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Basic Concepts and Distinctions1
Logic
Keith Burgess-Jackson
25 August 2016
1. Logic is the science (study) of validity.2
2. Validity (to be explained more fully below) is a property (i.e., feature
or characteristic) of deductive arguments. Some deductive arguments
possess this (desirable) property and some do not. Those that possess it
are said to be valid. Those that do not possess it are said to be invalid.
3. An argument (deductive or otherwise) consists of two or more propositions, one of which, the conclusion, is claimed (by the arguer) to follow
from the other or others, the premises. The premises are said to provide
reasons, evidence, grounds, or support for the conclusion. The act or process of arguing is known as argumentation.3
4. Every argument, by definition, involves a claim (by the arguer) that its
conclusion follows from its premise(s). Some claims are stronger than
others:
a. If one’s claim is that the conclusion cannot be false, given the
truth of the premises, then one’s argument is deductive.
b. If one’s claim is that the conclusion is unlikely to be false, given
the truth of the premises, then one’s argument is inductive.
Deduction is to necessity as induction is to probability.4 What follows is
a deductive argument (for almost certainly the arguer in question would
I thank my colleague Dan Giberman for helpful comments on this handout.
See the Appendix for alternative definitions of “logic.” There is no canonical definition of the term.
3 There is a close connection between inference and argumentation, but the processes (activities) are not identical. Inference, or reasoning, is the (psychological) process by
which one draws a conclusion from one or more premises. For example, I may infer from the
fact that the roads are icy this morning that UTA will be closed today. Argumentation, by
contrast, is interpersonal. To argue is to try to persuade (or convince) someone (perhaps
everyone) to accept or believe a particular proposition. When an inference is expressed (i.e.,
formulated) as an argument, it can be evaluated with the tools of logic. I will focus on arguments in the remainder of this handout, but much of what I say about arguments can also
be said about inferences.
4 D:N::I:P. We might also say that deduction is to induction as necessity is to
probability, or D:I::N:P.
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2
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claim that the truth of the premises necessitates the truth of the conclusion):
All mammals are animals.
All dogs are mammals.
Therefore,
All dogs are animals.
Here, by way of contrast, is an inductive argument (for almost certainly
the arguer in question would not claim that the truth of the premises
necessitates the truth of the conclusion):
Most professors are atheists.
Jones is a professor.
Therefore,
Jones is an atheist.
Strictly speaking, only deductive arguments are valid or invalid. We
may, if we like, say that all inductive arguments are invalid, but this is
misleading, for it implies that there is something wrong with inductive
arguments. There is nothing wrong with inductive arguments. Some
inductive arguments are cogent, in the sense that (a) they have true
premises and (b) their premises make their conclusions probable. Example: “Most dogs have a tail of at least one inch in length; Shelbie is a dog;
therefore, Shelbie has a tail of at least one inch in length.” Perhaps it’s
best to say that the terms “valid” and “invalid” don’t apply to inductive
arguments, i.e., that inductive arguments are neither valid nor invalid.
They belong in category 3 of the following taxonomy:5
5 A taxonomy is a classification scheme. The taxonomy in the text is both jointly
exhaustive and mutually exclusive (because of how the categories are defined). To say that
it is jointly exhaustive is to say that every argument goes in at least one of its three categories. To say that it is mutually exclusive is to say that no argument goes in more than one
of its three categories. It follows that every argument goes in exactly one of its three categories. Every argument, in other words, is either (1) a valid (i.e., truth-preserving) deductive
argument, (2) an invalid (i.e., non-truth-preserving) deductive argument, or (3) an argument
that is neither valid nor invalid. Category 3 could, of course, be subdivided, for it contains
all inductive arguments, some of which are cogent (such as the one in the text) and some of
which are not cogent (e.g., “Eighty-five percent of the readers of the New York Times oppose
capital punishment; therefore, 85% of Americans oppose capital punishment”).
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Arguments
Valid
1
Not Valid
Invalid
2
Not Invalid
3
5. Ideally, one would like one’s argument to have both of the following
properties: necessity and informativeness. That is, one would like one’s
conclusion (a) to follow necessarily from one’s premises and (b) to “go beyond” one’s premises in terms of the amount of information conveyed.
Unfortunately, no argument can have both of these properties. To get
necessity, one must forgo informativeness. To get informativeness, one
must forgo necessity. Let us see why. In a (valid) deductive argument,
the conclusion may contain either the same amount of information as the
conjunction of the premises or less information than the conjunction of
the premises, but it may not contain more information than the conjunction of the premises. Consider the following classic argument:
All men (i.e., human beings) are mortal.
Socrates is a man (i.e., a human being).
Therefore,
Socrates is mortal.
The conclusion, which is about a particular human being (namely, Socrates), contains less information than the conjunction of the premises. (The
first premise alone contains information about many human beings.)
Now consider a typical inductive argument:
On nine of the past 10 occasions in which weather conditions
were as they are now, it rained within six hours.
Therefore,
It will rain within six hours.
The conclusion contains more information than the premise. The premise is about the past; the conclusion is about the future. We might say
that deduction purchases necessity at the cost of informativeness,
whereas induction purchases informativeness at the cost of necessity.6
Both types of argument confer knowledge:
6 See Wesley C. Salmon, The Foundations of Scientific Inference (Pittsburgh: University of Pittsburgh Press, 1967), 8.
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a. The knowledge conferred by deduction consists in making explicit (in the conclusion) what is implicit (in the premises).
b. The knowledge conferred by induction consists in extrapolating
(in the conclusion) from what is already known (in the premises).
We may think of the choice between necessity and informativeness as the
tragedy of logic, because in one of its senses the word “tragedy” means “a
sad event” or “a calamity.”7 The calamity is that we can’t have everything
we want or need.8
6. Propositions have truth values. In this course, we shall assume that
there are two (and only two) truth values: ‘true’ and ‘false.’9 Here are two
Laws of Thought:10
a. Every proposition is either true or false, i.e., the truth values
‘true’ and ‘false’ are jointly exhaustive (this is known as the Law
of Excluded Middle or, more precisely, the Law of Bivalence); and
b. No proposition is both true and false, i.e., the truth values ‘true’
and ‘false’ are mutually exclusive (this is known as the Law of
Noncontradiction).11
Sentences, unlike propositions, are linguistic entities, which means that
they are always in particular languages, such as English, German, Swahili, or Latin. Propositions, which are in no particular language, are what
indicative (declarative) sentences express, assert, or signify.12 Two different indicative (declarative) sentences (e.g., “John loves Mary” and
“Mary is loved by John” [both of which are in English], or “It is raining”
and “Il pleut” [the former of which is in English, the latter of which is in
The Oxford American Dictionary and Language Guide (1999), 1068.
You may have heard the term “tragic choice.” Suppose two or more people need
life-saving medicine, but that I, a physician, have only enough medicine for one of them.
Obviously, I would like to save all the needy people, and would do so if I could, but the situation constrains me. I must make a tragic choice.
9 In what is called a “many-valued logic,” there are more than two truth values.
10 Strictly speaking, they are implications or entailments of the Laws of Thought,
but we will ignore that complication here.
11 While every proposition is either true or false and no proposition is both true
and false, it doesn’t follow that we always know which truth value a given proposition has.
Take the proposition that Abraham Lincoln thought about his son Robert five seconds before
he (Abraham) was shot. This proposition is either true or false and it is not both true and
false, but we will probably never know its truth value. (Query: What would constitute evidence for its truth? What would constitute evidence for its falsity?)
12 Sentence is to proposition as numeral is to number.
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8
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French]) can express, assert, or signify the same proposition. This is
called synonymy.13 A given indicative (declarative) sentence (e.g., “Jones
is happy”) can express, assert, or signify different propositions, depending
on such things as (a) when it is uttered (Jones may be happy at one time
but not at another) and (b) what its terms (e.g., “Jones”) refer to (Adam
Jones may be happy while Andruw Jones is not). This is called ambiguity.
7. A valid argument is a deductive argument that has the following (desirable) property (i.e., feature or characteristic): it is logically impossible
for its premises to be true while its conclusion is false. (If it is logically
possible for a given argument’s premises to be true while its conclusion is
false, then the argument is invalid.) Valid arguments are truth-preserving:
a. If the premises of a valid argument are true, then its conclusion
must be true as well.
b. If the conclusion of a valid argument is false, then at least one of
its premises must be false as well.
Note that there can be a valid argument with false premises as well as
an invalid argument with true premises. An example of the former is:
Texas is east of the Mississippi River.
Therefore,
Something is east of the Mississippi River.
An example of the latter is:
Something is east of the Mississippi River.
Therefore,
Texas is east of the Mississippi River.
Note also that, while all valid arguments are deductive, not all deductive
arguments are valid. What makes an argument deductive (as we saw
earlier) is the strength of the claim being made by the arguer. What
makes a deductive argument valid (we are now seeing) is the correctness
of the claim being made by the arguer. This course is concerned exclusively with deduction. If you wish to study induction, you should take
13
Some people refer to it as “synonymity.”
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Fundamentals of Reasoning (PHIL 1301).14
8. Validity is valuable (i.e., worthy of being valued, not merely capable
of being valued) for the sake of what it preserves, namely, truth. It is not
valuable for its own sake.15 Truth (i.e., true belief) is valuable because it
is essential to knowledge. (One can’t know falsehoods, though one can
believe falsehoods.) Knowledge is valuable because it is a component of
the good life. As the Greek philosopher Socrates long ago put it (according to his disciple Plato), “The unexamined life is not worth living.”16
9. Any argument that has a valid form is a valid argument. If one’s aim
is to produce valid arguments, therefore, it is in one’s interest to know as
many valid argument forms as possible. We will examine (i.e., you will
learn) 93 valid argument forms in this course: 35 in categorical logic, 40
in propositional logic, and 18 in predicate logic.
10. The form of an argument is its skeleton—the part that remains after
the flesh has been removed. For example, the form of the argument “No
dogs are cats; therefore, no cats are dogs” is “No Ø are Ψ; therefore, no
Ψ are Ø.” Another argument with the same form as this is “No animals
are dogs; therefore, no dogs are animals.” This shows that there can be
two or more arguments of (with) the same form. (We might say that each
of them instantiates the form.) Since validity has to do solely with the
form of an argument (and not with its content, matter, or substance), if
two arguments have the same specific form, then either both of them are
valid or both of them are invalid.17
14 The two courses may be taken in any order; neither, in other words, is a prerequisite for the other. Some people disparage Fundamentals of Reasoning (which at some
universities is known as Critical Thinking or Informal Logic) as “baby logic.” That’s like
saying that abnormal psychology is “baby psychology” or that microeconomics is “baby economics.” The two types of logic are distinct but equally important. I recommend that both
courses be taken by every student.
15 In other words, validity is extrinsically or instrumentally valuable, not intrinsically valuable. Validity is a means to an end, not an end in itself. In this respect, validity is
like money and unlike, say, friendship, knowledge, pleasure, or beauty.
16 See Plato’s Apology.
17 This is the basis of what is known as refutation by logical analogy. If one’s aim
is to refute a particular argument (call it “A”), one may do so by thinking up a second argument (call it “B”) that (1) has the same specific form as A but (2) has true premises and a
false conclusion. The fact that B has true premises and a false conclusion shows that it is
invalid, for, by definition, no valid argument has true premises and a false conclusion; the
fact that B has the same specific form as A shows that A, like B, is invalid.
Do not confuse refutation with rebuttal. To refute is to prove the invalidity of a
given argument. “Refutation” is a success term. One can attempt to refute an argument but
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11. A sound argument is a valid argument all of whose premises are
true. It follows from this definition that:
a. All sound arguments have true conclusions. (Note the difference
between x following from a definition and x being part of a definition.)18
b. Any argument that is invalid is unsound.
c. Any argument that has a false premise (even one) is unsound.
A given argument may be invalid and have a false premise. Such an
argument has two defects: one of them formal (invalidity) and the other
material or substantive (a false premise). Here is an example of a doubly
defective argument:
Keith’s automobile is green.
Therefore,
Everything is green.
As a matter of fact, Keith’s automobile is not green; but even if it were
green, it would not follow that everything is green.19 Here, in the form of
a taxonomy (of deductive arguments), is a summary of this section:
Valid
Invalid
All Premises True
Sound
Unsound
Not All Premises True
Unsound
Unsound
12. Validity and soundness are all or nothing, not matters of degree.
(Figuratively speaking, they are digital, not analog.) A given deductive
fail to do so, just as one can attempt to refute an argument and succeed in doing so. “Rebut,”
by contrast, is not a success term. To rebut is to push back against something—with no
implication that the pushing succeeds (or fails, for that matter). Refutation may be thought
of as successful rebuttal.
18 Suppose I put the following true-false question on an examination: “By definition, all sound arguments have true conclusions.” The answer to this question is “false.”
While it’s true that all sound arguments have true conclusions, it’s not true by definition.
The definition of “sound argument” makes no reference to the conclusion of the argument,
much less to the conclusion being true. The definition of “sound argument” makes reference
to two things: (1) validity; and (2) the truth of the premises. When you put these two components of the definition together, it follows that all sound arguments have true conclusions.
19 If one’s aim is to criticize (i.e., find fault with) another person’s argument, one
should point out all its defects, formal as well as material. Conversely, if one is making an
argument, one should ensure not only that one’s conclusion follows from one’s premises (i.e.,
that it has a truth-preserving form), but that one’s premises are (in fact) true.
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argument is either valid or invalid, sound or unsound. It makes no sense
to say, of a deductive argument, that it is “almost valid” or “almost
sound,” or that one deductive argument is “valider,” “more valid,”
“sounder,” or “more sound” than another. If a given deductive argument
has 1,000,000 premises and 999,999 of them are true (the remaining
premise being false), then the argument is unsound, just as it would be if
all 1,000,000 premises were false. As the old saying goes, “close” doesn’t
count except in horseshoes and hand grenades. Here, then, is our final
taxonomy of arguments:
Arguments
Deductive
Inductive
Valid
Sound
1
Invalid
Unsound
2
3
4
The taxonomy is jointly exhaustive in that every argument is in at least
one of its four categories. Every argument, in other words, is either (1)
sound (i.e., a valid deductive argument with true premises), (2) unsound
(i.e., a valid deductive argument with at least one false premise), (3) invalid (i.e., a deductive argument that is not truth-preserving), or (4) inductive (i.e., an argument that is not deductive). The taxonomy is mutually exclusive in that no argument is in more than one of its four categories (i.e., every argument is in at most one of its four categories). It follows
that every argument is in exactly one of the taxonomy’s four categories.
13. Specialists in logic are known as logicians.20 Logicians, as such, have
no expertise in determining which propositions are true and which
false—unless, of course, the propositions in question are true or false
simply by virtue of their form (such as “God exists or God does not exist,”
which is true by virtue of its form, and “God exists and God does not exist,” which is false by virtue of its form).21 Logicians are, however, expert
in determining which arguments (argument forms) are valid, for logic, as
we saw at the outset, is the science (study) of validity.22
20 Specialists in magic are known as magicians—which is not to say that logic has
anything to do with magic!
21 Propositions that are true by virtue of their form are called “tautologies.” Propositions that are false by virtue of their form are called “self-contradictions.” We will have
more to say about tautologies and self-contradictions in due course.
22 People who are not expert in a given field should defer to those who are. If you
are not a logician, then you should defer to logicians on matters within the scope of their
expertise, just as, if you are not a biologist, you should defer to biologists on matters within
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Appendix: Definitions of “Logic”
“The general science of inference.”
Simon Blackburn, The Oxford
Dictionary of Philosophy, 2d ed.
revised, Oxford Paperback Reference (New York: Oxford University Press, 2008), 212.
Gregory Pence, A Dictionary of
Common Philosophical Terms
(New York: McGraw-Hill, 2000),
31.
“[T]he branch of philosophy that
examines the correctness of rational inference, the ways we
think, and the limitations of such
inferences.”
“Loosely speaking, logic is the process of correct reasoning, and
something is logical when it
makes sense. Philosophers often
reserve this word for things having to do with various theories of
correct reasoning.”
“The scope of the term ‘logic’ has
varied widely from writer to
writer through the centuries. But
these varying scopes seem all to
enclose a common part: the logic
which is commonly described,
vaguely, as the science of necessary inference.”
“Logic is the study of principles of
reasoning. It is concerned not
with how people actually reason,
Robert M. Martin, The Philosopher’s Dictionary, 3d ed. (Orchard
Park, NY: Broadview Press,
2002), 182.
Willard Van Orman Quine, Elementary Logic, rev. ed. (Cambridge: Harvard University Press,
1980), 1.
Warren Goldfarb, Deductive Logic
(Indianapolis: Hackett Publishing
Company, 2003), xiii.
the scope of their expertise. Expertise does not automatically transfer from one field to another, so the fact that a particular individual is expert in field X does not mean that that
individual is expert in field Y. Many people (sadly) are expert in nothing. (A person of this
sort is said to be a “jack of all trades, master of none.”) Some people are expert in more than
one field. I have a friend (Robert “Bob” Schopp, who teaches at the University of Nebraska)
who has three advanced degrees: a Ph.D. in psychology, a Ph.D. in philosophy, and a J.D.
(law). He works at the intersection of these three fields—on such topics as the criminal
defenses of insanity and automatism. People who are expert in more than one field—especially people who are expert in multiple fields—are known as “polymaths.” Aristotle (384322 BCE), Gottfried Wilhelm Leibniz (1646-1716), and Immanuel Kant (1724-1804) were
polymaths.
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but rather with how people ought
to reason if they wish to ensure
the truth of their results.”
“Logic is the study of the methods
and principles used to distinguish
good (correct) from bad (incorrect)
reasoning.”
“Logic deals with arguments and
inferences. One of its main purposes is to provide methods for
distinguishing those that are logically correct from those that are
not.”
“Logical inference leads from
premises—statements assumed
or believed for whatever reason—
to conclusions which can be
shown on purely logical grounds
to be true if the premises are true.
Techniques to this end are a primary business of logic. . . .”
“Logic as a distinctive science is
concerned . . . with the relation of
implication between propositions.
Thus the specific task of logic is
the study of the conditions under
which one proposition necessarily
follows and may therefore be deduced from one or more others, regardless of whether the latter are
in fact true.”
“Logic may be defined as the organized body of knowledge, or science, that evaluates arguments.”
“[T]he subject matter of symbolic
logic is merely logic—the principles which govern the validity of
inference.”
“It is quite common for people to
concentrate on the individual
statements in an argument and
Irving M. Copi, Introduction to
Logic, 7th ed. (New York: Macmillan Publishing Company, 1986),
3.
Wesley C. Salmon, Logic, 3d ed.
(Englewood Cliffs, NJ: PrenticeHall, 1984), 1.
W. V. Quine, Methods of Logic, 4th
ed. (Cambridge: Harvard University Press, 1982), 53 (italics in
original).
Morris R. Cohen and Ernest
Nagel, An Introduction to Logic
(New York: Harcourt, Brace &
World, 1962), 8 (italics in original).
Patrick J. Hurley, A Concise Introduction to Logic, 11th ed. (Boston: Wadsworth, 2012), 1.
Clarence Irving Lewis and Cooper
Harold Langford, Symbolic Logic,
2d ed. (New York: Dover Publications, 1959), 3 (italics in original).
Stan Baronett, Logic, 3d ed. (New
York: Oxford University Press,
2016), 3.
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investigate whether they are true
or false. Since people want to
know things, the actual truth or
falsity of statements is important;
but it is not the only important
question. Equally important is
the question ‘Assuming the premises are true, do they support the
conclusion?’ This question offers
a glimpse of the role of logic,
which is the study of reasoning,
and the evaluation of arguments.”
“Logic is the study of reasoning.”
“Logic may be broadly defined as
the study of methods for determining whether or not a conclusion has been correctly drawn
from a set of assumptions.”
“Logic in general is the science
and art of right thinking.”
“The study of the validity of different kinds of inference.”
“[T]he science of reasoning, proof,
thinking, or inference.”
“Logic is concerned with the principles of valid inference. . . .”
“Logic is concerned with what
makes reasoning good and what
makes arguments valid.”
Daniel Bonevac, Simple Logic
(New York: Oxford University
Press, 1999), 2.
Joseph Bessie and Stuart Glennan, Elements of Deductive Inference: An Introduction to Symbolic
Logic (Belmont, CA: Wadsworth
Publishing Company, 2000), 1.
Raymond J. McCall, Basic Logic:
The Fundamental Principles of
Formal Deductive Reasoning, 2d
ed., College Outline Series (New
York: Barnes & Noble, 1952), xvii.
Boruch A. Brody, “Logical Terms,
Glossary of,” in The Encyclopedia
of Philosophy, ed. Paul Edwards
(New York: Macmillan Publishing
Company, 1967), 5:57-77, at 67.
The Oxford American Dictionary
and Language Guide (New York:
Oxford University Press, 1999),
583.
William Kneale and Martha
Kneale, The Development of
Logic (Oxford: Clarendon Press,
1962), 1.
Tom Tymoczko and Jim Henle,
Sweet Reason: A Field Guide to
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“The province of logic must be restricted to that portion of our
knowledge which consists of inferences from truths previously
known; whether those antecedent
data be general propositions, or
particular observations and perceptions. Logic is not the science
of Belief, but the science of Proof,
or Evidence. In so far as belief
professes to be founded on proof,
the office of logic is to supply a test
for ascertaining whether or not
the belief is well grounded. With
the claims which any proposition
has to belief on the evidence of
consciousness, that is, without evidence in the proper sense of the
word, logic has nothing to do.”
Modern Logic (New York: W. H.
Freeman and Company, 1995), 1.
John Stuart Mill, A System of
Logic Ratiocinative and Inductive: Being a Connected View of
the Principles of Evidence and the
Methods of Scientific Investigation, Collected Works of John
Stuart Mill, vol. VII (Toronto:
University of Toronto Press, 1974
[first published in 1843]), 9.
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