Intermediate Level Conditionals

Conditionals

Conditionals ‘If rabbits had not been deliberately introduced into New Zealand, there would be none there today.’ These are probably all true, and we can hear, think, and say such things without intellectual discomfort. Philosophy teaches us how to inspect familiar things from an angle that makes them look disturbing and problematic. ‘If Shackleton had known how to ski, then he would have reached the South Pole in 1909.’ ‘If the American ambassador had understood her instructions, Iraq would not have invaded Kuwait.’ ‘If the American ambassador had understood her instructions, Iraq would not have invaded Kuwait.’ ‘If Shackleton had known how to ski, then he would have reached the South Pole in 1909.’ ‘If rabbits had not been deliberately introduced into New Zealand, there would be none there today.’ These are probably all true, and we can hear, think, and say such things without intellectual discomfort. Philosophy teaches us how to inspect familiar things from an angle that makes them look disturbing and problematic.

Conditionals What are conditional sentences? If P then Q (1) If it’s a square, then it’s rectangle. (2) If you strike the match, it will light. (3) If you had struck the match, it would have lit. Role of conditionals in mathematical, practical and causal reasoning. What are conditional sentences? If P then Q (1) If it’s a square, then it’s rectangle. (2) If you strike the match, it will light. (3) If you had struck the match, it would have lit. Role of conditionals in mathematical, practical and causal reasoning.

Conditionals Antecedent and consequent (4) If P then Q P: antecedent, Q: consequent, Antecedent and consequent (4) If P then Q P: antecedent, protasis Q: consequent, apodosis

Conditionals MODUS PONENS In propositional logic, modus ponens (Latin for "mode that affirms") is a rule of inference. In propositional logic, modus ponens (Latin for "mode that affirms by affirming") is a rule of inference. It can be summarized as "P implies Q and P are both asserted to be true, so therefore Q must be true."

Conditionals An example of an argument that fits the form modus ponens: If today is Tuesday, then John will go to work. Today is Tuesday. Therefore, John will go to work. The argument form has two premises (hypothesis). The first premise is the "if–then" or conditional claim, namely that P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well. In artificial intelligence, modus ponens is often called forward chaining. An example of an argument that fits the form modus ponens: If today is Tuesday, then John will go to work. Today is Tuesday. Therefore, John will go to work.

Conditionals This argument is valid, but this has no bearing on whether any of the statements in the argument are true; for modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion. If today is Tuesday, then John will go to work. Today is Tuesday. Therefore, John will go to work. This argument is valid, but this has no bearing on whether any of the statements in the argument are true; for modus ponens to be a sound argument, the premises must be true for any true instances of the conclusion.

Conditionals An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. If today is Tuesday, then John will go to work. Today is Tuesday. Therefore, John will go to work. An argument can be valid but nonetheless unsound if one or more premises are false; if an argument is valid and all the premises are true, then the argument is sound. For example, John might be going to work on Wednesday. In this case, the reasoning for John's going to work (because it is Wednesday) is unsound. The argument is not only sound on Tuesdays (when John goes to work), but valid on every day of the week. A propositional argument using modus ponens is said to be deductive.

Conditionals A modus ponens argument has the following form: P1: If X, then Y. P2: X. C1: Therefore, Y. For example P1: If my friends are coming over tonight, I will bake a cake. P2: My friends are coming over tonight. C1: Therefore, I will bake a cake. Modus ponens ("mode of putting") is a logical argument, or rule of inference, based on an if-then statement. Modus ponens ("mode of putting") is a logical argument, or rule of inference, based on an if-then statement. Modus ponens is closely related to modus tollens ("mode of taking"); both argument forms are valid, and complementary to each other. A modus ponens argument has the following form: P1: If X, then Y. P2: X. C1: Therefore, Y. For example: P1: If my friends are coming over tonight, I will bake a cake. P2: My friends are coming over tonight. C1: Therefore, I will bake a cake. Modus ponens is also known as "affirming the antecedent". Confusing the directionality of the if-then statement in a modus ponens argument results in the fallacy of affirming the consequent, represented by the following invalid syllogism: P2: I will bake a cake. C1: Therefore, my friends are coming over tonight. Modus tollens ("mode of taking") is a logical argument, or rule of inference. (Compare with modus ponens, or "mode of putting.") As an argument A modus tollens argument has the following form: P2: Not Y. C1: Therefore, not X. P1: If it is raining, the ground is wet. P2: The ground is not wet. C1: Therefore, it is not raining.

Conditionals Modus ponens is also known as "affirming the antecedent". Confusing the directionality of the if-then statement in a modus ponens argument results in the fallacy of affirming the consequent, represented by the following invalid syllogism: P1: If my friends are coming over tonight, I will bake a cake. P2: I will bake a cake. C1: Therefore, my friends are coming over tonight. Modus ponens is also known as "affirming the antecedent". Confusing the directionality of the if-then statement in a modus ponens argument results in the fallacy of affirming the consequent, represented by the following invalid syllogism: P1: If my friends are coming over tonight, I will bake a cake. P2: I will bake a cake. C1: Therefore, my friends are coming over tonight.

Conditionals Modus tollens ("mode of taking") is a logical argument, or rule of inference. (Compare with modus ponens, or "mode of putting.") As an argument. A modus tollens argument has the following form: P1: If X, then Y. P2: Not Y. C1: Therefore, not X. For example: P1: If it is raining, the ground is wet. P2: The ground is not wet. C1: Therefore, it is not raining. Modus tollens ("mode of taking") is a logical argument, or rule of inference. (Compare with modus ponens, or "mode of putting.") As an argument A modus tollens argument has the following form: P1: If X, then Y. P2: Not Y. C1: Therefore, not X. For example: P1: If it is raining, the ground is wet. P2: The ground is not wet. C1: Therefore, it is not raining.

Conditionals Confusing the directionality of the if-then statement in a modus tollens argument results in the fallacy of denying the antecedent, represented by the following invalid syllogism: For example: P1: If it is raining, the ground is wet. P2: It isn’t raining C1: Therefore, the ground is not wet. Confusing the directionality of the if-then statement in a modus tollens argument results in the fallacy of denying the anticedent, represented by the following invalid syllogism: For example: P1: If it is raining, the ground is wet. P2: It isn’t raining C1: Therefore, the ground is not wet. Or

If you are a ski instructor, then you have a job. You are not a ski instructor Therefore, you have no job Conditionals If you are a ski instructor, then you have a job. You are not a ski instructor Therefore, you have no job. (Denying the Antecedent)

Resources

Bibliography Herbert B. Enderton, 2001, A Mathematical Introduction to Logic Second Edition, Harcourt Academic Press, Burlington MA, ISBN 978-0- 12-238452-3. Audun Jøsang, 2016, Subjective Logic; A formalism for Reasoning Under Uncertainty Springer, Cham, ISBN 978-3-319-42337-1 Alfred North Whitehead and Bertrand Russell 1927 Principia Mathematica to *56 (Second Edition) paperback edition 1962, Cambridge at the University Press, London UK. No ISBN, no LCCCN. Alfred Tarski 1946 Introduction to Logic and to the Methodology of the Deductive Sciences 2nd Edition, reprinted by Dover Publications, Mineola NY. ISBN 0-486-28462-X (pbk) Wikipedia--https://en.wikipedia.org/wiki/Modus_ponens