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o connect P with Q using the arrow ‘→’ in PL is to form the conditional P → Q. In any conditional, the formula occurring before the arrow is known as the antecedent of the conditional, and the formula occurring after the arrow is the consequent. So, in the case in question, P is the antecedent and Q the consequent. Here, I want to consider four different inferences which people may make using conditionals. All four inferences are commonplace in ordinary discourse—despite the fact that two of them are obviously invalid! To understand fully the four inferences in question we first need to understand the nature of negation. It is intuitive to think of negation as working by simply reversing truth-values. In other words, if the sentence ‘I am now reading this book’ is true then the negation of that sentence ‘I am not now reading this book’ must be false. Similarly, if the sentence ‘It’s raining in Old Aberdeen today’ is true then the negation of that sentence, namely ‘It’s not raining in Old Aberdeen today’, will, again, be false. But this is only half of the story. If the negation of a given sentence is true then the original sentence must be false, i.e. if the sentence ‘I am not now reading this book’ is true then the sentence ‘I am now reading this book’ is false. Equally, if the sentence ‘It’s not raining in Old Aberdeen today’is true, then the sentence ‘It’s raining in Old Aberdeen today’ is false.
These intuitions about negation are central to the account of negation given in PL. The kind of negation involved in PL is called classical negation.
Classical negation is denial and the net effect of negating a sentence is precisely to reverse the truth-value of that sentence. To deny that P is true is to assert that ~P is true. To deny that ~P is true is to assert that P is true. We can easily tabulate this state of affairs for an arbitrarily chosen formula, say P, and represent the reversal of truth-value which negation effects as follows:
With these points about negation in mind, we can now consider four basic inferences which people may make. Note carefully the way in which these differ just in terms of whether the antecedent or the consequent is affirmed or denied in the course of the inference. Crucially, note also how this affects the validity of each inference.
First, recall the original Blind Lemon Jefferson argument and its formal counterpart in PL. It is itself a prime example of one of the four inferences, i.e. recall the natural language argument:
1. If it’s a Blind Lemon Jefferson album then it’s a blues album.
2. It’s a Blind Lemon Jefferson album.
Therefore,
3. It’s a blues album.
As you know, this argument can be formally represented in PL as P → Q, P:
Q. In this case, given P → Q, we assert the antecedent, P, and then infer the consequent, Q. In more traditional logical terms, we affirmed the antecedent P and then concluded that Q. Does Q logically follow? It certainly does.
Indeed, just this kind of reasoning is represented by one of our rules of inference, modus ponens. So affirming the antecedent is a perfectly valid form of inference.
But suppose that, rather than affirm the antecedent, we affirm the consequent instead, i.e. that, given P → Q, we affirm Q and infer P. Is this inference a valid inference? It certainly is not! To affirm the consequent is to commit a traditional fallacy, i.e. to reason in a perfectly invalid way. The following counterexample demonstrates that this so:
1. If all cats are black then Tiffin is black. P → Q
2. Tiffin is black. Q
Therefore,
3. All cats are black. P
Hence, affirming the consequent is invalid. (You might remember that this particular example featured earlier in Exercise 1.1. In the course of that exercise you should have proved yourself that the form of argument involved is indeed an invalid one.)
P ~P
True False
False True
The two remaining inferences both involve negation and hence denial rather than assertion or affirmation. The first case involves denying the antecedent. For example, consider the following sequent: P → Q, ~P:~Q.
Is this a valid sequent of PL? Again, it certainly is not. Here is a counterexample:
1. If all cats are black then Zebedee is black. P → Q 2. It’s not the case that all cats are black. ~P
Therefore, ______
3. It’s not the case that Zebedee is black. ~Q
Again, we met this example in Exercise 1.1 and so you should already have proved there that this particular argument is indeed an instance of an invalid logical form. Hence, denying the antecedent is fallacious.
One last possibility remains, namely, that of denying the consequent. Is the following sequent valid in PL, P → Q, ~Q:~P? This sequent is indeed valid in PL. And again, we frequently make inferences of this type in natural language. For example:
1. If it’s a Blind Lemon Jefferson album then it’s a blues album. P → Q
2. It’s not a blues album. ~Q
Therefore, ______
3. It’s not a Blind Lemon Jefferson album. ~P
This particular argument is certainly a valid argument and it is also an instance of a valid logical form of argument. Moreover, like the first type of inference we considered in this section, inferences of this last type also have a traditional name, not modus ponens but modus tollens. Further, we can easily incorporate this mode of inference into our proof-theory for PL simply by adding a new rule. Logically enough, the new rule is called modus tollens and its annotation is just ‘MT’. Here is the rule-statement in full:
MT: Given a conditional on one line and the negation of its consequent on another, infer the negation of the antecedent.
Annotate the new line with the line numbers of both lines used and
‘MT’. The dependency-numbers of the new line are all those of both lines used.
Consider the simplest possible example of the rule MT in action in the proof of the sequent P → Q, ~Q ~P:
P → Q, ~Q ~P
{1} 1. P → Q Premise
{2} 2. ~Q Premise
{1,2} 3. ~P 1,2 MT
MT works well with our existing stock of rules. For example, consider the proof of the following sequent P → Q ~Q → ~P. Because the main connective in the conclusion is a conditional the strategy for proof is CP. So, assume the antecedent, ~Q, and try to derive the consequent, i.e. ~P. Given MT, ~P is easy to derive:
{1} 1. P → Q Premise
{2} 2. ~Q Assumption for CP
{1,2} 3. ~P 1,2 MT
{1} 4. ~Q → ~P 2,3 CP
In the next case (Exercise 3.1), MT works together with MP to give us the desired conclusion. Try this one yourself.
EXERCISE 3.1
1 Construct a proof of the following sequent (as ever, the number in brackets indicates the number of lines in my proof):
P, P → (Q → R), ~R : ~Q (5)