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An Exposition of Symbolic Logic
with Kalish-Montague derivations
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The system of logic used here is essentially that of Kalish & Montague 1964 and Kalish, Montague and Mar, Jovanovich, 1992. The principle difference is that written justifications are required for boxing and canceling: ‘dd’ for a direct derivation, ‘id’ for an indirect derivation, etc. This text is written to be used along with the UCLA Logic 2010 software program, but that program is not mentioned, and the text can be used independently (although you would want to supplement the exercises).
The system of notation is almost the same as KK&M; major differences are that the signs ‘∀’ and ‘∃’ are used for the quantifiers, name and operation symbols are the small letters between ‘a’ and ‘h’, and variables are the small letters between ‘i’ and ‘z’.
The exercises are new.
Chapters 1-3 cover pretty much the same material as KM&M except that the rule allowing for the use of previously proved theorems is now in chapter 2, immediately following the section on theorems. (Previous versions of this text used the terminology ‘tautological implication’ in section 2.11. This has been changed to ‘tautological validity’ to agree with the logic program.)
Chapters 4-6 include invalidity problems with infinite universes, where one specifies the interpretation of notation “by description”; e.g. “R(): ≤”. These are discussed in the final section of each chapter, so they may easily be avoided. (They are not currently implemented in the logic program.)
Chapter 4 covers material from KK&M chapter IV, but without operation symbols. Chapter 4 also includes material from KK&M chapter VII, namely interchange of equivalents, biconditional derivations, monadic sentences without quantifier overlay, and prenex form.
Chapter 5 covers identity and operation symbols.
Chapter 6 covers Fregean definite descriptions, as in KK&M chapter VI.
Version Aug 2013 of An Exposition of Symbolic Logic
is a lightly revised version of the August 2012 version of An Introduction to Symbolic Logic (also known as Terry-Text).
Copyrighted material Introduction — 2 Version of Aug 2013

Chapter One
Sentential Logic with ‘if’ and ‘not’
1 SYMBOLIC NOTATION
2 MEANINGS OF THE SYMBOLIC NOTATION
3 SYMBOLIZATION: TRANSLATING COMPLEX SENTENCES INTO SYMBOLIC NOTATION 4 RULES
5 DIRECT DERIVATIONS
6 CONDITIONAL DERIVATIONS
7 INDIRECT DERIVATIONS
8 SUBDERIVATIONS
9 SHORTCUTS
10 STRATEGY HINTS FOR DERIVATIONS
11 THEOREMS
12 USING PREVIOUSLY PROVED THEOREMS IN DERIVATIONS
Chapter Two
Sentential Logic with ‘and’, ‘or’, if-and-only-if’
1 SYMBOLIC NOTATION
2 ENGLISH EQUIVALENTS OF THE CONNECTIVES 3 COMPLEX SENTENCES
5 SOME DERIVATIONS USING RULES S, ADJ, CB 6 ABBREVIATING DERIVATIONS
7 USING THEOREMS AS RULES
8 DERIVED RULES
9 OFFICIAL CONDITIONS FOR DERIVATIONS
10 TRUTH TABLES AND TAUTOLOGIES
11 TAUTOLOGICAL VALIDITY
Chapter Three
Individual constants, Predicates, Variables and Quantifiers
1 INDIVIDUAL CONSTANTS AND PREDICATES
2 QUANTIFIERS, VARIABLES, AND FORMULAS
3 SCOPE AND BINDING
4 MEANINGS OF THE QUANTIFIERS
5 SYMBOLIZING SENTENCES WITH QUANTIFIERS 6 DERIVATIONS WITH QUANTIFIERS
7 UNIVERSAL DERIVATIONS
8 SOME DERIVATIONS
9 DERIVED RULES
10 INVALIDITIES
11 EXPANSIONS
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Introduction — 3
Version of Aug 2013

Chapter Four Many-Place Predicates
1 MANY-PLACE PREDICATES
2 SYMBOLIZING SENTENCES USING MANY-PLACE PREDICATES 3 DERIVATIONS
4 THE RULE “INTERCHANGE OF EQUIVALENTS”
5 BICONDITIONAL DERIVATIONS
6 SENTENCES WITHOUT OVERLAY OF QUANTIFIERS
7 PRENEX NORMAL FORMS
8 SOME THEOREMS
9 SHOWING INVALIDITY
10 COUNTER-EXAMPLES WITH INFINITE UNIVERSES
Chapter Five
Identity and Operation Symbols
1 IDENTITY
2 AT LEAST AND AT MOST, EXACTLY, AND ONLY
3 DERIVATIONAL RULES FOR IDENTITY
4 INVALIDITIES WITH IDENTITY
5 OPERATION SYMBOLS
6 DERIVATIONS WITH COMPLEX TERMS
7 INVALID ARGUMENTS WITH OPERATION SYMBOLS 8 COUNTER-EXAMPLES WITH INFINITE UNIVERSES
Chapter Six
Definite Descriptions
1 DEFINITE DESCRIPTIONS
2 SYMBOLIZING SENTENCES WITH DEFINITE DESCRIPTIONS
3 DERIVATIONAL RULES FOR DEFINITE DESCRIPTIONS: PROPER DESCRIPTIONS
4 SYMBOLIZING ORDINARY LANGUAGE
5 DERIVATIONAL RULES FOR DEFINITE DESCRIPTIONS: IMPROPER DESCRIPTIONS 6 INVALIDITIES WITH DEFINITE DESCRIPTIONS
7 UNIVERSAL DERIVATIONS
8 COUNTER-EXAMPLES WITH INFINITE UNIVERSES
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Introduction
Logic is concerned with arguments, good and bad. With the docile and the reasonable, arguments are sometimes useful in settling disputes. With the reasonable, this utility attaches only to good arguments. It is the logician’s business to serve the reasonable. Therefore, in the realm of arguments, it is the logician who distinguishes good from bad.
Kalish & Montague 1964 p. 1
1 DEDUCTIVE REASONING
Logic is the study of correct reasoning. It is not a study of how this reasoning originates, or what its effects are in persuading people; it is rather a study of what it is that makes some reasoning “correct” as opposed to “incorrect”. If you have ever found yourself or someone else making a mistake in reasoning, then this is an example of someone being taken in by incorrect reasoning, and you have some idea of what we mean by correct reasoning: it is reasoning that contains no mistakes, persuasive or otherwise.
It is typical in logic to divide reasoning into two kinds: deductive and inductive, or, roughly, “airtight” and “merely probable”. Here is an example of probable reasoning. You have just been told that Mary bought a new car, and you say to yourself:
In the past, Mary always bought big cars. Big cars are usually gas-guzzlers.
So she (probably) now has a gas-guzzler.
Your conclusion, that Mary has a gas-guzzler, is not one that you think of as following logically from the information that you have; it is merely a probable inference.
Inductive Logic, which is the study of probable reasoning, is not very well understood at present. There are certain rather special cases that are well developed, such as the application of the probability calculus to gambling games. But a general study has not met with great success. This is not a book about probable reasoning, but if you are interested in it, this is the place to start. This is because most studies of Inductive Logic take for granted that you are already familiar with Deductive Logic — the logic of “airtight” reasoning — which forms the subject matter of this book. So you have to start here anyway.
Here is an example of deductive reasoning. Suppose that you recall reading that either or was a president of the United States, but you can’t remember which one. Some knowledgeable person tells you that was never president (he was a famous inventor). Based on this information you conclude that Polk was a president.
The information that you have, and the conclusion that you draw from this information, is:
Either Polk or Whitney was a president. Whitney was not a president.
So Polk was a president.
Let us compare this reasoning with the other reasoning given above. They both have one thing in common: the information that you start with is not known for certain. In the first example, you have only been told that Mary bought a new car, and this may be a lie or a mistake. Likewise, you may be misremembering her past preferences for car sizes. The same is true in the second reasoning: you were only told that was not president — by someone else or by a history book — and your memory that either Polk or Whitney was a president may also be inaccurate. In both cases the information that you start with is not known for certain, and so in this sense your conclusions are only probable. Reasoning is always reasoning from some claims, called the premises of the reasoning, to some further claim, called the conclusion. If the premises are not known for certain, then no matter how good the reasoning is, the conclusion will not be known for certain either. (There are certain special exceptions to this; see the exercises below.) There is, however, a difference in the nature of the inferences in the two cases. In the
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Introduction

Introduction
second piece of reasoning the reasoning itself is airtight in the following sense: If the premises that you start with are true, then you are guaranteed that your conclusion is true too. That is, if you were right in thinking that either Polk or Whitney was a president, and if you were right in thinking that Whitney was not a president, then you must be right in thinking that Polk was a president. In this case, it is logically impossible for your premises to be true and your conclusion, nonetheless, to be false. This is an example of what is called validity. If your reasoning is valid, then, although you are not guaranteed that your conclusion is true, you are guaranteed that it is true if your premises are.
This guarantee is absent in the case of inductive reasoning. Suppose that Mary has indeed just bought a new car, and suppose that you are correct in believing that she always bought big cars in the past, and also correct in believing that big cars are usually gas-guzzlers. You could still be wrong in your conclusion that she now has a gas-guzzler. Maybe she decided this time to buy a smaller car. Or maybe she got a big one with some extraordinary new fuel economy equipment. These may be unlikely, but they are not ruled out, even assuming that all of your premises are true. The reasoning is not deductively valid because there is a logical possibility that the conclusion is false even if the premises are all true. In short, in the case of inductive reasoning, the inconclusiveness of the reasoning itself introduces further uncertainty in addition to the original uncertainty of the premises.
We rarely have certain knowledge, and a study of logic will not give it to us. Logic is not a method of achieving certainty in general, though it sometimes yields such knowledge as a by-product; instead, it is a study of the logical relationships among all our sentences, including those that are only probable.
2 TRUTH & VALIDITY
A principle unit of investigation in logic is called an argument. An “argument”, in its technical sense, consists of two parts: a set of sentences, called the premises, and a sentence called the conclusion. The term “argument” may suggest a dispute, but in logic something is called an argument whether or not any people ever have or ever will disagree about it. Likewise, the “premises” of such an argument may or may not have been believed or asserted by somebody, and it is sometimes useful to examine arguments whose “premises” would never be believed by any rational person. Likewise, by calling something a “conclusion” we do not suggest that anyone ever has or even should “conclude” this thing on the basis of the premises given. The point of the terminology is this: a major topic in the study of deductive logic is validity. This is a relationship between a set of sentences and another sentence; this relationship holds whenever it is logically impossible for there to be a situation in which all the sentences in the first set are true and the other sentence false. It turns out to be very useful to study this relationship in complete generality. That is, it is useful to have a theory which tells us when this relationship holds between any set of sentences and any other sentence. Since a major practical application of such a theory is to pieces of reasoning that people actually use, the tradition has arisen of calling the first set of sentences the “premises”, and the other sentence the “conclusion”. And since a practical application of logic is to situations in which people disagree, it is perhaps appropriate to call the whole thing an “argument”. But these are now technical terms. An argument is simply something that has two parts: a set of sentences called the premises, and another sentence called the conclusion. For logical purposes, any such combination counts as an argument.
In displaying arguments it is customary to write their premises first, and to indicate the conclusion by the word like ‘so’ or a symbol such as ‘∴’
Either Polk was a president or Whitney was a president. Whitney was not a president.
∴ Polk was a president.
The triangle made of three dots is an abbreviation of the word `therefore’, and is a way of identifying the conclusion of an argument. In order to save on writing, and also to begin displaying the form of the arguments under discussion, we will start abbreviating simple sentences by capital letters. For the time being we will abbreviate Polk was a president by `P’,
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Introduction
and Whitney was a president by `W’. We will abbreviate Whitney was not a president by `not W’. So the argument can be shortened to:
P or W not W ∴P
A major point of this book is to explore the notion of deductive validity. Since the deductive kind is the only one considered here, we simply refer to it as “validity”. In this section we will go over certain consequences of the following definition of validity:
• An argument is valid if, and only if, there is no logically possible situation in which all of its premises are true and its conclusion false.
When we talk about “truth” here we do not have anything deep or mysterious in mind. For example, we say that the sentence ‘There is beer in the refrigerator’ is true if there is beer in the refrigerator, and false if there isn’t beer in the refrigerator. That’s all there is to it.
We have already seen one case of a valid argument which has all of its premises true and its conclusion true as well:
P or W True not W True ∴P True
What other possibilities are there? Well, as we noted above, it is possible to have some of the premises false and the conclusion false too. (This is sometimes referred to as a case of the “garbage in, garbage out” principle.) Suppose we use `R’ to abbreviate . Lee was a president. Then this argument does not have all of its premises true, nor is its conclusion true:
R or W not W ∴ R
False True
Yet this argument is just as good, as far as its validity is concerned, as the first one. If its premises were true, then that would guarantee that its conclusion would be true too. There is no logically possible situation in which the premises are all true and the conclusion false. This argument, though it starts with a false premise and ends up with a false conclusion, has exactly the same logical form as the first one. This sameness of logical form lies at the foundation of the theory in this book; it is discussed in the following section.
Although false inputs can lead to false outputs, there is no guarantee that this will happen, for you can reason validly from false information and accidentally end up with a conclusion that is true. Here is an example of that:
P or not W True W False ∴P True
In this example, one of the premises is false, but the conclusion happens to be true anyway. Mistaken assumptions can sometimes lead to a true conclusion by chance.
The one combination that we cannot have is a valid argument which has all true premises and a false conclusion. This is in keeping with the definition given above: a deductively valid argument is one for which it is logically impossible for its conclusion to be false if its premises are all true.
We have seen that there are valid arguments of each of these sorts:
PREMISES all true not all true not all true CONCLUSION true false true
What about invalid arguments? (That is, what about arguments that are not deductively valid?) What combination of truth-values can the parts of invalid arguments have? The answer is that they can have any combination of truth-values whatsoever. Here are some examples:
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P or W True P True ∴ W False
P True not W True ∴ not R True
P or W True W False ∴ P True
W or R False P True ∴ R False
PREMISES ALL TRUE CONCLUSION FALSE
PREMISES ALL TRUE CONCLUSION TRUE
PREMISES NOT ALL TRUE CONCLUSION TRUE
PREMISES NOT ALL TRUE CONCLUSION FALSE
The moral of the story so far is that if you know that an argument is invalid, that fact alone tells you nothing at all about the actual truth-values possessed by its parts. And if you know that it is valid, all that that fact tells you about the actual truth-values of its parts is that it does not have all of its premises true plus its conclusion false.
However, there is more to be said. Suppose that you want to show that an argument is invalid, but the argument does not already have all true premises and a false conclusion. How can you do this? One approach is to appeal directly to the characterization of validity, and describe a possible situation in which the premises are all true and the conclusion false. For example, suppose someone has given this (invalid) argument:
Either Roosevelt or Truman (or perhaps both) was a president.
Truman was a president.
∴ Roosevelt was a president.
There is no mistake of fact involved here, but the argument is a bad one, and you would like to establish this. You could do so as follows. You say:
“Suppose that Truman had been a president, but not Roosevelt. In that situation the premises would have been true, but the conclusion false.”
This is enough to show the reasoning bad, that is, to show the argument invalid. We can do even more than this, as we will see in the next section.
Introduction
b. Lee wasn’t a president, and Polk was. Either Polk or Whitney was a president. ∴ Whitney was a president.
This book provides a stock of exercises as an aid to learning. They were written in the belief that the “hands on” approach to modern logical theory is the best way to master it. You will also be supplied with answers to many of the exercises. You should attempt every exercise on your own, and then check your efforts against the answers that are given. If you do not understand one or more of the exercises, ask for help!
Several of the exercises contain material that supplements the explanations in the body of the text. None of the exercises presuppose material that is not provided in the text or in the exercise itself.
1. Decide whether each of the following arguments is valid or invalid. If the argument is invalid then describe a possible situation in which its premises are all true and its conclusion false.
a. Either Polk or Lee was a president. Either Lee or Whitney was a president.
∴ Either Polk or Whitney was a president.
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c. Polk was a president and so was Lee. Whitney was a president.
∴ Polk was a president and so was Whitney.
d. Either Polk or Whitney was a president. Lee was not a president.
∴ Lee wasn’t a president and Polk was.
2. Which of these are true, and which are false:
a. Some valid arguments have false conclusions.
b. No invalid argument has all true premises and a false conclusion.
c. If an argument is valid, and you produce a new argument from it by adding one or more
premises to it, the resulting argument will still be valid.
d. If an argument is invalid, and you produce a new argument from it by adding one or more
premises to it, the resulting argument must still be invalid.
e. If an argument has an impossible premise, it is valid. (An example of an impossible
sentence is `Some giraffes aren’t giraffes’.)
f. If an argument has a necessarily true conclusion, it is valid. (An example of a necessarily
true sentence is `Every giraffe is a giraffe’.) g. If an argument has a false premise, it is valid.
3. An argument which is valid and which also has all of its premises true is called sound. Based on this definition, which of the following are true, and which false:
Copyrighted material Introduction

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