philosophers who
established the method of truth tables to verify the
truth value of reasoning.
o Working in philosophy — much like working in
architecture in many ways — is, in reality, a work on
oneself. On one's own interpretation. On one's own
way of seeing things and what one expects from
them.
o Wittgenstein, Ludwig. Culture and Value
o To delve deeper into the topic of connectors and truth tables, follow the video
at this link: http://bit.ly/29mgAHa
PRIMARY SOURCE

30. o Tautologies are propositional forms that correspond to
logically valid propositions that are unnecessarily repeated
in a single reasoning.
o Example: It is not true that Pedro traveled to Peru, and he did
not travel to Peru.
o Example: "If God is omnipotent, then He has unlimited power.“
o Explanation: This statement is a tautology because the definition of
omnipotence includes having unlimited power.
o Example: "Either God exists, or He does not exist.“
o Explanation: This is a tautology because it covers all possible cases;
it is always true, as it includes all possibilities.
o D. Echave in his book "Logic, Proposition, and Norm," says
that:
2.4. Tautologies, Contradictions, and
Contingencies

32. o Its Truth Table:
o Contradictions are propositional forms that correspond to logically invalid
propositions.
o For example, "¬p ∧ p" constitutes a propositional contradictory form: it is raining
and it is not raining. Its truth table:

33. o Contingencies are propositional forms that correspond to logically
undetermined propositions, that is, propositions that are true or false for factual
reasons, but not solely based on their logical form.
o For example, "¬p ∧ q" constitutes a contingent propositional form, which can be
true or false (a combination of tautology and contradiction) depending on the
values of the propositions it comprises.
o For instance: Camus was not born in France, but Sartre was born in France.
Here is its truth table:

35. o Symbolic logic has methods to determine whether a given reasoning is valid or
not.
o The associated conditional method allows mechanically verifying the validity or
invalidity of a reasoning by following these steps:
o • Given a reasoning, we abstract its logical proposition form, as in the following
example: If you have the right calculator, you will be able to complete the
exercise. ("p→q")
o You did not complete the exercise. (¬q)
o You do not have the right calculator. (¬p)
o The logical proposition form would be: [(p→q) ∧¬q] → ¬q.
o • Create the truth table for the obtained propositional form. If the result is a
tautology, the reasoning under analysis is valid. Otherwise, it is not.
o [ (p → q ) ∧ ¬q] → ¬q
Method for creating truth tables

36. o "Valid reasoning because it adheres to the tautology.“
o To delve into the concept of mathematical logic, follow the link to an
instructional video:
o https://goo.gl/50JeAH

37. 7. Define what truth value is in logic.
8. Explain the five fundamental operations with an example for each.
9. Formalize the following reasoning, first identify the propositions, assign them a variable (p, q, r), and
establish their logical form.
- Sunni Muslims are the majority in Qatar and the minority in Yemen.
- We are either more reflective, or we are less cautious.
- If you don't have accurate knowledge, you will make many mistakes.
- The signing of peace does not imply the cessation of violence.
- Being afraid is equivalent to being paralyzed.
- If you study and come to class, then you will have good results.
10. Create the truth table for the following reasoning:
¬(p→q) ∨ (¬p∧¬q) (p→q∧¬q) → ¬p
11Abstract the logical form of the following reasoning, create the truth table, and verify if they are valid:
- If Bernardo went to class, he met Cristina. But he didn't meet Cristina; therefore, Bernardo did not go to
class.
- In April, it either rains or is cold in Quito. It is not raining; therefore, it is cold.
- If the river overflows, there will be a flood. Therefore, if there is no flood, the river did not overflow.
- Democracy works if there is popular participation. There is no popular participation; therefore, democracy
does not work.
ACTIVITIES

38. To delve into the use of symbols in propositional logic, follow the link to
an educational video:
https://goo.gl/3hAqpQ

39. • Establish the logical form of the following
propositions:
• p: I study with consciousness.
• q: I will get good grades in Philosophy.
• r: I will do sports every weekend.
• s: I will feel very motivated.
• Express the following compound propositions in
words:
• r ⇒ s
• q ⇔ p
• p ⇒ (q ∧ s)
• (¬p ∧ r) ⇒ ¬q
• p ⇒ (q ∧ r)
• ¬ (q ∧ r)
• ¬ (p ⇒ r)
• ¬p ⇒ (¬q ∧ ¬r)
• Create truth tables for the following propositions:
p: Mortgage banks lower loan interest rates to 6%.
q: The sale of houses and apartments will experience
a significant increase.
r: The demand for renting houses and apartments will
decrease.
Mastering Propositional Logic Symbols

40. • Determine the logical form of each of the following arguments. Guide yourself with the example:
• Sunny days are perfect for going to the beach; today is not a sunny day, so today is not perfect for
going to the beach. (p ^ ¬q) → ¬p
• Not all philosophers were a product of their time, but Socrates was a philosopher and expressed his
time.
• Philosophers are either empiricists or rationalists. Hume was a philosopher and not a rationalist;
therefore, Hume was an empiricist.
• If some South American countries go to war, Venezuela will arm itself. If Venezuela arms itself,
Colombia reinforces its border. It is not true that any South American country is going to war. Therefore,
Colombia reinforces its border.
• If no economist is a philosopher, Nietzsche is not an economist.
• Baudelaire is a brilliant and bizarre poet, but there are poets who are neither brilliant nor bizarre.
• Some philosophy books are difficult to understand; "Sophie's World" is easy to understand; therefore, it
is not a philosophy book.
• If history has come to an end, then humanity is doomed to repeat itself. Indeed, history has come to an
end. Therefore, humanity is doomed to repeat itself.
• If the day is rainy, the soccer match will not take place, and then the fans will be upset.
To conclude:

41.  Answer the following questions:
 a. Let p∨qp∨q be a true statement; what value must q have for the variable p to be
necessarily true?
 b. Let p be a statement such that for any true statement q, the statement (p∨q)(p∨q)
is true; what truth value does p have?
 c. If the statement p→qp→q is false, which of the two statements p or q is false?
 d. If the statement p is false, determine the truth value of the following reasoning
−p→q−p→q.
 Given the following logical forms, determine an example with reasoning from
everyday life:
[(p→q)∧¬q]→¬q[(p→q)∧¬q]→¬q
p→(q∧¬p)→¬pp→(q∧¬p)→¬p
(p∧q)→¬q(p∧q)→¬q
(p∨q)∧(p∧q)(p∨q)∧(p∧q)
(p→¬q)∨(p∧q)(p→¬q)∨(p∧q)
(¬p→q)→(q∨p)(¬p→q)→(q∨p)