VIEN MIKKA F. CORCIEGA GRADE-VIII-YANG IF-THEN STATEMENT When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then its called deductive reasoning. We will explain this by using an example. If you get good grades then you will get into a good college. The part after the if: you get good grades - is called a hypotheses and the part after the then - you will get into a good college - is called a conclusion. Hypotheses followed by a conclusion is called an If-then statement or a conditional statement. This is noted as : This is read - if p then q. A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said if you get good grades then you will not get into a good college. If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional. Example : Our conditional statement is: if a population consists of 50% men then 50% of the population must be women. If we exchange the position of the hypothesis and the conclusion we get a converse statement: if a population consists of 50% women then 50% of the population must be men. If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing If we negate both the hypothesis and the conclusion we get a inverse statement: if a population do not consist of 50% men then the population do not consist of 50% women. The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse. We could also negate a converse statement, this is called a contrapositive statement: if a population do not consist of 50% women then the population do not consist of 50% men. The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true. A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism. Example : If we turn of the water in the shower, then the water will stop pouring. If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted: The law of syllogism tells us that if p → q and q → r then p → r is also true. This is noted: Example: If the following statements are true . If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we dont get wet any more (r) . Then the law of syllogism tells us that if we turn of the water (p) then we dont get wet (r) must be true . Converse, Inverse, Contrapositive © Copyright 1999, Jim Loy In the video Geometry, Part I, by the award winning Standard Deviants, we are told the following: If a statement is true, the inverse is also logically true. Likewise, when the converse is true, the contrapositive is also logically true. They are mistaken. What they should have said is: If a statement is true, the contrapositive is also logically true. Likewise, when the converse is true, the inverse is also logically true. We start with a simple statement of fact, like, (1) A triangle is a polygon, or (2) An even number is divisible by two. Each of these is actually an implication, an if-then statement: (1) If an object is a triangle then it is a polygon. (2) If a number is even then it is divisible by two. Here is a quick definition of converse, inverse, and contrapositive: • statement: if p then q • converse: if q then p • inverse: if not p then not q • contrapositive: if not q then not p I will now show the converse, inverse, and contrapositive of our examples involving triangles and even numbers: Converse: (1) If an object is a polygon then it is a triangle (false). A square is a polygon but not a triangle. (2) If a number is divisible by two then it is even (true). Of course the first one is false because not all polygons are triangles. Inverse: (1) If an object is not a triangle then it is not a polygon (false). A square is not a triangle, but is a polygon. (2) If a number is not even, then it is not divisible by two (true). Contrapositive : (1) If a number is not divisible by two then it is not even (true). The first one, being true, cannot be equivalent to the converse, which is false. The Standard Deviants error does not matter, for their purposes, as they proceed to ignore the inverse and contrapositive, because they are equivalent to the other two (statement and converse, even though they are equivalent in the wrong order). Most of the time, logicians and mathematicians also ignore the inverse and contrapositive, although sometimes it may be easier to prove the contrapositive, and then remark that that is equivalent to proving the original statement. Often the converse of a mathematical statement is true. Then it is handy to prove both the statement and its converse. The above can also be examined with the use of a boolean algeba (symbolic logic) or with truth tables. If we do that, then we must note that in logic an implication (as an if is called) is a statement which can be true or false. And when the condition is false, then the entire statement is true. If my shirt is red, then I am wearing blue jeans, is true when I wear a yellow shirt, regardless of whether I am wearing blue jeans or not. This is important so that our statements will be well defined, we can build a truth table. My contrapositive road sign: If you arent home already, then you dont live here. Deductive Reasoning An example of a deductive argument: All men are mortal. Deductive reasoning, also deductive logic or logical deduction or, informally, top-down logic,] is the process of reasoning from one or more general statements (premises) to reach a logically certain conclusion . Deductive reasoning links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true. Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information (and so induction can be used even in an open domain, one where there is epistemic uncertainty. (Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs - mathematical induction is actually a form of deductive reasoning.) 1. All men are mortal . 2. Aristotle is a man. 3. Therefore, Aristotle is mortal. The first premise states that all objects classified as men have the attribute mortal. The second premise states that Aristotle is classified as a man – a member of the set men. The conclusion then states that Aristotle must be mortal because he inherits this attribute from his classification as a man. Inductive Reasoning The philosophical definition of inductive reasoning is much more nuanced than simple progression from particular/individual instances to broader generalizations. Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms, discussed below). Though many dictionaries define inductive reasoning as reasoning that derives general principles from specific observations, this usage is outdated.[2] Inductive reasoning is inherently uncertain. It only deals in degrees to which, given the premises, the conclusion is credible according to some theory of evidence, for example a many-valued logic, Dempster–Shafer theory, or probability theory with rules for inference such as Bayes rule. Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases of epistemic uncertainty. (Technical issues with this may arise however; for example, the second axiom of probability is a closed-world assumption. A statistical syllogism is an example of inductive reasoning: 1. Almost all people are taller than 26 inches 2. Gareth is a person 3. Therefore, Gareth is almost certainly taller than 26 inches As a stronger example: 100% of biological life forms that we know of depend on liquid water to exist. Therefore, if we discover a new biological life form it will probably depend on liquid water to exist. This argument could have been made every time a new biological life form was found, and would have been correct every time; this does not mean it is impossible that in the future a biological life form that does not require water could be discovered. As a result, the argument may be stated less formally as: All biological life forms that we know of depend on liquid water to exist. All biological life probably depends on liquid water to exist.
Posted on: Tue, 12 Nov 2013 01:10:06 +0000
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