Chapter 13 Limitations of TFL

We have reached an important milestone: a test for the validity of arguments! However, we should not get carried away just yet. It is important to understand the limits of our achievement. We will illustrate these limits with four examples.

First, consider the argument:

• ​

  Daisy has four legs.

• ∴
  ​

  Daisy has more than two legs.

To symbolize this argument in TFL, we would have to use two different sentence letters—perhaps ‘$F$’ and ‘$T$’—for the premise and the conclusion respectively. Now, it is obvious that ‘$F$’ does not entail ‘$T$’. But the English argument is surely valid!

Second, consider the sentence:

1. 1.
   ​

   Jan is neither bald nor not bald.

To symbolize this sentence in TFL, we would offer something like ‘$\neg J\wedge \neg \neg J$’. This a contradiction (check this with a truth-table), but sentence 1 does not itself seem like a contradiction: we might have happily added ‘Jan is on the borderline of baldness’!

Third, consider the following sentence:

1. 2.
   ​

   It’s not the case that, if God exists, She answers malevolent prayers.

Symbolizing this in TFL, we would offer something like ‘$\neg (G\to M)$’. Now, ‘$\neg (G\to M)$’ entails ‘$G$’ (again, check this with a truth table). So if we symbolize sentence 2 in TFL, it seems to entail that God exists. But that’s strange: surely even an atheist can accept sentence 2, without contradicting herself!

One lesson of this is that the symbolization of sentence 2 as ‘$\neg (G\to M)$’ shows that sentence 2 does not express what we intend. Perhaps we should rephrase it as

1. 3.
   ​

   If God exists, She does not answer malevolent prayers.

and symbolize sentence 3 as ‘$G\to \neg M$’. Now, if atheists are right, and there is no God, then ‘$G$’ is false and so ‘$G\to \neg M$’ is true, and the puzzle disappears. However, if ‘$G$’ is false, ‘$G\to M$’, i.e., ‘If God exists, She answers malevolent prayers’, is also true!

In different ways, these four examples highlight some of the limits of TFL, symbolization of English in TFL, and the tests based on truth tables we’ve devised. Our first example showed that TFL is not expressive enough to symbolize everything we might want to in a way that allows us to apply our logical toolkit to even the best possible symbolization. We will see later (chapter 26) that we can properly symbolize ‘Daisy has four legs’ in the more expressive language FOL, and then the test we will devise for FOL will apply (and give the correct answer).

In sentence 1 we similarly had an example where the straightforward symbolization in TFL does not quite work. The best we can do in TFL is to symbolize ‘Jan is not bald’ using its own sentence letter, say, ‘$N$’ (contrary to our standing practice). But this also does not work. For instance, we would then symbolize ‘Jan is both bald and isn’t’ as ‘$J\wedge N$’. But ‘Jan is both bald and isn’t’ arguably is a self-contradiction, yet the alternate symbolization ‘$J\wedge N$’ is not. Some logicians have proposed that in cases of sentences like ‘Jan is bald’ which allow borderline cases, we have to adjust our semantics and allow that the sentence letter ‘$J$’ that symbolizes it should be allowed to take truth values other than T and F. But this is by no means universally accepted.

Sentence 2 also showed that symbolization is often tricky, and that the straightforward symbolization of and English sentence sometimes does not capture its intended truth conditions. But even our improvement, sentence 3, and its symbolization ‘$G\to \neg M$’ turned out to not be good enough. The phenomenon we encountered here is one of the so-called paradoxes of the material conditional. The paradox is that material conditionals like ‘$G\to \neg M$’ and ‘$G\to \neg M$’ are true whenever their antecedent ‘$G$’ is false, even though intuitively the two sentences ‘If God exists, She answers malevolent prayers’ and ‘If God exists, She does not answer malevolent prayers’ should not both be true. This indicates that the ‘if …then’ in these two examples is not properly captured by the TFL connective ‘$\to$’. The truth-functionality of TFL here is a real limitation. The solution would be to treat ‘if …then’ in this case as a subjunctive conditional (see section 10.3), but TFL cannot capture those.¹¹ 1 Logicians have devised logics that deal with subjunctive conditionals better. They are called ‘conditional logics’ or ‘logics of counterfactuals’.

The case of Jan’s baldness (or otherwise) raises the general question of what logic we should use when dealing with vague discourse. The case of the atheist raises the question of how to deal with the (so-called) paradoxes of the material conditional. Part of the purpose of this book is to equip you with the tools to explore these questions of philosophical logic. But we have to walk before we can run; we have to become proficient in using TFL, before we can adequately discuss its limits, and consider alternatives.