If . . . then? An Introduction to Logic.
Valid and Invalid Argument Forms

Valid argument forms

Before we continue, let's reflect for a moment on a central concept we introduced in the previous section - the concept of validity. We learned that arguments with valid inferences have a special property, that being that if the premises are true, then the conclusion must be true. Take note of the 'if' and the 'must be' in the previous sentence. A valid argument is not an argument that must have all true premises, but if it does something interesting happens, the truth of those premises is 'passed on' necessarily to the conclusion. In other words, if we know an argument is valid, and we also know all the premises are true, then we know automatically that the conclusion must be true (this is the importance of knowing whether an argument is sound). This connection between the truth of the premises and the truth of the conclusion is a logical connection. Let's explore this concept more by considering the following two propositions:

  1. The temperature outside is over 70 degrees right now.
  2. The temperature outside is over 70 degrees right now or the temperature outside is not over 70 degrees.

Experience tells us that the first sentence has the property of sometimes being true and sometimes not being true. In addition the truth of the sentence, whatever it may be, can not be determined by just thinking about what the sentence says or what the words mean. In the end, someone is going to have to read a thermometer or do some experiment to determine the answer. Propositions which are sometimes true and sometimes false are known as contingent propositions.

On the other hand the truth of sentence 2 can be determined without doing any experiment. Since the temperature outside will be 70 or some other temperature besides 70 degrees, one of the statements connected by the disjunction 'or' must be true. This means the entire sentence is true, no matter what conditions exist in a world where temperature measurement is meaningful. Propositions of this type are known as tautologies. We will examine both contingent propositions and tautologies in more detail in our formal analysis of logic, but for now let's state a fact which we will accept as fundamental (such a fact is known as an axiom).

 

Fact: Let p be any proposition, then the statement 'p or not p' is always true.

 

What our fact tells us is that any statement which has the form, 'p or not p' is always true. The idea that the truth of something can be determined by form alone is quite old in logic.

The amazing discovery is that the property of validity can also be determined by the form of an argument alone, and perhaps even more surprising is that invalidity can not. Aristotle was one of the first philosophers to clearly state this discovery and make us of it in logical reasoning. Let us now define what is meant by a valid form and give just a few examples, but first we will need to define another concept.

Definition: Let the p, q, r etc be statement variables. Any substitution or replacement of these variables with an actual or symbolic proposition is called a replacement instance or substitution instance of that variable.

In this section we will primarily be interested in replacement instances where the replacement is an actual proposition in English. For example, given the symbolic proposition "p or not p", if we let p be the statement, "Today is Monday", then this is a replacement instance for the variable p, and the replacement results in the following proposition, "Today is Monday or today is not Monday".

With this in mind, we now define what is meant by a valid argument form.

Definition: Let p, q, r, etc. stand for propositions. A valid argument form is an argument given in terms of p, q, r, such that the resulting argument is always valid for any choice of propositions for p, q, r etc.

This leads to the following fact, which we will call a theorem.

Theorem 1: Given any argument, if it is possible to assign statement variables to statements in the argument such that the resulting replacement results in a valid argument form, then the argument is valid. In other words, if an argument has a replacement instance which results in a valid argument form, then the argument is valid.

 

For now we will only consider four valid argument forms.

 

Four common valid argument forms

modus ponens/affirming the antecedent

If p then q

p

Therefore q

 

disjunctive syllogism / process of elimination

p or q

not p

Therefore q

 

modus tollens/denying the consequent

If p then q

not q

Therefore not p

 

hypothetical syllogism / chain of reasoning

If p then q

If q then r

Therefore if p then r

 

 

Theorem 1 suggests that we should think of valid argument forms as recipes for creating a valid argument. Here is how this recipe would work:

 

Example 3.0.1

Consider modus ponens, it has only two propositional variables p and q. Our 'recipe' allows us to assign actual propositions to both p and q - it does not matter if the propositions are truth functionally related or not- they can be any propositions - the result will be a valid argument.

For example:

Let p be , 'Travelers always arrive at their destinations excited but tired'

Let q be 'the central time zone is one hour behind the eastern time zone'.

We now combine these propositions using the form given by modus ponens to obtain:

 

modus ponens

If p then q

p

Therefore q

 

Argument created using modus ponens

If travelers always arrive at their destinations excited but tired then the central time zone is one hour behind the eastern time zone.

Travelers always arrive at their destinations excited but tired.

Therefore the central time zone is one hour behind the eastern time zone.

 

 

As odd as the argument seems, the fact that it was made using the 'directions' given by modus ponens guarantees it is a valid argument by Theorem 1!

The analogy with a recipe may be a bit misleading, since with recipes in cooking it matters which order you add the ingredients. However for valid argument forms the premises can come in any order. For example the following argument is the same as the previous one, but the order of the premises has been switched.

 

Travelers always arrive at their destinations excited but tired.

If travelers always arrive at their destinations excited but tired then the central time zone is one hour behind the eastern time zone.

Therefore the central time zone is one hour behind the eastern time zone.

 

What our recipe for modus ponens tells us is that the resulting argument is still valid.

For now we will not switch the order of the premises from what is given in the above table of valid argument forms, but store this fact away in your mind, as it will be useful to know when we study the method of deduction using rules of inference.

 

Exercise 3.0.2

Use the following substitutions for p and q to create an argument in the form of modus tollens.

(a)

Let p be, 'my car is still in the shop' and let q be 'I have to get a ride with a friend'.

(b)

Let p be, 'Bottled water is no more good for you than tap water' and let q be 'you should not buy bottled water'.

Solution: We want to use the p and q given above as replacements for the p and q in the following argument form (such use is called a replacement instance).

If p then q

not q

Therefore not p

Hence we have for (a)

If my car is still in the shop then I have to get a ride with a friend.

I don't have to get a ride with a friend.

Therefore my car is not still in the shop.

For (b) this gives:

If bottled water is no more good for you than tap water, then you should not buy bottled water.

You should buy bottled water.

Therefore bottled water is more good for you than tap water.

 

Notice that not p in case (b) is 'you should buy bottled water' as we use the convention in English that the negation of a negation (for example, "you should not not buy bottled water") is the same as an affirmation, so for example, the negation of 'I am not going' is 'I am going' and so forth.


Using a recipe for creating a valid argument is pretty easy. However many times we are presented with an argument already made - the challenge will then be to determine whether the argument has a valid form, and if so, which valid form. Here is an exercise that illustrates this.

 

Exercise 3.0.3

Determine whether the following argument has a valid form or not.

The computer graphics card is malfunctioning or the monitor is not working.

The computer graphics card is not malfunctioning.

Therefore the monitor is not working.

Solution: We want to determine whether the argument has a valid for or not.

One way to determine this is to examine the words (not letters) one finds in the above valid argument forms (ignoring the word ' therefore'), and look to see if they exist in a given argument (such words are known as logical operators). These words are, "if . . . then", "or", "and", and 'not' where 'if . . . then" is used in modus ponens and modus tollens, 'or' is used in disjunctive syllogism, 'and' is used in conjunction and 'not' is used in modus tollens and disjunctive syllogism.

Here is the above argument with the corresponding word(s) highlighted.

The computer graphics card is malfunctioning or the monitor is not working.

The computer graphics card is not malfunctioning.

Therefore the monitor is not working.

This suggests that if the argument has a valid form, it will be disjunctive syllogism. To make sure this is correct, we need to assign letters to propositions given in the actual argument. To this end, let p be 'the computer graphics card is malfunctioning' and let q be the monitor is not working', then 'not p' will be 'the computer graphics card is not malfunctioning'.

Now we just use these assignments and re-write the above to get:

p or q

not p

Therefore q

This is the exact form given for disjunctive syllogism, so we have shown that the argument does have a valid form.  


In deciding whether an argument has a valid form, we may have to re-write some sentences to make the fit be exact. As long as such re-writes are logically equivalent to the original or imply the original, then such re-writes are permitted. For example, suppose the conclusion to an argument were, "John is a lawyer and a student'. One might need to re-write the sentence to the equivalent form, 'John is a lawyer and John is a student'. This allows one to assign (without grammatical problems) the variable p to 'John is a lawyer' and the variable q to 'John is a student'. Another common re-write deals with changes in verb tense, "If it rains tomorrow, I will have to take the bus. It rained. Therefore I had to take the bus" is considered an instance of modus ponens.

Now test your ability at determining whether an argument has a valid form by answering the following questions.

 

Test your understanding: determine the valid form

   

 

Invalid argument forms

 

 Consider the following argument form:

p

q

Therefore r

If we let p be 'It is raining in the southeast', let q be 'increased rain usually helps crops produce a higher crop yield' and r be 'crops in California will produce more' then the resulting argument is not valid (check to make sure you see a possible way to have all true premises and a false conclusion).

On the other hand, if we let p be 'If travelers always arrive at their destinations excited but tired then the central time zone is one hour behind the eastern time zone', q be 'travelers always arrive at their destinations excited but tired' and r be 'the central time zone is one hour behind the eastern time zone' then we have the exact same argument as given in Example 3.0.1, and thus is a valid argument. This example tells us that some argument forms can result in arguments which are not valid or arguments which are valid depending on the propositions used to replace the letters used in the argument form. This outcome is impossible for valid argument forms. Valid argument forms always produce valid arguments irrespective of the propositions chosen to replace the variable letters used in the argument form.

This leads to the following definition.

Definition: Let p, q, r, etc. stand for propositions. An invalid argument form is an argument given in terms of p, q, r, such that the resulting argument may be invalid or may be valid depending on the propositions used to replace the variables p, q, r, etc.

 

Notice that the definition for an invalid argument form is just the negation of the definition of "valid argument form". The surprise occurs when we negate the phrase, "… the resulting argument is always valid for any choice of propositions for p, q, r etc." where the negation of "always" is "at least one is not", which together with the above observation leads to the given definition.

There are many invalid argument forms. However some invalid forms are very similar to valid forms and such similarity historically mislead some to think the resulting form was actually valid. These forms have been traditionally called formal deductive fallacies, but for this text we will use the more descriptive term 'pseudo-valid argument form' to emphasize their similarity to valid argument forms.

Definition: A pseudo-valid argument form is an invalid argument form which is similar in form to valid argument forms.

Here we give a small list of pseudo-valid argument forms, comparing them to valid argument forms given on the left.

 

Valid and pseudo-valid argument forms

Valid argument form

Pseudo-valid argument form

modus ponens/affirming the antecedent

If p then q

p

Therefore q

 

denying the antecedent

If p then q

not p

Therefore not q

 

modus tollens/denying the consequent

If p then q

not q

Therefore not p

 

affirming the consequent

If p then q

q

Therefore p

 

disjunctive syllogism / process of elimination

p or q

not p

Therefore q

 

false dilemma

p or q

p

Therefore not q

 

 

 

Due to the similarity between the valid form and the pseudo-valid form, historically some people assumed the pseudo-valid from was valid. This error in reasoning many times is called a deductive fallacy.

There is one technicality we should note here that applies to both disjunctive syllogism and false dilemma. The word 'or' has two different meanings in English. Those meanings are given the names 'inclusive or' and 'exclusive or'.

Two different meanings of 'or'

 

The phrase 'inclusive or' means 'p and/or q'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition even when both p and q are true. An example would be, "John is a student or John is a father'. If John is both a student and a father, then the entire sentence is still true.

 

The phrase 'exclusive or' means 'p or q but not both'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition only when exactly one of p or q is true. An example would be, "Today is Monday or today is Wednesday'. Here the meanings of the weekdays exclude the possibility that both p and q can be true and the sentence is true only when it is really Monday or really Wednesday.

 

The word 'or' will always be understood to be the inclusive version. When we want to refer to the exclusive or, we will write 'xor' or use the additional phrases like, 'p or q but not both', or 'exactly one of p or q is true'.

 Let's examine the concept of argument form in more detail by comparing two different arguments, one has a valid form while the other has an invalid form.

Challenge: Verify that the following argument form is valid

(which means it is impossible to have all true premises and a false conclusion).

 

p xor q

p

Therefore not q

 

 

 

Example 3.1.1

(a)

If it rains, then the game will be cancelled.

It is raining.

Therefore the game will be cancelled.

 

(b)

If it rains, then the game will be cancelled.

It did not rain.

Therefore the game will not be cancelled.

 

Notice that (a) is an example of modus ponens, and hence must be valid.

However example (b) is an instance of denying the antecedent, so the resulting argument form may be valid or invalid. How do we know which? For now we can use the two step method for invalidity.

One final point before you get a chance to test your understanding. Since pseudo-valid argument forms are forms which were thought to be valid forms, when an argument has such a form and is indeed invalid, we will classify it as an erroneous proof in our classification scheme for arguments.

Test your understanding: determine the argument form

   

 

Summary Section 3

In this section we introduced and defined the following terms and concepts:

valid argument form is an argument given in terms of pqr, such that the resulting argument is always valid for any choice of propositions for pqr etc.

Propositions that have the property of being sometimes true and sometimes false are known as contingent propositions.

Propositions that are always true are known as tautologies.

A fundamental proposition whose truth seems to be self-evident and is not proven by use of more fundamental propositions is known as an axiom.

Letters of the alphabet which stand for propositions are known as propositional variables.

When an actual proposition replaces a variable given in an argument form, such substitution is called a replacement instance (of the variable x).

Given any argument, if it is possible to assign statement variables to statements in the argument such that the resulting replacement results in a valid argument form, then the argument is valid.

An invalid argument form is an argument given in terms of pqr, such that the resulting argument may be invalid or may be valid depending on the propositions used to replace the variables pqr, etc.

pseudo-valid argument form is an invalid argument form which is similar in form to valid argument forms. Some texts refer to these forms as invalid deductive arguments.

The term 'inclusive or' means 'p and/or q'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition even when both p and q are true.

The term 'exclusive or' means 'p or q but not both'. Specifically given two propositions p and q, then the resulting proposition 'p or q' means 'p or q' is a true proposition only when exactly one of p or q is true. 

We introduced four valid argument forms and compared them to three pseudo-valid argument forms (see table below).

Valid argument forms can be used as 'recipes' to create valid arguments. Also some existing arguments can be shown to have valid forms by isolating certain terms in the argument, looking for a match of corresponding terms in known valid argument forms, and assigning variables to propositions, giving the same variable to the same propositions.

Replacement instances for the variables used in invalid argument forms may or may not produce an invalid argument. To determine which is the case, we use the two step method for determining invalidity.

We noted that in logic the meaning of 'or' is always inclusive, unless otherwise indicated.

 

Valid argument form

Pseudo-valid argument form

modus ponens/affirming the antecedent

If p then q

p

Therefore q

 

denying the antecedent

If p then q

not p

Therefore not q

 

modus tollens/denying the consequent

If p then q

not q

Therefore not p

 

affirming the consequent

If p then q

q

Therefore p

 

disjunctive syllogism / process of elimination

p or q

not q

Therefore p

 

false dilemma

p or q

p

Therefore not q

 

 

hypothetical syllogism / chain of reasoning

If p then q

If q then r

Therefore if p then r

 

 

(no corresponding pseudo-valid form)