Chapter 6 Sentences of TFL

We have seen that there are three kinds of symbols in TFL:

Define an expression of TFL as any string of symbols of TFL. So: write down any sequence of symbols of TFL, in any order, and you have an expression of TFL.

Given what we just said, ‘$(A\wedge B)$’ is an expression of TFL, and so is ‘$\neg )(\vee ()\wedge (\neg \neg ())((B$’. But the former is a sentence, and the latter is gibberish. We want some rules to tell us which TFL expressions are sentences.

Obviously, individual sentence letters like ‘$A$’ and ‘$G_{13}$’ should count as sentences. (We’ll also call them atomic sentences.) We can form further sentences out of these by using the various connectives. Using negation, we can get ‘$\neg A$’ and ‘$\neg G_{13}$’. Using conjunction, we can get ‘$(A\wedge G_{13})$’, ‘$(G_{13}\wedge A)$’, ‘$(A\wedge A)$’, and ‘$(G_{13}\wedge G_{13})$’. We could also apply negation repeatedly to get sentences like ‘$\neg \neg A$’ or apply negation along with conjunction to get sentences like ‘$\neg (A\wedge G_{13})$’ and ‘$\neg (G_{13}\wedge \neg G_{13})$’. The possible combinations are endless, even starting with just these two sentence letters, and there are infinitely many sentence letters. So there is no point in trying to list all the sentences one by one.

Instead, we will describe the process by which sentences can be constructed. Consider negation: Given any sentence $𝒜$ of TFL, $\neg 𝒜$ is a sentence of TFL. (Why the funny fonts? We return to this in section 8.3.)

We can say similar things for each of the other connectives. For instance, if $𝒜$ and $ℬ$ are sentences of TFL, then $(𝒜\wedge ℬ)$ is a sentence of TFL. Providing clauses like this for all of the connectives, we arrive at the following formal definition for a sentence of TFL :

Definitions like this are called inductive . Inductive definitions begin with some specifiable base elements, and then present ways to generate indefinitely many more elements by compounding together previously established ones. To give you a better idea of what an inductive definition is, we can give an inductive definition of the idea of an ancestor of mine. We specify a base clause:

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  My parents are ancestors of mine.

and then offer further clauses like:

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  If $x$ is an ancestor of mine, then $x$’s parents are ancestors of mine.

Finally, we stipulate that only what the base and inductive clauses say are ancestors of mine will count as such.

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  Nothing else is an ancestor of mine.

Using this definition, we can easily check to see whether someone is my ancestor: just check whether she is the parent of the parent of…one of my parents. And the same is true for our inductive definition of sentences of TFL. Just as the inductive definition allows complex sentences to be built up from simpler parts, the definition allows us to decompose sentences into their simpler parts. Once we get down to sentence letters, then we know we are ok.

Let’s consider some examples.

Suppose we want to know whether or not ‘$\neg \neg \neg D$’ is a sentence of TFL. Looking at the second clause of the definition, we know that ‘$\neg \neg \neg D$’ is a sentence if ‘$\neg \neg D$’ is a sentence. So now we need to ask whether or not ‘$\neg \neg D$’ is a sentence. Again looking at the second clause of the definition, ‘$\neg \neg D$’ is a sentence if ‘$\neg D$’ is. So, ‘$\neg D$’ is a sentence if ‘$D$’ is a sentence. Now ‘$D$’ is a sentence letter of TFL, so we know that ‘$D$’ is a sentence by the first clause of the definition. So for a compound sentence like ‘$\neg \neg \neg D$’, we must apply the definition repeatedly. Eventually we arrive at the sentence letters from which the sentence is built up.

Next, consider the example ‘$\neg (P\wedge \neg (\neg Q\vee R))$’. Looking at the second clause of the definition, this is a sentence if ‘$(P\wedge \neg (\neg Q\vee R))$’ is, and this is a sentence if both ‘$P$’ and ‘$\neg (\neg Q\vee R)$’ are sentences. The former is a sentence letter, and the latter is a sentence if ‘$(\neg Q\vee R)$’ is a sentence. It is. Looking at the fourth clause of the definition, this is a sentence if both ‘$\neg Q$’ and ‘$R$’ are sentences, and both are!

Ultimately, every sentence is constructed nicely out of sentence letters. When we are dealing with a sentence other than a sentence letter, we can see that there must be some sentential connective that was introduced last, when constructing the sentence. We call that connective the main logical operator of the sentence. In the case of ‘$\neg \neg \neg D$’, the main logical operator is the very first ‘$\neg$’ sign. In the case of ‘$(P\wedge \neg (\neg Q\vee R))$’, the main logical operator is ‘$\wedge$’. In the case of ‘$((\neg E\vee F)\to \neg \neg G)$’, the main logical operator is ‘$\to$’.

As a general rule, you can find the main logical operator for a sentence by using the following method:

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  If the first symbol in the sentence is ‘$\neg$’, then that is the main logical operator

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  Otherwise, start counting the brackets. For each open-bracket, i.e., ‘(’, add $1$; for each closing-bracket, i.e., ‘$)$’, subtract $1$. When your count is at exactly $1$, the first operator you hit (apart from a ‘$\neg$’) is the main logical operator.

(Note: if you do use this method, then make sure to include all the brackets in the sentence, rather than omitting some as per the conventions of section 6.3!)

The inductive structure of sentences in TFL will be important when we consider the circumstances under which a particular sentence would be true or false. The sentence ‘$\neg \neg \neg D$’ is true if and only if the sentence ‘$\neg \neg D$’ is false, and so on through the structure of the sentence, until we arrive at the atomic components. We will return to this point in part III.

The inductive structure of sentences in TFL also allows us to give a formal definition of the scope of a negation (mentioned in section 5.2). The scope of a ‘$\neg$’ is the subsentence for which ‘$\neg$’ is the main logical operator. Consider a sentence like:

which was constructed by conjoining ‘$P$’ with ‘$(\neg (R\wedge B)\leftrightarrow Q)$’. This last sentence was constructed by placing a biconditional between ‘$\neg (R\wedge B)$’ and ‘$Q$’. The former of these sentences—a subsentence of our original sentence—is a sentence for which ‘$\neg$’ is the main logical operator. So the scope of the negation is just ‘$\neg (R\wedge B)$’. More generally:

Strictly speaking, the brackets in ‘$(Q\wedge R)$’ are an indispensable part of the sentence. Part of this is because we might use ‘$(Q\wedge R)$’ as a subsentence in a more complicated sentence. For example, we might want to negate ‘$(Q\wedge R)$’, obtaining ‘$\neg (Q\wedge R)$’. If we just had ‘$Q\wedge R$’ without the brackets and put a negation in front of it, we would have ‘$\neg Q\wedge R$’. It is most natural to read this as meaning the same thing as ‘$(\neg Q\wedge R)$’, but as we saw in section 5.2, this is very different from ‘$\neg (Q\wedge R)$’.

Strictly speaking, then, ‘$Q\wedge R$’ is not a sentence. It is a mere expression.

When working with TFL, however, it will make our lives easier if we are sometimes a little less than strict. So, here are some convenient conventions.

First, we allow ourselves to omit the outermost brackets of a sentence. Thus we allow ourselves to write ‘$Q\wedge R$’ instead of the sentence ‘$(Q\wedge R)$’. However, we must remember to put the brackets back in, when we want to embed the sentence into a more complicated sentence!

Second, it can be a bit painful to stare at long sentences with many nested pairs of brackets. To make things a bit easier on the eyes, we will allow ourselves to use square brackets, ‘[’ and ‘]’, instead of rounded ones. So there is no logical difference between ‘$(P\vee Q)$’ and ‘$[P\vee Q]$’, for example.

Combining these two conventions, we can rewrite the unwieldy sentence

rather more clearly as follows:

The scope of each connective is now much easier to pick out.