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The Logic of Arguments
Supplement to: Paradoxes: Their Diagnosis, Treatment and Cure

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
http://www.pitt.edu/~jdnorton

• The Elements of an Argument
• Deduction
• Validity and Soundness
• Reductio ad Absurdum
• Rules of Inference

This supplement provides the barest minimum needed to treat the logic of arguments responsibly. The literature on logic is enormous and can be consulted with profit. Standard texts include those written by Irving M.  Copi, Introduction to Logic and Symbolic Logic and Merrilee H. Salmon, Introduction to Logic and Critical Thinking. Both works have been published in multiple editions. P. D. Magnus, ForallX is an open access text, freely available online.

The Elements of an Argument

The basic unit of an argument is the proposition. It is a statement that asserts some state of affairs and is either true or false.

An argument consists of a list of propositions connected as follows.

Each proposition in the list is either:

• a premise (a proposition introduced by supposition)
  or
• a conclusion derived validly from propositions earlier in the list.

The entries in the list are commonly numbered for ease of reference. Each is marked as a premise or as derived, with an indication of the earlier propositions from which it is derived and, if possible, the basis of the justification.

The familiar philosopher's example is

1. All men are mortal. (premise)
2. Socrates is a man. (premise)
Therefore, 3. Socrates is mortal (derived from 1 and 2)

This is one of the simplest cases. More elaborate arguments may have many more premises and many intermediate conclusions.

Deduction

The key component of the above is the concept of "validly derived." The most important case, and the one we will need most, is deduction or deductive inference.

A valid deductive inference is one in which the truth of the premises assures the truth of the conclusion.

The Socrates argument above is a valid deduction. If premises 1. and 2. are true, then the conclusion 3. cannot fail to be true.

Logicians have put considerable effort into cataloging just which inferences are valid. Their methods generally depend upon identifying the form of a valid argument. What results are schemas (examples below) from which many valid arguments can be generated. For example, here is the rule known as "simplification."

A and B
Therefore, A

We can substitute any propositions we like for A and for B and recover a valid argument. For example:

(Elephants are grey.) AND (Pigs cannot fly.)
Therefore (Elephants are grey.)

As a general matter with standard logics, the deductive validity of the resulting arguments depends on the meaning of the terms used. The simplest of these terms are the connectives:

AND, OR, NOT, IF...THEN...

Consider someone who accepts the truth of the premise (Elephants are grey.) AND (Pigs cannot fly.) but denies the truth of the conclusion (Elephants are grey.). We would surely wonder whether that person understands what "AND" means. In this context, AND tells us that each of the conjuncts (Elephants are grey.), (Pigs cannot fly.) are true individually. Hence we can extract one as a true conclusion.

This approach works well for the standard systems of logic, based on connectives like these and a few extra terms. However practical argumentation requires more. We eventually find cases in which an inference is deductive, but its validity does not depend on the meaning of connectives. For example:

This book is blue and small.
Therefore, this book is not red and not large.

This is a valid deductive inference in the context of informal reasoning since the truth of the premise assures the truth of the conclusion. That assurance derives from the ordinary meaning of the terms blue, red, small and large. Something that is blue cannot also be red; and something that is small cannot also be large.

There are many examples like this in which attention to the meaning of terms provides crucial elements of the inference. An important case arises in mathematical derivations. For example, for real numbers a, b and c, we can infer validly that:

a = b x c
Therefore, a = c x b

The validity of the deduction depends on a, b and c being real numbers. Then they have a commutative property under multiplication: b x c = c x b. We all know that 5 x 2 = 2 x 5.

If, however a, b and c are not real numbers, then this property can fail. If for example, they are vectors and "x" is the cross product, then they anticommute: b x c = - c x b.

A related problem arises when we apply mathematical notions to the world. It is an uncontested piece of arithmetic that

1 + 1 = 2

We use it to infer

x = 1 + 1
Therefore, x = 2

We apply this without much thought to simple examples in the real world when we put things together: 

1 liter of water + l liter of water = 2 liters of water

Careful! Sometimes, putting things together does not behave in ways we would expect. If instead we mix alcohol and water, the combination of the two substances leads to a slight contraction:

1 liter of water + 1 liter of alcohol = 1.93 liters of mixture

We risk a fallacy:

Volume = 1 liter of water + 1 liter of alcohol
Therefore, Volume = 2 liters of mixture  (fallacy)

The correct inference is:

Volume = 1 liter of water + 1 liter of alcohol
Therefore, Volume = 1.93 liters of mixture

Considerable attention is also given in logic texts to inductive inference. These are inferences in which the truth of the premises does not assure the truth of the conclusion, but just makes it likely.

For example: Most birds can fly. Tweety is a bird. Therefore Tweety can fly

A popular approach to inductive inference is to treat them all as if they were statements in the probability calculus. In my view, this is too narrow a construal. There are cases that do not conform with the probability calculus. My overall principle is that inductive inferences are warranted by the truth of background assumptions in the pertinent domain. There are no universal schemas for inductive inference. For more, see my Material Theory of Induction.

Validity and Soundness

A valid argument is one whose individual inferences from premises to conclusions are all valid.

The most important case is deductive validity. The truth of the premises in a deductively valid argument guarantees the truth of the conclusion. The most direct way of affirming that an argument is deductively valid is to find that it conforms with one of the deductive argument schemas of a known logic. Some examples of these schemas are listed at the end of this chapter.

What is missing from this definition is the requirement that the premises must be true. A valid argument can have false premises. The following is a valid argument.

(Elephants are grey.) AND (Pigs CAN fly.)
Therefore, (Pigs CAN fly.)

IF--and that is a big IF--the premise is true, then the conclusion must be true. Of course in this example, the premises are not true.

A more curious result is that a valid argument can have false premises and a true conclusion. The following is also a valid argument:

(Elephants are grey.) AND (Pigs CAN fly.)
Therefore, (Elephants are grey.)

The premise is false, but the conclusion is true. Examples like this set a trap for the unwary. That a valid argument has true conclusions does not assure us that the premises are true.

Validity can fail. Then we have:

A fallacy is an argument whose inferences are not valid. For example, this is a fallacy:

(Elephants are grey.)
Therefore, (Elephants are grey.) and (Pigs cannot fly.)

The truth of the premise (Elephants are grey.) does not guarantee the truth of the conclusion. The conclusion happens to be true, but that truth is not guaranteed by the truth of the premises. We can see that failure in a similar fallacy in which the conclusion is false:

(Elephants are grey.)
Therefore, (Elephants are grey.) and (Pigs CAN fly.)

What we seek is the best case:

A sound argument is one whose premises are all true and whose inferences are all valid.

Reductio ad Absurdum

A reductio ad absurdum (reduction to absurdity) or just reductio is a common argument form and one that appears in the context of paradoxes. It proceeds indirectly. One posits some proposition as an hypothesis. It is usually the negation of the result sought. A contradiction is derived. That shows the falsity of the original hypothesis. Its negation is concluded.

For example:

1. There is a smallest real number "x" greater than zero. (hypothesis for reductio)
2. Therefore, there is no number smaller than x but greater than zero. (from the meaning of "smallest")
3. Whatever x is, x/2 is smaller than x but greater than zero. (Premise)
4. Contradiction (2. and 3. contradict)
5. The hypothesis 1. is false (by reductio)

From the falsity of the hypotheses, we conclude that there is no smallest number x greater than zero.

A trap for the unwary is that there is no guarantee that the hypothesis proposed for the reductio is the culprit producing the contradiction. In common cases, we can safely blame it since we have selected the hypothesis as a proposition that we already find dubious. We have been careful to introduce further premises whose truth is secure.

Sometimes, all this goes wrong. The reductio above works for real numbers, that is, numbers with arbitrarily long decimal expressions, like pi and the square root of two. However, had we tried to run the analogous reductio for the counting numbers 1, 2, 3, ..., we would have risked falling into the trap. We would have written:

1. There is a smallest counting number "x" greater than zero. (hypothesis for reductio)
2. Therefore, there is no counting number smaller than x but greater than zero. (from the meaning of "smallest")
3. Whatever x is, x/2 is smaller than x but greater than zero. (Premise)
4. Contradiction (2. and 3. contradict)
5. The hypothesis 1. is false (by reductio)

We still arrive at the contradiction in 5. However the source of the contradiction is the premise 3. It is false for counting numbers. Of course it is true for some counting numbers 100/2 = 50; 4/2=2, etc. However it fails for the smallest counting number greater than zero, 1. For 1/2 is a fraction and is no longer a counting number.

The correct analysis of this last reductio is that premise 3 is false. There is a smallest counting number x greater than zero. It is x=1.

Rules of Inference

Here are some common rules of deductive inference. If the P's, Q's, etc. are replaced by propositions, they generate valid deductive inferences; or licit replacements.

(From I. Copi, Symbolic Logic.)

These rules apply to the simplest case of propositional logic; that is, a logic whose most fundamental element is the proposition. Richer logics introduce structure into the propositions. Quantified logics structure them internally with properties and individuals. Simple propositions in a quantified logic might be:

For all x, P(x)              "For all x, x has the property P."
There exists x, P(x)      "There is an x that has property P"

There are more elaborate rules of inference for quantified logics. For example:

Universal instantiation
For all x, P(x)
Therefore, P(a)

This rule tells us that we can infer from the universal generalization that there is an individual a with the property P. There is a rule that moves in the reverse direction:

Existential generalization
P(a)
Therefore, there exists x, P(x)

June 1, September 7, 2021. January 12, 2023.
Copyright, John D. Norton