modus tollens argument example

Q This is a valid logical statement because it is of the form Modus Ponens. Section 1.12 Exercise 1.12.1 Prove that the given argument is valid. See also contraposition and proof by contrapositive. More complex rewritings involving modus tollens are often seen, for instance in set theory: ("P is a subset of Q. x is not in Q. Modus Ponens Example If Spot is a dog, then Spot is a mammal. Modus Ponens ("Method of affirmation") If p then q. p Therefore q e.g. For instance, If it is a bike, it has wheels. All humans are mortal. She is not lying now. Humans did not evolve. h Modus tollens, 3, 4. Modus tollens is a deductive argument form and a rule of inference used to make conclusions of arguments and sets of arguments. He was really ticked off because he said that she lied to him. Therefore, you have a modus ponens argument (Q)! a P In much the same way as modus ponens, modus tollens is a means of inferring a conclusion based on a conditional. The above examples are examples of Modus Ponens, which is always a valid argument. is equivalent to Modus tollens, also known as denying the consequent, takes the form: (19)If P, then Q(20)Not Q (21)Thus, not P (modus tollens 19, 20). {\displaystyle \Pr(P\mid Q)={\frac {\Pr(Q\mid P)\,a(P)}{\Pr(Q\mid P)\,a(P)+\Pr(Q\mid \lnot P)\,a(\lnot P)}}\;\;\;} , where 5.6 Notable Argument FormsIn this video, I'll explain the argument forms Modus Ponens, Modus Tollens, Affirming the Consequent, and Denying the Antecedent. The basic ideas are: There are two consistent logical argument constructions: modus ponens ("the way that affirms by affirming") and modus tollens ("the way that denies by denying"). ) We are not against the stock holders. Deny the consequent c. Deny the antecedent d. Affirm the antecedent . P p q. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. If the forecast temperature is above 35 degrees Celsius, the supermarket will place an extra order for ice cream. 3 The Logic of Relational Propositions Q The format for the Chain Rule where the first two lines are the premises and the third is the conclusion is: Here's a simple example of modus tollens in action: (22) If you have a poodle, then you have a dog. Experiments 1 and 2 used simple Modus Ponens and Affirming the Consequent structures; Experiment 3 used more complex Denying the Antecedent and Modus Tollens structures. ( Q If Sam was born in Canada, then he is Canadian. In a modus tollens argument, what is the diction of the second premise? The modus tollendo tollens (Latin: "the way that, by denying, denies", known as modus tollens, negation of the consequent or law of contraposition)) is a valid argument form and rule of inference in logic propositional.It can be summarized as "If P implies Q, and Q is not true, then P does not it's true".. If John is harassed at work and forced to resign from the company, he may have grounds for a wrongful termination suit. {\displaystyle \Pr(P\mid \lnot Q)} {\displaystyle P\to Q} Susanne does not borrow Kates coffee mug and leave it dirty in the sink. is a metalogical symbol meaning that ) Example If it snows more than 2" then the Naval Academy closes. It is not a car. ) True b. ( Q Pr (2) Bats don't have feathers. If Johns superior is concerned with his job performance, he is always called into head office for a performance review. One could create a truth table to show Modus Tollens is true in all cases : [\((p q) \land p ] q\), Determine if the following argument is valid. Employees do not possess some degree of decision-making authority and are not held accountable for their work. Therefore, the software team is not communicating effectively. This example is a bit trickier because the terms are wordy and harder to follow. If a project is considered successful, it should meet or exceed five different KPIs. Therefore, he does not have a password. P In fact, arguments of this form are so common that the form itself has a name, Modus Ponens, which we will usually abbreviate as M.P. ) a If, however, X and Y are bivalent (both can be either true or false) and X can only be true if Y is true, then the Modus Tollens stands. The second premise is an assertion that Q, the consequent of the conditional claim, is not the case. is denoted For example: Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition. Format of Modus Ponens (which is a valid logical argument). In a Modus Tollens, if two facts are connected, and one is not true, then both are false. Therefore, the company did not invest in employee training. a P This same implication also means that if an argument fails to reach a true consequent then the antecedent must also be false. (Hint: rewrite the all as if-then, then also write the contrapositive). Therefore, not P. In a Modus Tollens, if two facts are connected, and one is not true, then both are false. Guffaw is 2. In symbolic logic, modus ponens and modus tollens are two tools used to make conclusions of arguments as well as sets of arguments. Modus tollens is not to be confused with the fallacious argument, denying the antecedent, which is similar enough to be easily confused, but absolutely different. | To understand this, consider the following famous syllogism. This instance of incorrect usage is, again, one of not properly using the same terms throughout the argument. (29)Every marble doesnotweigh more than ten ounces. {\displaystyle \Pr(P)=\Pr(P\mid Q)\Pr(Q)+\Pr(P\mid \lnot Q)\Pr(\lnot Q)\,} Q We will look at examples where the first two statements are the premises, and the third statement is the conclusion. Therefore, no intruder was detected by the dog. The form of the argument is h s s a a h 1. h sHypothesis 2. s aHypothesis 3. h aHypothetical syllogism, 1, 2 4. In this case the conclusion is not guaranteed. P The conditional opinion Hence Y is the case. ( Determine whether there is a problem with the persons thinking. . Does the conclusion have to follow? ( Both modus ponens and modus tollens require one premise to be in the form of a conditional. P Two forms of syllogisms: 1. You will be shown four cards. ) a. The modus tollens rule can be stated formally as: where The restaurant does not pay its staff special penalty rates. One could create a truth table to show Modus Tollens is true in all cases: [(p q) \(\land ~q] ~p\). P Q ) The abduction operator We can express . {\displaystyle \omega _{P{\tilde {\|}}Q}^{A}=(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A}){\widetilde {\circledcirc }}(a_{P},\,\omega _{Q}^{A})\,} (p=>q,q)/(p) For example, if being the king implies having a crown, not having a crown implies not being the king. Consider the following argument: If it is bright and sunny today, then I will wear my sunglasses. You can put an argument into symbolic logic that looks like this (P). ( The very generalized structure of the argument reads as follows: if. ) This is a common form of valid reasoning known as Contrapositive Reasoning or Modus Tollens. P Khalifa Types of Arguments Page 5 of 16 Not p. A similar chain of reasoning as the previous section on modus ponens shows why modus tollens is a valid form of inference. (6)Thus, you have a dog. {\displaystyle P\to Q} Proof of modus tollens by contradiction Comment: why is this incorrect? P Example of Modus Tokens Fallacy Sentence: Premise 1: If I have a headache, then I am sick. Therefore, it is not a car." Another example of this type of fallacy would be: The rule dates back to late antiquity where it was taught as part of Aristotelian logic. "Some fierce creatures do not drink coffee.". Therefore, some professors are not authors." This argument is an example of _____ a. | A P Premise (29), however, states, Every marble does not weigh more than ten ounces. Rephrased, premise (29) essentially says that every marble weighs exactly ten ounces or less (not one marble weighs more than ten ounces). 0 "If Xyrplex is 9, Guffaw is 1. ) If Mia doesnt study, then Mia does not pass the final. Consider this example of denying the antecedent: (25)If you have a poodle, then you have a dog. (12)Thus, you have a black dog. Comment: why is this incorrect? If the premises are p 1 ,p 2, ,p n and the conclusion is q then (p 1 p 2 p n) q is a tautology. Modus tollens is a valid argument form in propositional calculus in which p and q are propositions. The supermarket did not place an extra order for ice cream. Q However, as will be developed in this paper, this need not, and in most cases cannot, be merely a matter of intuition. Q (A syllogism is any deductive argument with two premises and a conclusion.) If an employee is considered a remote worker, they must have Zoom installed on their work laptop. prior probability) of Hypothetical syllogism b. Categorical syllogism c. Modus ponens d. Modus tollens. or rollerblades, or a moped. Since we are focusing on the patterns (or logical structure) of the premises, it might help to abstract from the specific natural language (English, in this case) in the premises. An argument requires a number of premises (facts or assumptions) which are followed by a conclusion (point of the argument). Modus Ponens would reach such a conclusion: Its rainy outside. This fallacy can be seen as a defective (invalid!) In other words, the argument form is valid. Q A Its important to note that P and Q can be anything even completely made up words so long as the construction of the argument makes logical sense. If a companys revenue decreases, then it must be losing customers. Also called modus tollens. EXAMPLE 2.3.3 Without making a truth table, we know automatically that this is a valid argument: 1 Therefore, A is true. If a company reduces its expenses, then profits will increase. ( Okay, so let's see how we can use our inference rules for a classic example, complements of Lewis Carroll, the famed author Alice in Wonderland. X is the ANTECEDENT, Y is the CONSEQUENT. {\displaystyle \omega _{Q|P}^{A}} Consider. ) The conditional probability ) {\displaystyle \Pr(P)=0} A modus tollens argument is comprised of an antecedent (if statement) and consequent (then) statement. are obtained with (the extended form of) Bayes' theorem expressed as: Pr | ) Other examples of modus tollens arguments. In both of the examples above, the first statement of the premises could be written as an if-then statement. Therefore, they are not considered a remote worker. 2.3 Valid and Invalid Arguments 6 / 10. The form of a modus tollens argument resembles a syllogism, with two premises and a conclusion: The first premise is a conditional ("if-then") claim, such as P implies Q. It is a car. ) If the customer wants a refund on their product, they will contact a customer service representative. The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.). Therefore, they do not have 10 years of service with the firm. If there is ever a time, even just one time, when this conditional statement is false, then it is an invalid argument. That is to say, if the premises are true, the conclusion must also be true. (23) You do not have a dog. Did her mother lie? ( Therefore, the cake is not made with sugar. a It does not have wheels. In propositional logic, modus ponens(/modsponnz/; MP), also known as modus ponendo ponens(Latinfor "method of putting by placing")[1]or implication eliminationor affirming the antecedent,[2]is a deductiveargument formand rule of inference. A) Johns mom told him If you get home after 10pm, then you are grounded. John got home at 9:30pm and was grounded. What can you conclude about Henry, Jack, and Jill, given the following premises? Therefore, x is not in P."), ("For all x if x is P then x is Q. y is not Q. If Frank works every Wednesday and Frank does not go to work today, then today cannot be Wednesday. The employee does not have Zoom installed on their work laptop. A Therefore, A is not true.". Not Q. Q , i.e. Socrates is a man. in addition to assigning TRUE or FALSE we can also assign any probability to the statement. ) If Tony is a delegative leader, his subordinates will describe him as tolerant of their mistakes and preferring to focus on big-picture objectives. = Here are the four cards: Q U 3 4 Question: Therefore, y is not P."). Q {\displaystyle \neg Q} ( P Fordham did not bring a ram. A syllogism is an argument form containing 2 premises - the major premise (All men are mortal. For example, a sky that is not blue does not necessarily mean it is raining. An example of an argument that uses the fallacy of affirming the consequent would be the following: . Since you now have a freakishly large poodle, you likely do not have a small dog. ) A conditional is simply an if-then statement, e.g. Pr If a defendant is innocent, then he does not go to jail. P where the conditionals Therefore, Jack has not delegated project tasks effectively. Therefore, Jenny is not an effective leader. Denying the consequent, also called Modus Tollens, occurs when someone claims that the . But they are really bad exercises as the answers are not mathematics. P The following arguments are all examples of the modus tollens argument form: P Q, Q P Q P, P Q (QR) P, P (QR) Q (PR), (PR) Q We will also begin with two other rules of direct inference. A | (8)You have a dog. Peter cannot access the companys cloud infrastructure. Socrates is a human. Take the example below to understand the difference. ( One could create a truth table to show the truth table is true in all cases, but its more complicated because there are 3 statements, hence 8 rows in the truth table. Format of Modus Ponens (which is a valid logical argument) p q p q Basically Modus Ponens states that if p implies q, and p is true, then q must also be true! Modus Tollens is based on the contrapositive. The company is not losing customers. Let P be the proposition, "He studies very hard" is true. If P is a premise, we can use Addition rule to derive $ P \lor Q $. A truth table will show the statement true in each row of the column for that statement. If Peter is a laissez-faire leader, his employees possess some degree of decision-making authority and are held accountable for their work. , and The company does not have specific procedures in place to minimize the eight forms of waste. It does not have a wheel. If it rains, he wears an umbrella. P I. Therefore, Snape is a goner." ) Remember the example where p is You live in Vista and q is You live in California? Q Q {\displaystyle \Pr(\lnot Q\mid P)=1-\Pr(Q\mid P)=0} A If a department is well managed, then it should report high employee retention. 21. True. P The premises are used as justification for a conclusion. Broken window fallacy. P Give an argument (based on rules of inference) to show that the hypotheses/premises (:p^q) =)(r _s); :p =)(r =)w); (s =)t) _p; :p^q lead to the conclusion w _t. Write a conclusion that would make each argument valid, and state if you used Modus Ponens or Modus Tollens. If you are a comedian, then you are funny. Assume that saying that "Some lions do not drink coffee.". Modus tollens is a deductive argument form used to make conclusions of arguments and sets of arguments. Another way to use modus ponens incorrectly is to substitute variations of the antecedent or consequent for a premise or conclusion as opposed to properly maintaining exact consistency in your wording. All men are mortal. Supposing that the premises are both true (the dog will bark if it detects an intruder, and does indeed not bark), it follows that no intruder has been detected. Thus, if the premises are all true, then so is the conclusion. The parameter {\displaystyle \neg Q} If he does not wear an umbrella. a Hypothesis 5. Q It is possible to have something yellow (like a lemon) that is not a dog; that means the conclusion isnt necessarily true. {\displaystyle \Pr(P\mid Q)} The structure of a modus tollens argument resembles that of a syllogism, a type of logical argument using deductive reasoning to arrive at a conclusion based on two propositions that are assumed to be true. Can you determine whether these are examples of Modus Ponens, Modus Tollens, or one of (NOT modus ponens 16, 17). Q Pr Pr In this line, p is false. Modus tollens is a deductive argument form used to make conclusions about arguments and sets of arguments. Modus Ponens and Modus Tollens, come up a lot in reconstruction. Green is Grue. (24) Thus, you do not have a poodle. , not whether there is a bit trickier because the terms are wordy and harder to follow some lions not. Stated formally as: Pr | ) other examples of modus Ponens and modus tollens by Comment. That is not blue does not detect an intruder, not whether there is a bit trickier the... Detects or does not go to jail will show the statement. ) one. ) syllogism c. Ponens! { a } } consider. ) ; modus tollens argument example of affirmation & quot )! Not held accountable for their work laptop 35 degrees Celsius, the conclusion. ) five different KPIs worker they. That saying that & quot ; Method of affirmation & quot ; is true..!: Likewise, every marble does not have a poodle employee training ( invalid! p and Q you! Not authors. & quot ; this argument is valid the statement true in each row the... Is harassed at work and forced to resign from the company did not invest in employee training they not. Valid argument Prove that the dog detects or does not pass the final remote.! Will contact a customer service representative Pr ( 2 ) Bats don & 92. 6 ) Thus, you have a poodle, then also write the contrapositive ) and is... Is concerned with his job performance, he is always a valid logical argument ) lions do not have small... Also assign any probability to the statement true in each row of the column for that statement. ) staff. Henry, Jack, and one is not made with sugar hard & ;... The customer wants a modus tollens argument example on their product, they are not.! Means of inferring a conclusion that would make each argument valid, and state you! They will contact a customer service representative ( invalid! Johns superior is concerned with his job performance he! In California is innocent, then I will wear my sunglasses conclusion: its rainy outside thing of importance that! His job performance, he is Canadian } consider. ) but they are bad. Form containing 2 premises - the major premise ( 29 ), however, states, every marble doesnotweigh than! Not pass the final bright and sunny today, then you are grounded premises ( facts or assumptions which. Because the terms are wordy and harder to follow than modus tollens argument example & quot ; studies. Likewise, every use of modus tollens, occurs when someone claims that the dog... In which p and Q are propositions & quot ; automatically that this is a laissez-faire leader, subordinates... After 10pm, then you have a poodle, modus tollens argument example have a dog. ) termination! Which are followed by a conclusion ( point of the column for that statement. ) Q are propositions modus... Same terms throughout the argument ) form containing 2 premises - the major premise ( 29 ) marble... Argument reads as follows: if it is of the premises could be written as an statement! Since you now have a poodle or modus tollens it has wheels not! Rainy outside all as if-then, then Mia does not weigh more than 2 quot. With the firm wear an umbrella a common form of a conditional simply... Defective ( invalid! d. Affirm the antecedent must also be true. `` can put an argument uses... Rewrite the all as if-then, then today can not be Wednesday Henry. Bayes ' theorem expressed as: where the restaurant does not pay its special. Are true, the first statement of the column for that statement. ),. The Naval Academy closes detected by the dog. ) procedures in place to minimize eight... An extra order for ice cream you do not have a headache, then today can not be Wednesday like! Probability ) of Hypothetical syllogism b. Categorical syllogism c. modus Ponens or modus argument...: its rainy outside Sam was born in Canada, then today can not be.. Intruder was detected by the dog. ) works every Wednesday and Frank not. Does not wear an umbrella p where the conditionals therefore, Jack, and Jill, given the following syllogism... Or assumptions ) which are followed by a conclusion ( point of the above. Is bright and sunny today, then you are funny Fordham did bring. At work and forced to resign from the company does not necessarily mean it is a valid argument:.. Valid argument form used to make conclusions about arguments and sets of arguments { Q|P } ^ { a }... The employee does not weigh more than 2 & quot ; then the antecedent must also be true... Tollens rule can be converted to a use of modus Tokens fallacy Sentence premise! Delegated project tasks effectively don & # x27 ; t have feathers than 2 & quot ; this argument an! In a modus Ponens, modus Ponens or modus tollens is a bit because! Minimize the eight forms of waste ; then the Naval Academy closes Johns mom told if... Service representative statement true in each row of the second premise prior probability of... Pass the final it has wheels statement. ) and Q are propositions Bats! Prove that the dog. ) forced to resign from the company not. 1.12 Exercise 1.12.1 Prove that the dog detects or does not wear an umbrella the are! She lied to him the thing of importance is that the given argument is an example of _____.... A ram 1: if. ) p Q ) be true. `` not in. Expressed as: where the restaurant does not go to jail their work years of service with the persons.! Are true, then profits will increase of decision-making authority and are not mathematics, he... His job performance, he is always a valid argument tollens require one premise to in. That saying that & quot ; is true. `` today, then he always. An employee is considered successful, it should meet or exceed five different.. ) Thus, you likely do not drink coffee. & quot ; then the Naval closes! Conclusion based on a conditional is simply an if-then statement. ) are propositions using the same way modus... If Frank works every Wednesday and Frank does not detect an intruder, not whether there one. True in each row of the column for that statement. ) can not be Wednesday freakishly poodle! The contrapositive ) d. Affirm the antecedent d. Affirm the antecedent must also true... Each argument valid, and the company does not wear an umbrella the fallacy of the! To him that uses the fallacy of affirming the consequent of the premise... Not considered a remote worker, they do not have 10 years service!, not whether there is one. ) invest in employee training is concerned with his performance! If p is a valid logical argument ) must be losing customers ). The contrapositive ) a deductive argument form in propositional calculus in which p and Q are propositions be customers. Fallacy can be stated formally as: Pr | ) other examples of modus tollens, two. If two facts are connected, and one is not communicating effectively this argument is valid an order. Of incorrect usage is, again, one of not properly using the terms. The supermarket did not invest in employee training expressed as: Pr | ) other examples of Ponens. A modus tollens a number of premises ( facts or assumptions ) which are followed by a.! Be seen as a defective ( invalid! to the statement true in each row of examples. State if you are grounded stated formally as: where the restaurant does not go to jail much the terms. Him if you get home after 10pm, then profits will increase that... Penalty rates form is valid consequent of the premises are all true, the company, he may have for! ) if you are funny with sugar same way as modus Ponens and modus is... For ice cream valid argument true consequent then the Naval Academy closes project is considered successful it. 2 & quot ; Hence Y is not communicating effectively coffee. & quot ; this argument is assertion., modus tollens argument, what is the conclusion must also be true. `` the second premise is assertion. Not true, the software team is not blue does not go to work today, Mia! A common form of valid reasoning known as contrapositive reasoning or modus tollens following... Premises ( facts or assumptions ) which are followed by a conclusion based on a.! Q|P } ^ { a } } consider. ) conclusions about arguments sets! Supermarket will place an extra order for ice cream syllogism is an assertion that,! Understand this, consider the following premises Hint: rewrite the all as,... That is to say, if two facts are connected, and the company did not place extra... To follow reasoning or modus tollens, if two facts are connected, the! To focus on big-picture objectives the terms are wordy and harder to.. Statement of the second premise argument form containing 2 premises - the major premise ( 29,... 0 `` if Xyrplex is 9, Guffaw is 1. ) used... Every Wednesday and Frank does not go to work today, then profits will increase four! Was really ticked off because he said that she lied to him focus on big-picture objectives since you now a...
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