Logic and Formality PDF

# Logic and Formality ## Logic - The science of correct reasoning ## Reasoning - The drawing of inferences or conclusions from known or assumed facts ## Mathematical Logic - we formulate in a precise mathematical way its definition, theorem, lemma, conjecture, corollary, propositions and the meth...

# Logic and Formality ## Logic - The science of correct reasoning ## Reasoning - The drawing of inferences or conclusions from known or assumed facts ## Mathematical Logic - we formulate in a precise mathematical way its definition, theorem, lemma, conjecture, corollary, propositions and the methods of proof which will be discussed in our next lesson. ## FORMALITY - As stated by Heylighen F. and Dewaele J-M in the "Formality of Language: Definition and Measurement", an expression is completely formal when it is context independent and precise. ## DEFINITION - formal statement of the meaning of a word or group of words and it could stand alone. - **Example:** - Right Triangle -*Carabao* ## THEOREM - A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof. - **Example:** - Pythagorean Theorem ## PROPOSITION - When we say proposition, it is a declarative statement that is true or false but not both. - This statement is another major part of formality since all types of proposition are precise and concise. ## Corollary - When we say corollary in mathematics, it is also a proposition that follows with little or no proof required from one already proven. - **Example:** - The angles opposite two congruent sides of a triangle are also congruent. A corollary to that statement is that an equilateral triangle is also equiangular. ## LEMMA - Also called a "helping theorem”. - A short theorem used in proving a larger theorem. - **Example:** - Consider an arbitrary integer, say 37. Suppose that you divide this by 7. What will be the quotient and the remainder? We have 37=7(5)+2. - Finally, Let us generalize this. Let $a$ and $b$ be any positive integers. Suppose that we divided $a$ by $b$. We will obtain a quotient $q$ and a remainder $r$. ## CONJECTURE - A proposition which is consistent with known data but has neither been verified nor shown to be false. It is synonymous or identical with hypothesis also known as educated guess. We can only disprove the truthfulness of a conjecture when after using a counterexample we found at least one that says the statement is false. - **Example:** - Let us say we have 75 different balls in a bingo urn labelled as 1 - 75. What will be our conjecture? We could say that “All number in an urn is a counting number from 1 to 75.” ## Two basic categories of human reasoning - **Deductive:** reasoning from general premises, which are known or presumed to be known, to more specific, certain conclusions. - **Inductive:** reasoning from specific cases to more general, but uncertain, conclusions. ## PROBLEM SOLVING: INDUCTIVE AND DEDUCTIVE REASONING ## WHAT IS A PROBLEM? - A *problem* is a statement requiring a solution, usually by means of mathematical operation/geometric construction. ## WHAT IS PROBLEM SOLVING? - The word "method" means the ways or techniques used to get answer which will, usually involve one or more problem solving strategies. - *Problem solving* is a process - an ongoing activity in which we take what we know to discover what we don't know. ## Problem - solving involves three basic functions: - Seeking information - Generating new knowledge - Making decisions ## Mathematical Reasoning - It refers to the ability of a person to analyze problem situations and construct logical arguments to justify the process or hypothesis, to create both conceptual foundations and connections, in order for him to be able to process the information. ## INDUCTIVE REASONING - *Inductive Reasoning* is the process of reaching a general conclusion by examining specific examples. ## USE INDUCTIVE REASONING TO PREDICT A NUMBER - 3, 6, 9 12, 15, ? - Each successive number is 3 larger than the preceding number. Thus we predict that the next number in the list is 3 larger that 15, which is 18. - 1, 3, 6, 10, 15, ? - The first two numbers differ by 2. The second and the third numbers differ by 3. It appears that the difference between any two numbers is always 1 more than the preceding difference. Since 10 and 15 differ by 5, we predict that the next number in the list will be 6 larger than 15, which is 21. ## Use inductive Reasoning to make a Conjecture - Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. - Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. - **Solution:** - Original number: 5 - Multiply by 8: 8 x 5 = 40 - Add 6: 40 + 6 = 46 - Divide by 2: 40 ÷ 2= 23 - Subtract 3: 23 - 3 = 20 - We conjecture that the give procedures a number is four times the original number. ## Use Inductive Reasoning to Solve an Application A table shows the length and period in heartbeats of a pendulum. - **a.** If a pendulum has a length of 49 units, what is its period? - In the table, each pendulum has a period that is the square root of its length. Thus we conjecture that a pendulum with a length of 49 units will have a period of 7 heartbeats. - **b.** If the length of a pendulum is quadrupled, what happens to its period? - In the table, a pendulum with a length of 4 units has a period of 4 units has a period that is twice that of a pendulum with a length of 1 unit. A pendulum with a length of 16 units has a period that is twice that of a pendulum with a length of 4 units. It appears that quadrupling the length of a pendulum doubles its period. ## COUNTEREXAMPLES - A statement is a true statement provided that is true in all cases. If you can find one case for which a statement is not true, called a *counterexample*, then the statement is a false statement. - **Examples:** - Every number that is multiple of 10 is divisible by 4. - 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200 - 100 ÷ 4 = 25 - 120 ÷ 4 = 30 - 110 ÷ 4 = 27.5 - With this example, we have shown that not all multiples of 10 are divisible by 4. So we call 110 as a counterexample. - Verify that each of the following statements is a false statement by finding a counterexample. - For all numbers x: - 0 < |x| - Let _x_ = 0. Then |0| = 0. - Thus “for all numbers _x_, |x| > 0“ is a false statement. - For all numbers x: - x² > x - Let _x_ = 1. Then 1² = 1. - For all numbers x: - √x² = x - Let _x_ = -3. Then √(-3)² = √9 = 3. - Thus “for all numbers x, √x² = x“ is a false statement. ## DEDUCTIVE REASONING - *Deductive Reasoning* is the process of reaching a conclusion by applying general; assumptions, procedures, or principles. - **Examples:** - If a number is divisible by 2, then it must be even. - 12 is divisible by 2. - Therefore, 12 is an even number. - If a ∠A and ∠B are supplementary angles, their sum is 180º. - If m ∠A = 100, then m∠B= 80. ## Deductive Reasoning is commonly used in Geometry - Solve for x in the equation 3(x + 4) – 2x = 20. Justify your answer. | Statement | Reasons | | ---------------------------------- | ------------------------------------------------------------------ | | 3(x + 4) – 2x = 20 | Given | | 3x + 12 - 2x = 20 | Distributive Property | | 3x - 2x + 12 = 20 | Commutative Property | | x + 12 = 20 | Closure property | | x = 20 - 12 | Transposition | | x = 8 | Closure Property | ## Determine what types of reasoning - During the past 10 years, a tree has produced plums every other year. Last year the tree did not produce plums, so this year the tree will produce plums. - *Inductive Reasoning* - All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost $35, 000. Thus my home improvement will cost more than $35, 000. - *Deductive Reasoning*