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LOGICAL REASONING by Melanie Espejo What is LOGIC? LOGIC is often defined as the science of thinking and reasoning correctly. It was first studied extensively by Aristotle in approximately 400 B.C. Some famous contributors to the development of logic were Leonhard Euler(1707-1783), George Boole(18151864), John Venn(1834-1923), and Bertrand Russel(1872-1970). Symbolic logic enables us to solve complex problems. The symbols in logic, like those in algebra, are important aids to our thinking. In preparing for any task, we must first equip ourselves with the proper tools required to do the job. The first thing we must be able to do is identify and symbolize sentences. In logic we concern ourselves only with those sentences that are either true or false, but not both. STATEMENT is a declarative sentence that is either TRUE or FALSE (but not both true and false). COMPOUND STATEMENT is the result of joining two or more statements with connective words such as and, or, not, and if/then. When is a compound statement true? Let us study first the various ways in which the statement can be a negation, conjunction, disjunction, conditional or any combination thereof. Negation( ̴p) The negation of a statement is the denial statement and is represented by the symbol ̴ It is frequently formed by inserting the word “not”. Negation( ̴p) EXAMPLE: p: Today is Thursday. ̴p: Today is not Thursday. CONJUNCTION (p^q) A conjunction consist of two or more statements connected by the word and. We use the symbol to represent the word and; thus the conjunction “p^q” represents the compound statement “p and q”. CONJUNCTION (p^q) EXAMPLE: p: Logic is easy. q: Algebra is difficult. Therefore: 1. Logic is easy and Algebra is difficult. (p^q) 2. Logic is easy and Algebra is not difficult. (p^ ̴q) DISJUNCTION(pvq) A disjunction is consists of two or more statements connected by the word or. We use the symbol v to represent the word or. Thus, the disjunction “pvq” represents the compound statement “p or q”. DISJUNCTION(pvq) EXAMPLE: p: Gustav is honest. q: Iraz is honest . Therefore: Gustav is honest or Iraz is honest. (pvq) CONDITIONALS (p→q) Any statement of the form “If p, then q” is called a conditional (or an implication); p is called the hypothesis (or premise) of the conditional, and q is called the conclusion of the conditional. CONDITIONALS (p→q) EXAMPLE: P: I exercise regularly. q: I am healthy. Therefore: If I execise regularly, then I am healthy. (p→q) Can simply be stated as: “If I execise regularly, I am healthy”. (p→q) Alternatively, the conditional “ If p, then q” may be phrased as “q if p”. (“I am healthy if I exercise regularly”) The table below summarizes the logical connectives and symbols discussed: LOGICAL CONNECTIVES STATEMENTS SYMBOL Negation Conjunction ̴ ^ Disjunction v Conditional implication → READ AS... not and (inclusive) or If.... then... OTHER INFOS THAT YOU NEED TO REVIEW: INDUCTIVE REASONING is a process of observing data, recognizing patterns, and making generalizations. DEDUCTIVE REASONING is a kind of reasoning from general to particulars. It involves the stipulation of a specific statement based on a general statement that has been accepted to be true. To deduce means to reason from known facts. When you prove a theorem, you are using deductive reasoning- using existing structures to deduce new parts of the structure. In deductive reasoning, you assume that the hypothesis is true, and then write a series of statements that leads to the conclusion. Each statement is supported by a reason that justifies it. The set of statements and reasons consists the proof. A simple syllogism is an argument made up of three statements: a major premise, a minor premise (both of which are accepted as true), and a conclusion. Definition is a statement of the meaning of a word, or term, or phrase which made use of previously defined terms. Postulate is a statement which is accepted as true without proof. Theorem is any statement that can be proved true. Corollary is a statement that follows easily from a previously proved theorem. The CONVERSE of the conditional statement is formed by interchanging the hypothesis and conclusion. For instance, the converse of p→q is q→p. The converse may also be true or false. EXAMPLES: 1. If m<A=45, then <A is acute. Ans: The given statement is true because 45<90. Its converse “If <A is acute, then m<A=45” is false. Some acute angles do not measure 45. EXAMPLES: 2. If m<B=90, then <B is a right triangle. Ans: Both are true. EXAMPLES: 3. If today is Sunday, then it is a weekend day. Ans: The statement is true because Sunday is a weekend day. The converse “If it is a weekend day, then it is a Sunday” is false because Saturday (a counterexample) is also a weekend day. INVERSE of the statement: The inverse of an implication, “If p then q” is “If not p, then not q”. In symbols, the inverse of p→q is ̴p→ ̴q EXAMPLE: If two sides of a triangle are equal, then the angles opposite those sides are equal. Ans: If two sides of a triangles are unequal, then the angles opposite those sides are unequal. CONTRAPOSITIVE (counterexample) of the statement: The contrapositive of an implication is the converse of its inverse. To form the contrapositive of a statement, we interchange the hypothesis and the conclusion and negate each. In symbols, the contrapositive of p→q is ̴q→ ̴p EXAMPLE: If two angles are opposite angles of a parallelogram, then they are equal. Ans: If two angles of a parallelogram are unequal, then they are not opposite angles. Write the converse, inverse, and counterexample of the statement below: STATEMENT: If two angles are opposite angles of a parallelogram, then they are equal. CONVERSE: If two angles of a parallelogram are equal, then they are opposite angles. INVERSE: If two angles of a parallelogram are not opposite angles, then they are unequal. CONTRAPOSITIVE: If two angles are of a parallelogram are unequal, then they are not opposite angles. TRUTH TABLES: TABLE 1P Q P→Q T T T T F F F T T F F T TABLE 2P Q P↔Q T T T T F F T F F T F F SU TABLE 3P Q ̴Q ̴ QvP ( ̴ Q v P)→Q T T F T T F F T T F TABLE 4P Q P^ Q T T T T F F F T F F F F TABLE 5P Q Pv Q T T T T F T F T T F F F TABLE 6P Q ̴P ̴P^Q T F F F TABLE 7P Q ̴P ̴Q ̴P ^ ̴Q ̴( ̴P ^ ̴Q) F F T T T F The Thinker (French: Le Penseur) is a bronze and marble sculpture by Auguste Rodin held in the Musée Rodin in Paris. It depicts a man in sobermeditation battling with a powerful internal struggle. It is often used to represent philosophy. F “ Creative minds have always been known to survive any kind of bad training.” -Anna Freud (1895-1982) “ Exercise your mind as you F exercise your body”. MAG-ARAL!