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Mathematics LibreTexts

1.3: Arguments

Logic is the study of the methods and principles of reasoning. An argument is a set of facts or assumptions, called premises, used to support a conclusion. For a logical argument to be valid, it is the case that, if the premises are true then the conclusion must be true.

Argument

Definition

An argument is a set of statements called premises together with a conclusion. An argument consisting of two premises and a conclusion is called a syllogism.

 

Example \(\PageIndex{1}\):

  1. Either a leopard is a cat or a bird.  (First premise)

It is not the case that a leopard is a bird. (Second premise)

Therefore a leopard is a cat. (Conclusion)

 

  1. Either a leopard is a fish or a bird.  (First premise)

It is not the case that a leopard is a bird. (Second premise)

Therefore a leopard is a fish. (Conclusion)

 

  1. If the Moon is a planet then the Sun is an asteroid.

    The Moon is a planet.

    So the Sun is an asteroid.

 

  1. If the Earth is a planet then it is larger than a protoplanet.

The Earth is larger than a protoplanet.

Therefore, a leopard is a cat.

 

  1. If the Earth is a planet then it is larger than a protoplanet.

The Earth is larger than a protoplanet.

Therefore, the Earth is a planet.

 

Arguments can be discredited if any of the premises are false (or their truth is uncertain). This is, however, not the only way an argument can be discredited. Argument 4 has true premises and a true conclusion but, nevertheless, is a poor argument. Argument 5 is an inadequate argument as well but it is harder to spot. This is because this form allows for true premises to lead to a false conclusion. Here’s an example of an argument with the same form as argument 5 that we recognize is a bad argument:

Example \(\PageIndex{2}\):

  1. If the Sun is a planet then it is larger than a protoplanet.

The Sun is larger than a protoplanet.

Therefore, the Sun is a planet.

Valid/Invalid argument

The logical form of an argument is the second method for evaluating arguments.  An argument is valid if and when all the premises are true. If this is the case then the conclusion must be true (i.e. if you accept the truth of the premises it forces you to accept the truth of the conclusion). We can check the validity of an argument with a truth table. When you are given a valid argument and you know the premises are true, the argument proves the conclusion to be true.

Consider the first argument arguing a leopard is a cat.  It has the form:

p or q.

Not q.

Therefore p.

 

Is it a valid argument?  Well, it’s valid if whenever the premises (p or q) and (not q) are true, so the conclusion p must be true.

Here is the truth table with "p or q" and "not q" illustrated.

 

Table 1.3.1: Truth Table
\(p\) \(q\) Premise 1: \(p \vee q\) Premise 2: \(\neg q\) Conclusion: \(p\)
T T T F T
T F T T T
F T T F T
F F F T F

The second row of truth values is the only row where both premises are true.  When both premises were true, the conclusion was true as well.  This means the argument is valid.  Note that we have not determined if the actual premises are correct – we’ve just noted that, if the premises are true it must be the case the conclusion is true as well. 

The second argument is also valid because it has exactly the same form as the first (and hence the same truth table).  Because its first premise is false, though, this argument is not logically sound. 

Argument 4 has the form:

If p then q.

q

Therefore r.

 

Here is the truth table that illustrates premises and conclusion's possible truth values all at once.  Since there are 3 simple statements (p,q and r) involved it is bigger (twice as big!) than the previous truth tables.

Table 1.3.2: Truth Table
p q r Premise 1: If p then q. Premise 2: q Conclusion: r
T T T T T T
T T F T T F
T F T F F T
T F F F F F
F T T T T T
F T F T T F
F F T T F T
F F F T F F

The first, second, fifth and sixth rows of truth values all satisfy the condition that both premises are true.  But row 2 and row 6 have the premises true yet the conclusion false.  This is an invalid argument.  Note that as soon as we see the second row of truth values we can conclude that the argument was invalid without reading through the rest of the truth table.

Of course, it gets a little tiresome to have to construct truth tables for every argument form that arises.  Now that we understand the validity and are satisfied that we can determine the validity or invalidity of simple arguments, we can note classical forms of valid arguments (known as the Rules of Inference):

Table 1.3.3: Rules of Inference I
Argument type: A.Modus ponens (affirming the hypothesis) B.Modus tollens (denying the conclusion) C.Disjunctive syllogism D.Hypothetical syllogism
Premise 1 If p then q If p then q p or q If p then q
Premise 2 p Not q Not q If q then r
Conclusion Therefore q Therefore not p Therefore p Therefore if p then r

 

Table 1.3.4: Rules of Inference II
Argument type: E. Conjunction F. Addition G.Simplification
Premise 1 p p p and q
Premise 2 q    
Conclusion Therefore p and q Therefore p or q (this argument has only one premise) Therefore p

At this point we will also point out two common invalid argument types:

Table 1.3.5: common invalid argument
Argument type: H. Affirming the Conclusion I. Denying the Hypothesis
Premise 1 If p then q If p then q
Premise 2 q Not p
Conclusion Therefore p Therefore not q.

Both of these arguments are invalid due to the nature of conditional statements as one-directional implications.

With the valid and invalid argument forms above in mind let us consider the six arguments given, to begin with. Arguments 1 and 2 are disjunctive syllogisms and hence valid arguments. Argument 3 is modus tollens and hence is valid. Arguments 5 and 6 are examples of the affirming the conclusion argument type and hence are invalid arguments. Argument 4 is none of the above but we determined it was invalid by the truth table.

 

Example \(\PageIndex{3}\):

Determine if the following arguments are valid just by looking at their form.  Note that some are a combination of two or more argument types.

  1. If one is a wuzzle then one is a woozle

    If one is a woozle then one is a finkle

    Therefore if one is a wuzzle then one is a finkle.

 

  1. If p is a prime number larger than 2 then p is odd.

     p is odd.

    Therefore p is a prime number.

 

  1. You either like Coke or you like Pepsi.

     You like Coke.

     So you don’t like Pepsi.

 

  1. Lizzlestipes and quadrinons.

     If Lizzlestipes then fizbots.

     If quadrinons then apoplexis.

     Therefore fizbots and apoplexies.

 

  1. Paul likes Banff and Jasper.

     Anyone who likes Banff likes Yellowstone.

     If you like Yellowstone then you like the Grand Canyon.

     Therefore Paul likes the Grand Canyon.

 

  1. If heavier objects always fall faster than lighter ones then a lead ball will fall faster than a wooden one of the same size.

     It is not the case that a lead ball falls faster than a wooden one of the same size.

     Therefore it is not the case that heavier objects always fall faster than lighter ones.

 

  1. What is the origin of the Moon? 

     (Experts believe) Either the Moon formed elsewhere and was drawn into its present orbit by Earth’s gravity OR the Moon is a companion planet to the Earth formed from the same nebula at the same time OR the Moon was once part of the Earth.

     If the Moon formed elsewhere then the composition of the two objects should be dissimilar.

     This is not the case.

     Therefore the Moon did not form elsewhere.

     If the Moon is a companion planet to the Earth then they should have similar compositions and in particular the same proportion of iron.

     This is not the case.

     Therefore the Moon was once part of the Earth.

Deductive argument

Note that the discussion of arguments so far has dealt with deductive arguments. In a deductive argument, the validity/invalidity of the argument can be determined and if we can establish with certainty that the premises are true then we know that the conclusion must be true as well. This form of reasoning, while excellent in certain circumstances, is sometimes fallible in practical applications:

  1. It is often the case that there is some degree of uncertainty to the premises. For example, consider the argument for the origin of the Moon. It is a valid argument but the premises (although reasonable and widely believed among experts to be correct) can be disputed.  For example, one could ask if there is a fourth explanation for the Moon’s origin the experts missed.  One could also point out that it is theoretically possible the Earth and Moon were formed in two completely different places but by a coincidence, the Moon happens to have the same composition as the Earth’s crust.
  2. It is possible to construct deductive arguments that do not apply to the general case. Consider the common childhood argument "If an animal has four legs, it is a cat." This argument is logically sound, but leads to many false conclusions.

In deductive arguments, one tends to start with general premises and arrives at a specific conclusion.  Here is another example:

Example \(\PageIndex{4}\):

All prime numbers larger than two are odd.

1098074983729749873982798649876  is an even number

It is also bigger than two.

Therefore it is not prime

Inductive argument

If one makes a case for a general conclusion from more specific premises then one is giving an inductive argument.  Example:

Example \(\PageIndex{5}\):

Let \(x \in \mathbb{Z} \mid x > 2, x\) is prime

Since prime numbers have only themselves and 1 as factors (they are divisible only by 1 and themselves),

And \(x\) is prime, \(2 \nmid x\).

Since all even numbers are divisible by 2,

Thus, \(x\) is odd.

Example \(\PageIndex{6}\):

In study after study it has been observed that people who smoke have a higher rate of lung cancer than those who don’t, even with sex, socioeconomic status, genetics and age taken into account.

Studies have also shown that the more one smokes, the higher one's likelihood of getting cancer. 

Studies on rats have shown cigarette smoke is associated with a higher cancer rate.

Biologists experimenting with cell cultures and chemicals in cigarette smoke have discovered the basic process by which the chemicals can create cancer-causing mutations.

Therefore, smoking causes lung cancer.

Inductive arguments are a necessary part of science and life in general. Inductive arguments cannot prove (without a doubt) that their conclusion is true, but we can evaluate their strength. For example, research has made such a compelling case that smoking causes lung cancer that it is accepted as fact.