Introductory Lecture
Introductory Lecture
Kayla Ann Marie
This is a course in basic formal logic Formal logic is the study of an argument’s form. When we reason we argue from a collection of known things: the premise of our argument. To conclude to another thing that is not so well known. The forms arguments take can be good or bad. EX: good o I have five toes on my left foot. The left and right sides of animals are generally approximate reflections of one another. I have five toes on my right foot. EX: bad o I have five toes on my left foot. Stephen Harper is the Prime Minister of Canada. Different arguments can follow a common form or pattern: All A’s are B’s. All B’s are C’s. All A’s are C’s. Some A’s are B’s. Some B’s are C’s. Some A’s are C’s. a, b, and c are A’s. d is like a, b, and b. d is an A If P then Q. But not-P, Not-q. If P then Q. P, Q. a is a B. Most B’s are C’s. a is a C. What separates good and bad arguments? Good argument forms are of two kinds: o Deductive o Inductive We are concerned with deductively good arguments in this course. o Inductively good argument forms establish the likelihood of the truth of the conclusion. When an argument form is deductively good, the form itself makes
it impossible to have all true premises and a false conclusion. So, when an argument form is deductively good, it is impossible to think of a substitution instance (an argument with that form) that does have either: at least one false premises OR a true conclusion. EX: o All A’s are B’s. All B’s are C’s. All A’s are C’s. o All lawyers are greedy. All greedy people are untrustworthy. All lawyers are untrustworthy
o All cats are felines. All felines are mammals. All cats are mammals. Method for proving an argument form is bad: a counter example, think up a single argument that shares that form that has all true premises and a false conclusion. EX: o Some cats weigh over 12 pounds. Some animals weighing over 12 pounds are dogs. Therefore, some cats are dogs. o How do you prove that a deductive argument form is a good one?
First step: logicians construct a formal language that represents just those features of natural language that are relevant to the forms of arguments. Most basic and general formal elements have to do with the way entire sentences are compounded in arguments using logically relevant “connective” expressions such as “and,” “or,” and “not.” This course begins with the study of that logic, sentence logic.
Why the study of formal deductive logic is important: Constructing formal language that represents logical features of natural language tells us something about how natural languages work. It also tells us how to construct machines that will compute the right results. Tells us how to understand how the mind processes information. It was once thought that mathematics could be reduced to 2nd order predicate logic, the study of formal logic contributes to understanding the foundations of mathematics.
Formal logic gives us an example of an especially rigorous and precise way of approaching any topic. The way you learn to pove results in formal logic is highly instructive for effectively proving results in any other field. F.L has been the model for discourse in contemporary analytic philosophy and a basic understanding of it is required to understand papers on a wide range of philosophical topics, from philosophy of language to mathetics.
When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true. To make a claim that is generally regarded as trust is to say or write a certain kind of sentences: a “declarative” sentence. Sentences of this sort are the most basic thing that logic is concerned with. Exercise 1.3#1: Declarative Setences When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true. Arguments are therefore used to persuade and to make discoveries. They are to be distinguished from collections of sentences that merely serve to report a number of facts without attempting to justify any of them. Also to be distinguished from collections of sentences that make claims that are generally not well known to be true and appeal to those claims to explain why something that is much better known came to be. In this courses, arguments will be put in “standard form.” This means numbering and listing premises first and placing the conclusion last under a bar. Deductive Validity: An argument form is deductively valid if and only if each substitution instance of that form either has at least one false premise or has a true conclusion.
An arugment form is deductively invalid as long as there is even on substitution instance of that form that has all true premises and a false conclusion. An argument is deductively valid if and nly if it has a deductively valid form. An argument is deductively invalid if and only if it does not have a deductively valid form. Deductively Valid Arguments can have:
o True premises and true conclusions. o False Premises and true conclusions. o False premises and false conclusions. o False premises and true conclusion. Deductively Invalid Arguments can have: o True premises and true conclusions. o True premises and false conclusions. o False premises and true conclusions. o False premises and false conclusions. Deductive Soundness: Objects
An argument is deductively sound if and only if it both has a deductively valid form and it has all true premises. An argument is not deductively sound if and only if it either has at least one false premise or has deductively invalid form. Deductively sound arguments must have true conclusions. This is not necessarily the case for deductively valid arguments of Classical Formal Logic: Sentences Sets of sentences Arguments Sentence of Classical Logic: a set of sentences of classical logic must have one or the other, but not both, of exactly two truth values, true (T) and false (F). Other sentences do not exist as far as classical logic is concerned. Set of sentences of classical logic: a set of sentences of classical logic is a list of from none to infinitely many sentences of classical logic placed between braces ({,}). A set with no members , is the empty or null set. A set with exactly one member is a unit set.
Argument: an argument is a collection of two or more sentences, one of which is identified or intended as a conclusion. The Properties and Relations of these Objects: Logical truth, falsity, and indeterminacy Logical equivalence Logical entailment Deductive Validity and invalidity
Logical truth, falsity, and indeterminacy are properties of sentences. o It is improper to describe arguments or sets as true, false, or indeterminate. Logical equivalence is a relation between certain pairs of sentences. Logical consistency and inconsistency are properties of sets of sentences. o It is improper to describe sentences or arguments as consistent or inconsistent. Logical entailment is a relation that holds between certain sets and
certain sentences. Deductive validity and invalidity are properties of arguments. o It is improper to describe sentences or sets as valid or invalid. These properties and relations can be defined in terms of: Form (just done for deductive validity) Non-contradiction (handout definitions) Possibility (textbook definitions) Contradiction: a contradiction is either a sentence of the form “P but not-P” or two sentences, one of the form “P,” and the other of the form, “not-P.” o You are caught in a contradiction when you end up saying something that reduces to a contradiction. E.g., denying that all fish are fish reduces to saying that there something that is a fish but that thing is not a fish. Definitions in terms of contradiction: A sentence of classical logic is logically true if and only if you would get caught in a contradiction were you to deny it. E.g., “All
fish are fish.” A sentence of classical logic is logically false if and only if you would get caught in a contradiction were you to affirm it. E.g., “Some fish are not fish.” A sentence of classical logic is logically indeterminate if and only if there is no contradiction in either affirming or denying it. E.g., “There are more than 80 people in this room.”
A pair of sentences of classical logic is logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “Some hawks are osprey.” “Some osprey are hawks.” A pair of sentences of classical logic is not logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “All whales are mammals.” “All mammals are whales.” A set of sentences of classical logic is logically consistent if and only if there is no contradiction in supposing all the sentences in the set are true. o E.g., {“Stephen Harper is the current Prime Minister of Canada”, “Bob Rae is the current Prime Minister of Canada”} A set of sentences of classical logic is logically inconsistent if and only if there is a contradiction in supposing all the sentences in the set are true. o E.g., {“Stephen Harper is the current Prime Minister of Canada”, “Bob Rae is the current Prime Minister of Canada”, “Stephen Harper is not Bab Rae”} A set of sentences of classical logic {P1, P2, P3, …} logically entails a sentence of classical logic, C, if and only if there is a contradiction in supposing that all of P1, P2, P3, … are true and that C is false. o (P1, P2, P3, … could be as few as 0 and as many as infinitely many sentences.) o E.g., {All whales are fish, All fish are birds} logically entails “All whales are birds” {P1, P2, P3 …} does not logically entail C if and only if there is no contradiction in supposing that all of P1, P2, P3, … are true and that C is false. o E.g., {Stephen Harper is Prime Minister of Canada, All whales are mammals} does not logically entail “Grass is green.” An argument is deductively valid if and only if the set of its premises logically entails its conclusions, that is, if and only if there is a contradiction in supposing its premises are true and its conclusion is false.
o E.g., 1. All whales are fish. 2. All fish are birds. 3. All whales are birds. An argument is deductively invalid if and only if the set of its premises does not logically entail its conclusion, that is, if and only if there is no contradiction in supposing that it’s premises are true and its conclusion is false. o E.g., 1. Stephen Harper is Prime Minister of Canada. 2. All whales are mammals. 3. Grass is green. Alternative Definitions: These logical properties and relations can also be described in
terms of possibility or in terms of substitution instances of forms. Why this amounts to the same thing: o Because “possible,” in these definitions does not mean “likely” or “allowed by the laws of physics.” It means “thinkable without running into a contradiction.” o Because what produces contradictions (and so makes things logically impossible) is the form (pattern, structure) of sentences, sets, and arguments. The job of classical deductive logical to identify the forms that produce logical truth, falsity, indeterminacy, equivalence, inconsistency, and entailment, as well as deductive validity or invalidity.
Exercise 1.3#1: Declarative Sentences 8/21/2013 10:33:00 AM Exercise 1.3#1: Declarative Sentences A) George Washington was the second president of the United States. B) The next president of the United States will be a Republican. C) Turn in your homework on time or not at all. D) Would that John Kennedy had not been assassinated E) Two is the smallest prime number. F) One is the smallest prime number. G) George Bush senior was the immediate predecessor to George W. as president.
H) On January 15, 1134, there was a snowstorm in what is now Manhattan, at 3:00pm EST. I) Sentence m below is true. M) This sentence is false. J) May you live long ad prosper. K) Never look a gift horse in the mouth. L) Who created this screwy examples? N) Beware of Greeks bearing gifts.
Exercise 1.3 #2
8/21/2013 10:33:00 AM
Exercise 1.3 #2 A. When Mike, Sharon, Sandy and Vicky are all out of the office, no important decisions get made. Mike is off skiing, Sharon is in Spokane, Vicky is in Olympia, and Sandy is in Seattle. So no decisions will be made today. B. Our press releases are always crisp and upbeat. That’s because, though Jack doesn’t like sound bites, Mike does. And Mike is the press officer. C. Shelby and Noreen are wonderful in dealing with irate students and faculty. Stephanie is wonderful at managing that Chancellor’s
very demanding schedule, and Tina keeps everything moving and cheers everyone up. D. This is a great office to work in. Shelby and Noreen are wonderful in dealing with irate students and faculty. Stephanie is wonderful at managing the chancellor’s very demanding schedule, and Tina keeps everything moving and cheers everyone up. E. The galvanized nails, both common and finishing, are in the first drawer. The plain nails are in the second drawer. The third drawer contains Sheetrock screws of various sizes, and the fourth drawer contains wood screws. The bottom drawer contains miscellaneous hardware. F. The galvanized nails, both common and finishing, are in the first drawer. The plain nails are in the second drawer. The third drawer contains Sheetrock screws of various sizes, and the fourth drawer contains wood screws. The bottom drawer contains miscellaneous hardware. So we should have everything we need to repair the broken deck chair. G. The weather is perfect; the view is wonderful; and we’re on vacation. So why are you unhappy? H. The new kitchen cabinets are done, and the installers are scheduled to come Monday. But there will probably be a delay of at least a week, for the old cabinets haven’t removed, and the carpenter who is to do the removal is off for a week of duck hunting in North Dakota. I. Wood boats are beautiful but they require too much maintenance. Fiberglass boats require far less maintenance, but they tend to be
more floating bathtubs than real sailing craft. Steel boats are hard to find, and concrete boats never caught on. So there’s no boat that will please me. J. Sarah, John, Rita, and Bob have all worked hard and all deserve promotion. But the company is having a cash flow problem and is offering those over 55 a $5,000 bonus if they will retire at the end of this year. Sarah, John, and Bob are all over 55 and will take early retirement. So Rita will be promoted. K. Everyone from anywhere who’s anyone knows Barrett. All those who know her respect her and like her. Friedman is from Minneapolis and Barrett is from Duluth. Friedman doesn’t like anyone from Duluth. Therefore, either Friedman is a nobody or Minneapolis is a nowhere. L. I’m not going to die today. I didn’t die yesterday, and I didn’t die the day before that, or the day before that, and so on back some fifty years. M. Having cancer is a good, for whatever is required by something that is a good is itself a good. Being cured of cancer is a good, and being cured of cancer requires having cancer. N. The Soviet Union disintegrated because the perceived need for the military security offered by the union disappeared with the end of the cold war and because over 70 years of union had produced few economic benefits. Moreover the Soviet Union never successfully addressed the problem of how to inspire loyalty to a single state by peoples with vastly different cultures and histories. O. Only the two-party system is compatible both with effective governance and with the presenting and contesting of dissenting views, for when there are more than two political parties, support tends to split among the parties, with no party receiving the support of the majority of voters. And no party can govern effectively without majority support. When there is only one political party, dissenting views are neither presented nor contested. When there are two or more viable parties, dissenting views are presented and contested. P. Humpty Dumpty sat on a wall. Humpty Dumpty had a great fall. All the king’s horses and all the king’s men couldn’t put Humpty
together again. So they made him into an omelette and had a great lunch.
Some Corollaries
8/21/2013 10:33:00 AM
Prove each of the following by giving an argument or example, as appropriate. 1. All logically true sentences are logically equivalent to one another, e.g., “All fish are fish,” “Either I am over 5 ft. tall or I am not.” 2. All logically false sentences are logically equivalent to one another, e.g., “Some fish are not fish,” “I am over 5 ft. tall but I am not.” 3. No logically indeterminate sentence is logically equivalent to any logically true sentence or to any logically false sentence. 4. Some logically indeterminate sentences are logically equivalent to one another; others are not. 5. The unit set of any logically true sentence is logically consistent. 6. So is the unit set of any logically indeterminate sentence. 7. The unit set of any logically false sentence is logically inconsistent. 8. The set consisting of two logically equivalent sentences need not be logically consistent. 9. The set consisting of two logically equivalent logically indeterminate sentences must be logically consistent. 10. The null set is considered to be logically consistent. Bonus Exercise: Write a short critical essay explaining why. 11. A logically inconsistent set logically entails any sentence, including any logically false sentence. 12. A set that contains a logically false sentence logically entails any sentence, including any logically false sentence.
13. A logically true sentence is logically entailed by any set, including the null set. 14. An argument with logically inconsistent premises or a logically false premise must be deductively valid. 15. An argument with a logically true conclusion must be deductively valid. 16. The set consisting of the premises and the conclusion of a deductively valid argument need not be logically consistent. 17. The set consisting of the premises and the conclusion of a deductively invalid argument need not be logically inconsistent. 18. The set consisting of the premises and the negation of the conclusion of a deductively valid argument must be logically inconsistent.
Kayla Ann Marie
This is a course in basic formal logic Formal logic is the study of an argument’s form. When we reason we argue from a collection of known things: the premise of our argument. To conclude to another thing that is not so well known. The forms arguments take can be good or bad. EX: good o I have five toes on my left foot. The left and right sides of animals are generally approximate reflections of one another. I have five toes on my right foot. EX: bad o I have five toes on my left foot. Stephen Harper is the Prime Minister of Canada. Different arguments can follow a common form or pattern: All A’s are B’s. All B’s are C’s. All A’s are C’s. Some A’s are B’s. Some B’s are C’s. Some A’s are C’s. a, b, and c are A’s. d is like a, b, and b. d is an A If P then Q. But not-P, Not-q. If P then Q. P, Q. a is a B. Most B’s are C’s. a is a C. What separates good and bad arguments? Good argument forms are of two kinds: o Deductive o Inductive We are concerned with deductively good arguments in this course. o Inductively good argument forms establish the likelihood of the truth of the conclusion. When an argument form is deductively good, the form itself makes
it impossible to have all true premises and a false conclusion. So, when an argument form is deductively good, it is impossible to think of a substitution instance (an argument with that form) that does have either: at least one false premises OR a true conclusion. EX: o All A’s are B’s. All B’s are C’s. All A’s are C’s. o All lawyers are greedy. All greedy people are untrustworthy. All lawyers are untrustworthy
o All cats are felines. All felines are mammals. All cats are mammals. Method for proving an argument form is bad: a counter example, think up a single argument that shares that form that has all true premises and a false conclusion. EX: o Some cats weigh over 12 pounds. Some animals weighing over 12 pounds are dogs. Therefore, some cats are dogs. o How do you prove that a deductive argument form is a good one?
First step: logicians construct a formal language that represents just those features of natural language that are relevant to the forms of arguments. Most basic and general formal elements have to do with the way entire sentences are compounded in arguments using logically relevant “connective” expressions such as “and,” “or,” and “not.” This course begins with the study of that logic, sentence logic.
Why the study of formal deductive logic is important: Constructing formal language that represents logical features of natural language tells us something about how natural languages work. It also tells us how to construct machines that will compute the right results. Tells us how to understand how the mind processes information. It was once thought that mathematics could be reduced to 2nd order predicate logic, the study of formal logic contributes to understanding the foundations of mathematics.
Formal logic gives us an example of an especially rigorous and precise way of approaching any topic. The way you learn to pove results in formal logic is highly instructive for effectively proving results in any other field. F.L has been the model for discourse in contemporary analytic philosophy and a basic understanding of it is required to understand papers on a wide range of philosophical topics, from philosophy of language to mathetics.
When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true. To make a claim that is generally regarded as trust is to say or write a certain kind of sentences: a “declarative” sentence. Sentences of this sort are the most basic thing that logic is concerned with. Exercise 1.3#1: Declarative Setences When we reason we make claims that are generally well known to be true, and then draw conclusions from those claims that are less well known to be true. Arguments are therefore used to persuade and to make discoveries. They are to be distinguished from collections of sentences that merely serve to report a number of facts without attempting to justify any of them. Also to be distinguished from collections of sentences that make claims that are generally not well known to be true and appeal to those claims to explain why something that is much better known came to be. In this courses, arguments will be put in “standard form.” This means numbering and listing premises first and placing the conclusion last under a bar. Deductive Validity: An argument form is deductively valid if and only if each substitution instance of that form either has at least one false premise or has a true conclusion.
An arugment form is deductively invalid as long as there is even on substitution instance of that form that has all true premises and a false conclusion. An argument is deductively valid if and nly if it has a deductively valid form. An argument is deductively invalid if and only if it does not have a deductively valid form. Deductively Valid Arguments can have:
o True premises and true conclusions. o False Premises and true conclusions. o False premises and false conclusions. o False premises and true conclusion. Deductively Invalid Arguments can have: o True premises and true conclusions. o True premises and false conclusions. o False premises and true conclusions. o False premises and false conclusions. Deductive Soundness: Objects
An argument is deductively sound if and only if it both has a deductively valid form and it has all true premises. An argument is not deductively sound if and only if it either has at least one false premise or has deductively invalid form. Deductively sound arguments must have true conclusions. This is not necessarily the case for deductively valid arguments of Classical Formal Logic: Sentences Sets of sentences Arguments Sentence of Classical Logic: a set of sentences of classical logic must have one or the other, but not both, of exactly two truth values, true (T) and false (F). Other sentences do not exist as far as classical logic is concerned. Set of sentences of classical logic: a set of sentences of classical logic is a list of from none to infinitely many sentences of classical logic placed between braces ({,}). A set with no members , is the empty or null set. A set with exactly one member is a unit set.
Argument: an argument is a collection of two or more sentences, one of which is identified or intended as a conclusion. The Properties and Relations of these Objects: Logical truth, falsity, and indeterminacy Logical equivalence Logical entailment Deductive Validity and invalidity
Logical truth, falsity, and indeterminacy are properties of sentences. o It is improper to describe arguments or sets as true, false, or indeterminate. Logical equivalence is a relation between certain pairs of sentences. Logical consistency and inconsistency are properties of sets of sentences. o It is improper to describe sentences or arguments as consistent or inconsistent. Logical entailment is a relation that holds between certain sets and
certain sentences. Deductive validity and invalidity are properties of arguments. o It is improper to describe sentences or sets as valid or invalid. These properties and relations can be defined in terms of: Form (just done for deductive validity) Non-contradiction (handout definitions) Possibility (textbook definitions) Contradiction: a contradiction is either a sentence of the form “P but not-P” or two sentences, one of the form “P,” and the other of the form, “not-P.” o You are caught in a contradiction when you end up saying something that reduces to a contradiction. E.g., denying that all fish are fish reduces to saying that there something that is a fish but that thing is not a fish. Definitions in terms of contradiction: A sentence of classical logic is logically true if and only if you would get caught in a contradiction were you to deny it. E.g., “All
fish are fish.” A sentence of classical logic is logically false if and only if you would get caught in a contradiction were you to affirm it. E.g., “Some fish are not fish.” A sentence of classical logic is logically indeterminate if and only if there is no contradiction in either affirming or denying it. E.g., “There are more than 80 people in this room.”
A pair of sentences of classical logic is logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “Some hawks are osprey.” “Some osprey are hawks.” A pair of sentences of classical logic is not logically equivalent if and only if there would be a contradiction in supposing that they have different truth values. E.g., “All whales are mammals.” “All mammals are whales.” A set of sentences of classical logic is logically consistent if and only if there is no contradiction in supposing all the sentences in the set are true. o E.g., {“Stephen Harper is the current Prime Minister of Canada”, “Bob Rae is the current Prime Minister of Canada”} A set of sentences of classical logic is logically inconsistent if and only if there is a contradiction in supposing all the sentences in the set are true. o E.g., {“Stephen Harper is the current Prime Minister of Canada”, “Bob Rae is the current Prime Minister of Canada”, “Stephen Harper is not Bab Rae”} A set of sentences of classical logic {P1, P2, P3, …} logically entails a sentence of classical logic, C, if and only if there is a contradiction in supposing that all of P1, P2, P3, … are true and that C is false. o (P1, P2, P3, … could be as few as 0 and as many as infinitely many sentences.) o E.g., {All whales are fish, All fish are birds} logically entails “All whales are birds” {P1, P2, P3 …} does not logically entail C if and only if there is no contradiction in supposing that all of P1, P2, P3, … are true and that C is false. o E.g., {Stephen Harper is Prime Minister of Canada, All whales are mammals} does not logically entail “Grass is green.” An argument is deductively valid if and only if the set of its premises logically entails its conclusions, that is, if and only if there is a contradiction in supposing its premises are true and its conclusion is false.
o E.g., 1. All whales are fish. 2. All fish are birds. 3. All whales are birds. An argument is deductively invalid if and only if the set of its premises does not logically entail its conclusion, that is, if and only if there is no contradiction in supposing that it’s premises are true and its conclusion is false. o E.g., 1. Stephen Harper is Prime Minister of Canada. 2. All whales are mammals. 3. Grass is green. Alternative Definitions: These logical properties and relations can also be described in
terms of possibility or in terms of substitution instances of forms. Why this amounts to the same thing: o Because “possible,” in these definitions does not mean “likely” or “allowed by the laws of physics.” It means “thinkable without running into a contradiction.” o Because what produces contradictions (and so makes things logically impossible) is the form (pattern, structure) of sentences, sets, and arguments. The job of classical deductive logical to identify the forms that produce logical truth, falsity, indeterminacy, equivalence, inconsistency, and entailment, as well as deductive validity or invalidity.
Exercise 1.3#1: Declarative Sentences 8/21/2013 10:33:00 AM Exercise 1.3#1: Declarative Sentences A) George Washington was the second president of the United States. B) The next president of the United States will be a Republican. C) Turn in your homework on time or not at all. D) Would that John Kennedy had not been assassinated E) Two is the smallest prime number. F) One is the smallest prime number. G) George Bush senior was the immediate predecessor to George W. as president.
H) On January 15, 1134, there was a snowstorm in what is now Manhattan, at 3:00pm EST. I) Sentence m below is true. M) This sentence is false. J) May you live long ad prosper. K) Never look a gift horse in the mouth. L) Who created this screwy examples? N) Beware of Greeks bearing gifts.
Exercise 1.3 #2
8/21/2013 10:33:00 AM
Exercise 1.3 #2 A. When Mike, Sharon, Sandy and Vicky are all out of the office, no important decisions get made. Mike is off skiing, Sharon is in Spokane, Vicky is in Olympia, and Sandy is in Seattle. So no decisions will be made today. B. Our press releases are always crisp and upbeat. That’s because, though Jack doesn’t like sound bites, Mike does. And Mike is the press officer. C. Shelby and Noreen are wonderful in dealing with irate students and faculty. Stephanie is wonderful at managing that Chancellor’s
very demanding schedule, and Tina keeps everything moving and cheers everyone up. D. This is a great office to work in. Shelby and Noreen are wonderful in dealing with irate students and faculty. Stephanie is wonderful at managing the chancellor’s very demanding schedule, and Tina keeps everything moving and cheers everyone up. E. The galvanized nails, both common and finishing, are in the first drawer. The plain nails are in the second drawer. The third drawer contains Sheetrock screws of various sizes, and the fourth drawer contains wood screws. The bottom drawer contains miscellaneous hardware. F. The galvanized nails, both common and finishing, are in the first drawer. The plain nails are in the second drawer. The third drawer contains Sheetrock screws of various sizes, and the fourth drawer contains wood screws. The bottom drawer contains miscellaneous hardware. So we should have everything we need to repair the broken deck chair. G. The weather is perfect; the view is wonderful; and we’re on vacation. So why are you unhappy? H. The new kitchen cabinets are done, and the installers are scheduled to come Monday. But there will probably be a delay of at least a week, for the old cabinets haven’t removed, and the carpenter who is to do the removal is off for a week of duck hunting in North Dakota. I. Wood boats are beautiful but they require too much maintenance. Fiberglass boats require far less maintenance, but they tend to be
more floating bathtubs than real sailing craft. Steel boats are hard to find, and concrete boats never caught on. So there’s no boat that will please me. J. Sarah, John, Rita, and Bob have all worked hard and all deserve promotion. But the company is having a cash flow problem and is offering those over 55 a $5,000 bonus if they will retire at the end of this year. Sarah, John, and Bob are all over 55 and will take early retirement. So Rita will be promoted. K. Everyone from anywhere who’s anyone knows Barrett. All those who know her respect her and like her. Friedman is from Minneapolis and Barrett is from Duluth. Friedman doesn’t like anyone from Duluth. Therefore, either Friedman is a nobody or Minneapolis is a nowhere. L. I’m not going to die today. I didn’t die yesterday, and I didn’t die the day before that, or the day before that, and so on back some fifty years. M. Having cancer is a good, for whatever is required by something that is a good is itself a good. Being cured of cancer is a good, and being cured of cancer requires having cancer. N. The Soviet Union disintegrated because the perceived need for the military security offered by the union disappeared with the end of the cold war and because over 70 years of union had produced few economic benefits. Moreover the Soviet Union never successfully addressed the problem of how to inspire loyalty to a single state by peoples with vastly different cultures and histories. O. Only the two-party system is compatible both with effective governance and with the presenting and contesting of dissenting views, for when there are more than two political parties, support tends to split among the parties, with no party receiving the support of the majority of voters. And no party can govern effectively without majority support. When there is only one political party, dissenting views are neither presented nor contested. When there are two or more viable parties, dissenting views are presented and contested. P. Humpty Dumpty sat on a wall. Humpty Dumpty had a great fall. All the king’s horses and all the king’s men couldn’t put Humpty
together again. So they made him into an omelette and had a great lunch.
Some Corollaries
8/21/2013 10:33:00 AM
Prove each of the following by giving an argument or example, as appropriate. 1. All logically true sentences are logically equivalent to one another, e.g., “All fish are fish,” “Either I am over 5 ft. tall or I am not.” 2. All logically false sentences are logically equivalent to one another, e.g., “Some fish are not fish,” “I am over 5 ft. tall but I am not.” 3. No logically indeterminate sentence is logically equivalent to any logically true sentence or to any logically false sentence. 4. Some logically indeterminate sentences are logically equivalent to one another; others are not. 5. The unit set of any logically true sentence is logically consistent. 6. So is the unit set of any logically indeterminate sentence. 7. The unit set of any logically false sentence is logically inconsistent. 8. The set consisting of two logically equivalent sentences need not be logically consistent. 9. The set consisting of two logically equivalent logically indeterminate sentences must be logically consistent. 10. The null set is considered to be logically consistent. Bonus Exercise: Write a short critical essay explaining why. 11. A logically inconsistent set logically entails any sentence, including any logically false sentence. 12. A set that contains a logically false sentence logically entails any sentence, including any logically false sentence.
13. A logically true sentence is logically entailed by any set, including the null set. 14. An argument with logically inconsistent premises or a logically false premise must be deductively valid. 15. An argument with a logically true conclusion must be deductively valid. 16. The set consisting of the premises and the conclusion of a deductively valid argument need not be logically consistent. 17. The set consisting of the premises and the conclusion of a deductively invalid argument need not be logically inconsistent. 18. The set consisting of the premises and the negation of the conclusion of a deductively valid argument must be logically inconsistent.
