Basic Concepts and Distinctions
Logic
Keith Burgess-Jackson
1 January 2016
1. Logic is the science (study) of validity.1
2. Validity (to be explained more fully below) is a property (characteristic) of deductive arguments. Some deductive arguments possess this
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 (in general) 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.
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. If one’s claim is that the conclusion cannot be false, given the
truth of the premises (a strong claim indeed!), then one’s argument is deductive. If one’s claim is that the conclusion is unlikely to be false, given
the truth of the premises (a weaker claim, though not necessarily a weak
claim), then one’s argument is inductive. Deduction is to necessity as
induction is to probability. Here is a deductive argument (for almost certainly the arguer would 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 is an inductive argument (for almost certainly the arguer would not
claim that the truth of the premises necessitates the truth of the conclusion):
Most professors are atheists.
1
See the Appendix for alternative definitions of “logic.”
1
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 of
them 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. In
other words, inductive arguments are neither valid nor invalid.
5. Ideally, one would like one’s argument to have two 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 informativeness. 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 no more information than the conjunction of the premises. It may contain the same
amount of information or less information, but not more information.
Consider the 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, 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 12 hours.
Therefore,
It will rain within 12 hours.
2
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.
Both types of argument confer knowledge. The knowledge conferred by
deduction consists in making explicit (in the conclusion) what is implicit
(in the premises). 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.”2 The world is such that we can’t
have everything we want or need.3
6. Propositions have truth values. There are two truth values: true and
false. Every proposition is either true or false (this is known as the law
of excluded middle) and no proposition is both true and false (this is
known as the law of noncontradiction).4 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 or assert.5 Two different indicative (declarative) sentences (e.g., “John loves Mary” and “Mary is loved by John,” [both in English] or “It is raining” and “Il pleut” [the former in English, the latter in
French]) can express or assert the same proposition. This is called synonymy. A given indicative (declarative) sentence (e.g., “Jones is happy”)
can express or assert 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
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 of them, and would if I could, but the situation constrains
me from doing so. I must make a tragic choice.
4 While every proposition (by definition) is either true or false, and no proposition
(by definition) 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 just moments 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?)
5 Sentence is to proposition as numeral is to number.
2
3
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property (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 that
argument is invalid.) Valid arguments are truth-preserving. If the premises of a valid argument are true, then its conclusion must be true as well.
If the conclusion of a valid argument is false, then at least one of its premises must be false. There can be a valid argument with false premises
and there can be 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 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 Fundamentals of Reasoning (PHIL 1301).6
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.7 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 part of the good
life. As the Greek philosopher Socrates put it (according to his disciple
Plato), “The unexamined life is not worth living.”8
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 discover
(or devise) as many valid argument forms as one can. The more valid
argument forms there are, the more likely it is that any particular argument one makes (or encounters) will have one of those truth-preserving
6 The two courses may be taken in any order; neither is a prerequisite for the other.
Some people disparage Fundamentals of Reasoning (also known as Critical Thinking or Informal Logic) as “baby logic,” which is like saying that abnormal psychology is “baby psychology” or that microeconomics is “baby economics.” I recommend that both courses be
taken.
7 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, it is like
money and unlike, say, friendship, knowledge, or beauty.
8 See Plato’s Apology.
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forms—which is not to say that most arguments are valid! 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 of the argument 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 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 form, then either both of them
are valid or both of them are invalid.9
11. A sound argument is a valid argument all of whose premises are
true. It follows from this definition that all sound arguments have true
conclusions. (Note the difference between x following from a definition
and x being part of a definition.)10 Any argument that is invalid is, by
definition, unsound. Any argument that has a false premise is, by definition, unsound. A given argument may be invalid and have a false
premise. Such an argument has two defects: one formal (invalidity) and
one material or substantive (a false premise). Here is an example of a
9 This is the basis of what is known as refutation by logical analogy. If one’s aim
is to refute a particular argument, one may do so by thinking up a second argument that (1)
has the same form as the first argument but that (2) has true premises and a false conclusion. The fact that the second argument 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 the second argument has the same form as the first argument shows that the
first argument, like the second argument, is invalid. Do not confuse refutation with rebuttal.
To refute is to prove the invalidity of. “Refutation” is a success term. One can attempt to
refute an argument but 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—with no implication that the pushing succeeds (or fails, for that matter). Refutation may be thought of as successful rebuttal.
10 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.
Do not be surprised to see this question on an examination.
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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, it would not follow that everything is green.11
12. Validity and soundness are all or nothing, not matters of degree.
(Figuratively speaking, they are digital, not analog.) A given deductive
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 all but one of them are true, 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.
13. Specialists in logic are known as logicians. Logicians, as such, have
no expertise in determining which propositions are true and which
false—unless 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).12 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.13
11 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 or substantive. 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.
12 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.
13 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, then you should defer to biologists on matters
within the scope of their expertise. Expertise does not 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.
For example, I have both a law degree (J.D.) and a doctoral degree (Ph.D.) in philosophy. I
have a friend (Robert “Bob” Schopp) who has three advanced degrees: a Ph.D. in psychology,
a Ph.D. in philosophy, and a J.D. He works at the intersection of these 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 (384-322 BCE) and Immanuel Kant (1724-1804) were polymaths.
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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,
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
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.
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.
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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
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,
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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which is the study of reasoning,
and the evaluation of arguments.”
“Logic is the study of reasoning.”
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
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.
“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.”
“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,
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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.”
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