Better Reasoning V: Valid Argument Forms II
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In the last post in this series, the conjunction and simplification valid argument forms were introduced. Also, a convention for using letters to stand for entire propositions, that is to say, things that can be said as complete statements, was explained. This convention will continue to be used here.
We have also been made aware that there are certain special logical words, that have very predictable effects upon the sentences in which they appear. Last time, we looked at the words 'Not' and 'And', which fall into this class of terms. These are not the only examples, however.
Another important logical word is the word 'Or'. There is a minor complexity that arises with 'Or', though. This is that natural language contains two logically distinct versions of 'Or'.
The first kind of 'Or' is called 'inclusive Or'. Sentences that contain an inclusive Or are true when either of the disjuncts (that is to say, the propositions either side of the Or) are true, as well as when both disjuncts are true. Consider the sentence
"Sammy will bring ham or cheese to the potluck dinner".
This sentence would not be false, if it turned out that Sammy brought a ham and cheese plate. In this case, 'Sammy will bring ham' is one disjunct, while 'Sammy will bring cheese' is the other disjunct. This illustrates the way 'inclusive Or' functions in language. Generally speaking, it is usually assumed that this is the default type of 'Or' people use, when speaking in natural language.
The second kind of 'Or' is known as 'exclusive Or'. For the purpose of disambiguation, this is usually written 'Xor'. This practice will be followed here. Sentences which have an Xor between their disjuncts are only true when one of their disjuncts are true, but not when they are both true. In common speech, the use of Xor is often marked by the use of the phrase "Either...,or...". Suppose one was to go out to dinner and the server was to say,
"Your meal comes with either a salad, or coleslaw. Which would you like?"
There would certainly be some surprise if one was to ask for both salad AND coleslaw! For the most part, in what follows, the inclusive use of 'Or' will be assumed.
Now attention can be turned to valid patterns of inference that involve the use of Or. The first of these to consider is the so-called 'addition' inference. Following the conventions introduced earlier, this kind of inference has two variants, which can be represented as follows:
P
Thus,
P or Q
Q
Thus,
P or Q
On the face of it, this may appear a bit of an odd inference, albeit a valid one. If one knows that a particular proposition, P is true, then we can infer that 'P or Q' is also true (actually, the same holds with false sentences -- valid inferences are 'truth-value preserving', in more technical language). So, if one knows that, for example "The sky is blue", it is valid to infer that "The sky is blue, or grass is green".
The apparent oddity of addition inferences notwithstanding, there are occasions when this kind of inference can be useful to make. There are a number of cases in the philosophical literature where inferences of this kind have been crucial to developing philosophical objections to positions. Perhaps the most famous of these arises in the argument for the so-called Gettier Problem (see especially case II, in this account).
A much more common valid inference involving sentences containing 'Or' are 'Disjunctive Syllogism' inferences. Disjunctive syllogism inferences also come in two varieties. The forms of this kind of inference are as follows:
P or Q
Not P
Thus,
Q
P or Q
Not Q
Thus,
P
This kind of inference is quite important in everyday life. Suppose that one is hoping to meet a friend, but one is unsure where they will be. One might reason as follows,
"At this time of day, Robin will be in the office, or in the cafe. I just called the office and there was no reply, so I had better go and look in the cafe."
In this case, P corresponds to 'Robin will be in the office'. Q corresponds to 'Robin will be in the cafe'. The fact that the office phone went unanswered, suggests that 'It is not the case that Robin is in the office' (i.e. Not P). So, we are led to the conclusion that 'Robin will be in the cafe' (i.e. Q). It is relatively easy to see that this kind of inference is valid, as we are most likely very familiar with it. However, we now have a much deeper understanding of why this is a 'good' inference.
Another important class of valid inferences occur with 'If..., then...' statements. If we think about how sentences with 'if..., then...' in them function, it becomes clear that provided some condition is met (the bit after the 'If...,'), some other thing (namely, the bit after the 'then...'), follows, or will happen. It is useful to have some special terminology to talk about the different parts of this kind of sentence. By tradition, the part after the 'If...,' is known as the 'antecedent' and the part after the 'then...' part is known as the 'consequent'. With this terminology, we can now say that with sentences of this kind, provided that the antecedent condition is met, the consequent result will follow.
Probably the best known kind of inference involving if...then... sentences is an inference known as 'Modus Ponens'. The form of a Modus Ponens inference is as follows;
If P, then Q
P
Thus,
Q
Consider as an example the inference that,
"If the telephone is ringing, then there is someone trying to call. The telephone is indeed ringing. Thus, there is someone trying to call."
There are many similar inferences we can think of. Any inference that follows this pattern though is going to be valid, as all Modus Ponens inferences are valid.
One thing that is important to keep an eye on though is that the various parts of the argument are all in the correct places. For instance, the inference "If P, then Q, Q, thus P", is invalid. This would be an instance of the fallacy of 'affirming the consequent'. This is not a truth preserving inference. This can be seen by substituting the phrase 'it is raining' for P, and 'the streets are wet', for Q. In this case, while it is often true that 'If it is raining, then the streets are wet', we can think of instances when the the streets are indeed wet, yet there could be another cause (for instance, they could be cleaning the streets, with water jets). This shows that affirming the consequent is not a valid kind of inference. However, if one reflects a little about the valid instance, with the same phrases substituted for P and Q, then the truth preserving nature of the inference is quite apparent.
Another important kind of valid inference involving 'if...,then...' are so-called 'Modus Tollens' inferences. These inferences take the following form,
If P, then Q
Not Q
Thus,
Not P
On the face of it, it may appear surprising that this kind of inference is valid, given that in many ways it appears similar to inferences that involve affirming the consequent. A more concrete example may help here. Consider once again the case when P is the phrase 'it is raining' and Q is the phrase 'the streets are wet'. In this case the inference would be,
"If it is raining, then the streets are wet. The streets are not wet. Thus, it is not raining."
This is notably different from the affirming the consequent cases, due to the inclusion of the 'not'. It is also the case that, intuitively, this seems like a pretty reasonable inference.
One thing to realise is that it is not absolutely necessary that the word 'not' appears in the second premise. What really matters is that the second premise and the conclusion are opposite (in terms of their negated, or unnegated status), from the values the same letters take in the first premise. We can see this by considering the following example, which is hopefully somewhat intuitive. Suppose a parent were to say to their child,
"If you do not clean up your bedroom, then you will not get your allowance."
This would form the first premise of the Modus Tollens inference. We can the imagine the child reasoning,
"I want my allowance, thus I must clean up my bedroom."
It is worth mentioning in closing that some people find Modus Tollens inferences notoriously hard to teach, compared to Modus Ponens inferences. Thus, it may be worth spending a little time thinking about this kind of inference, to enure that it is sufficiently well understood.
The final classic class of inferences involving 'If...,then...' that are valid are so-called 'Hypothetical Syllogism' inferences. These inferences have the following form,
If P, then Q
If Q, then R
Thus,
If P, then R
This inference type is much more obvious than Modus Tollens inferences. The following little argument would be a case of a valid Hypothetical Syllogism inference.
"If there is rain tomorrow, then the picnic will be cancelled. If the picnic is cancelled, then my salad will go to waste. Thus, if there is rain tomorrow, then my salad will go to waste."
This inference is valid and also quite intuitive.
In this post, we have continued looking at valid argument forms. The great thing about knowing about these is that they enable us to pretty quickly spot valid arguments in natural language. If an argument matches one of these forms, it is valid! Simple as that. However, before we can become really skilled at spotting these kinds of valid arguments in the real world, there is one further complication that needs to be addressed, concerning missing premises and conclusions. This will be the topic of the next post in this series.
The CP
- Previous Post in the Series
- Combat Philosopher Home Page
- Next Post in the Series
- Combat Philosopher Home Page
- Next Post in the Series
In the last post in this series, the conjunction and simplification valid argument forms were introduced. Also, a convention for using letters to stand for entire propositions, that is to say, things that can be said as complete statements, was explained. This convention will continue to be used here.
We have also been made aware that there are certain special logical words, that have very predictable effects upon the sentences in which they appear. Last time, we looked at the words 'Not' and 'And', which fall into this class of terms. These are not the only examples, however.
Another important logical word is the word 'Or'. There is a minor complexity that arises with 'Or', though. This is that natural language contains two logically distinct versions of 'Or'.
The first kind of 'Or' is called 'inclusive Or'. Sentences that contain an inclusive Or are true when either of the disjuncts (that is to say, the propositions either side of the Or) are true, as well as when both disjuncts are true. Consider the sentence
"Sammy will bring ham or cheese to the potluck dinner".
This sentence would not be false, if it turned out that Sammy brought a ham and cheese plate. In this case, 'Sammy will bring ham' is one disjunct, while 'Sammy will bring cheese' is the other disjunct. This illustrates the way 'inclusive Or' functions in language. Generally speaking, it is usually assumed that this is the default type of 'Or' people use, when speaking in natural language.
The second kind of 'Or' is known as 'exclusive Or'. For the purpose of disambiguation, this is usually written 'Xor'. This practice will be followed here. Sentences which have an Xor between their disjuncts are only true when one of their disjuncts are true, but not when they are both true. In common speech, the use of Xor is often marked by the use of the phrase "Either...,or...". Suppose one was to go out to dinner and the server was to say,
"Your meal comes with either a salad, or coleslaw. Which would you like?"
There would certainly be some surprise if one was to ask for both salad AND coleslaw! For the most part, in what follows, the inclusive use of 'Or' will be assumed.
Now attention can be turned to valid patterns of inference that involve the use of Or. The first of these to consider is the so-called 'addition' inference. Following the conventions introduced earlier, this kind of inference has two variants, which can be represented as follows:
P
Thus,
P or Q
Q
Thus,
P or Q
On the face of it, this may appear a bit of an odd inference, albeit a valid one. If one knows that a particular proposition, P is true, then we can infer that 'P or Q' is also true (actually, the same holds with false sentences -- valid inferences are 'truth-value preserving', in more technical language). So, if one knows that, for example "The sky is blue", it is valid to infer that "The sky is blue, or grass is green".
The apparent oddity of addition inferences notwithstanding, there are occasions when this kind of inference can be useful to make. There are a number of cases in the philosophical literature where inferences of this kind have been crucial to developing philosophical objections to positions. Perhaps the most famous of these arises in the argument for the so-called Gettier Problem (see especially case II, in this account).
A much more common valid inference involving sentences containing 'Or' are 'Disjunctive Syllogism' inferences. Disjunctive syllogism inferences also come in two varieties. The forms of this kind of inference are as follows:
P or Q
Not P
Thus,
Q
P or Q
Not Q
Thus,
P
This kind of inference is quite important in everyday life. Suppose that one is hoping to meet a friend, but one is unsure where they will be. One might reason as follows,
"At this time of day, Robin will be in the office, or in the cafe. I just called the office and there was no reply, so I had better go and look in the cafe."
In this case, P corresponds to 'Robin will be in the office'. Q corresponds to 'Robin will be in the cafe'. The fact that the office phone went unanswered, suggests that 'It is not the case that Robin is in the office' (i.e. Not P). So, we are led to the conclusion that 'Robin will be in the cafe' (i.e. Q). It is relatively easy to see that this kind of inference is valid, as we are most likely very familiar with it. However, we now have a much deeper understanding of why this is a 'good' inference.
Another important class of valid inferences occur with 'If..., then...' statements. If we think about how sentences with 'if..., then...' in them function, it becomes clear that provided some condition is met (the bit after the 'If...,'), some other thing (namely, the bit after the 'then...'), follows, or will happen. It is useful to have some special terminology to talk about the different parts of this kind of sentence. By tradition, the part after the 'If...,' is known as the 'antecedent' and the part after the 'then...' part is known as the 'consequent'. With this terminology, we can now say that with sentences of this kind, provided that the antecedent condition is met, the consequent result will follow.
Probably the best known kind of inference involving if...then... sentences is an inference known as 'Modus Ponens'. The form of a Modus Ponens inference is as follows;
If P, then Q
P
Thus,
Q
Consider as an example the inference that,
"If the telephone is ringing, then there is someone trying to call. The telephone is indeed ringing. Thus, there is someone trying to call."
There are many similar inferences we can think of. Any inference that follows this pattern though is going to be valid, as all Modus Ponens inferences are valid.
One thing that is important to keep an eye on though is that the various parts of the argument are all in the correct places. For instance, the inference "If P, then Q, Q, thus P", is invalid. This would be an instance of the fallacy of 'affirming the consequent'. This is not a truth preserving inference. This can be seen by substituting the phrase 'it is raining' for P, and 'the streets are wet', for Q. In this case, while it is often true that 'If it is raining, then the streets are wet', we can think of instances when the the streets are indeed wet, yet there could be another cause (for instance, they could be cleaning the streets, with water jets). This shows that affirming the consequent is not a valid kind of inference. However, if one reflects a little about the valid instance, with the same phrases substituted for P and Q, then the truth preserving nature of the inference is quite apparent.
Another important kind of valid inference involving 'if...,then...' are so-called 'Modus Tollens' inferences. These inferences take the following form,
If P, then Q
Not Q
Thus,
Not P
On the face of it, it may appear surprising that this kind of inference is valid, given that in many ways it appears similar to inferences that involve affirming the consequent. A more concrete example may help here. Consider once again the case when P is the phrase 'it is raining' and Q is the phrase 'the streets are wet'. In this case the inference would be,
"If it is raining, then the streets are wet. The streets are not wet. Thus, it is not raining."
This is notably different from the affirming the consequent cases, due to the inclusion of the 'not'. It is also the case that, intuitively, this seems like a pretty reasonable inference.
One thing to realise is that it is not absolutely necessary that the word 'not' appears in the second premise. What really matters is that the second premise and the conclusion are opposite (in terms of their negated, or unnegated status), from the values the same letters take in the first premise. We can see this by considering the following example, which is hopefully somewhat intuitive. Suppose a parent were to say to their child,
"If you do not clean up your bedroom, then you will not get your allowance."
This would form the first premise of the Modus Tollens inference. We can the imagine the child reasoning,
"I want my allowance, thus I must clean up my bedroom."
It is worth mentioning in closing that some people find Modus Tollens inferences notoriously hard to teach, compared to Modus Ponens inferences. Thus, it may be worth spending a little time thinking about this kind of inference, to enure that it is sufficiently well understood.
The final classic class of inferences involving 'If...,then...' that are valid are so-called 'Hypothetical Syllogism' inferences. These inferences have the following form,
If P, then Q
If Q, then R
Thus,
If P, then R
This inference type is much more obvious than Modus Tollens inferences. The following little argument would be a case of a valid Hypothetical Syllogism inference.
"If there is rain tomorrow, then the picnic will be cancelled. If the picnic is cancelled, then my salad will go to waste. Thus, if there is rain tomorrow, then my salad will go to waste."
This inference is valid and also quite intuitive.
In this post, we have continued looking at valid argument forms. The great thing about knowing about these is that they enable us to pretty quickly spot valid arguments in natural language. If an argument matches one of these forms, it is valid! Simple as that. However, before we can become really skilled at spotting these kinds of valid arguments in the real world, there is one further complication that needs to be addressed, concerning missing premises and conclusions. This will be the topic of the next post in this series.
The CP
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Labels: Better Reasoning
posted by The Combat Philosopher at 1:16 PM
1 Comments:
"If the telephone is ringing, then there is someone trying to call. The telephone is indeed ringing.
Thus, there is someone trying to call."
This is not valid. For example, telephone repairmen can trigger a phone to ring via sending a signal to the telephone; in effect, wiring and a switch is 'calling'.
Likewise, a short can cause a phone to ring, though in a short burst rather than a long ring... such as 'half a ring' and no human or machine needs to be involved to create a short, therefore, no one is 'calling'.
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