Introduction to Philosophical Logic/Complex sentences and sentence functors
The reductio ad absurdum method can be extended to complex sentences. A complex sentence is a sentence made up of smaller sentences, for example "Sarah can swim but she cannot dive" is made up of the declarative sentences "Sarah can swim" and "She cannot dive", linked by the conjunction "but". Such words and phrases that link other declarative sentences are sentence functors. A functor is the part of a language that stands for a function. For instance "y=sin(x)" is a statement (in a particular mathematical notation) about the function sine and "sin()" is the functor that stands for this function in that statement. Function is not defined here. If you are unsure of its meaning, please refer to Wikipedia.
A 'sentence function' takes declarative sentences as input and yields declarative sentences as output. Sentence functors stand for such functions. A sentence functor is then a string of words and sentence variables that becomes a declarative sentence if each sentence variable is replaced by a declarative sentence (here "string" just means a series of symbols). This is true for English, French, Spanish, German, Greek, Latin and so on, but all words and declarative sentences must be in the same language.
The composition of sentences
Sentences comprise parts known as constituents. A constituent will be defined as a string of symbols that is meaningful by itself. This definition may be regarded as slightly empty and the meaning of constituent is probably best understood intuitively and by example. The following are constituents of the sentence "The cat sat on the mat":
sat on the mat
on the mat
the cat sat on the mat
All of the above are meaningful in themselves. Words are meaningful by themselves and so all individual words are constituents of any sentence that contains them (words can be considered atomic parts of language, that is they cannot be broken down further into other constituents - inflections, prefixes and suffixes are ignored here). The meaning of each constituent shown is the same as its meaning as a part of the sentence. The meaning of the constituent must be the same as it meaning in the sentence of which it is part, otherwise (although a string identical to the constituent may appear in the sentence) it is not a constituent of the sentence. For example, consider the sentence "the man who wrote on the blackboard was old".
The string "the blackboard was old" is meaningful, however its meaning is no part of the sentence above, which is asserting that the man, not the blackboard, was old. Hence, "the blackboard was old" is not a constituent of that sentence. Similarly, and perhaps less clearly, "the man" is not a constituent of the sentence. The meaning of "the man" is not the meaning conveyed in the sentence. Were "the man" to be a constituent, it would suggest that a particular man had already been identified. However, this is not the case since the restriction "who wrote on the blackboard" is needed. So, "the man" is not a constituent, but "the man who wrote on the blackboard" is.
An ambiguous sentence is one that has more than one meaning. Crudely speaking, there are two types of ambiguity: structural and lexical. A lexical ambiguity arises where one word (or perhaps a phrase) has more than one meaning; for example "fast" in the sentences "He drove very fast" and "He was fast asleep". A structural ambiguity arises in a constituent because it is unclear what that constituent's constituents are. To clarify this, the notion of scope is introduced.
The scope of a constituent is defined as the smallest constituent containing that constituent and something else besides. So, in the above example, the scope of "blackboard" is "the blackboard"; the scope of the first "the" is "the man who wrote on the blackboard".
Consider the sentence "He told me to be careful this evening". Was this warning discussed in this sentence issued this evening or was it about this evening? It is unclear exactly what the constituents of this sentence are: it is unclear what the scope of "this evening" is. Is the scope of "this evening" "to be careful this evening", or is it the whole sentence "He told me to be careful this evening"? It can be seen that a new language might be devised to clear up such ambiguities: "He told me [to be careful this evening]"; "[He told me to be careful this evening]". Propositional calculus is an entirely unambiguous language, using a bracketing system as shown here. This will be seen in the next part of the book.
To repeat the definition given earlier, sentence functors are strings of words and sentence variables such that if all the sentence variables are replaced by any declarative sentences, the whole becomes a declarative sentence. To fully understand this definition, it is necessary to know what a sentence variable is. A sentence variable is something (usually represented by the Greek letters psi, phi or chi) that can be assigned as its value any declarative sentence.
One example of a sentence functor was discussed in the section on consistency: it is not the case that phi
Other examples of a sentence functors are:
phi and psi I know that phi It is obvious that phi Either phi, or psi and chi
The following are not English sentence functors (consider whether the string obtained by replacing the sentence variable with a declarative sentence is in itself a declarative sentence):
Mary and phi Is it true that phi? phi is true (but "phi" is true is a sentence functor) Whomever phi should stand up for themselves
The last example forms a declarative sentence when phi is replaced by some declarative sentences (for example, if replaced with "Jack is bullying"), but not all (e.g. "the sky is blue"), so it is not a sentence functor.
The number of places of a sentence functor is the number of different sentence variables it contains.
Either phi, or psi and chi is a 3-place sentence functor. Either phi or psi is a 2-place sentence functor. Either it is the case that psi or it is not the case that psi is a 1-place sentence functor.
An n-place sentence functor is satisfied by certain ordered n-tuplets of declarative sentences. An ordered n-tuplet of declarative sentences is a list of n different declarative sentences in a particular order. In particular, those ordered n-tuplets of declarative sentences that satisfy an n-place sentence functor yield a true declarative sentence when they, in order, replace the sentence variables of a sentence functor.
The ordered pair (grass is green, snow is black) satisfies it is the case that phi, but not that psi; whereas the ordered pair (snow is black, grass is green) does not.
A sentence is a sentence functor with no sentence variables, i.e. a sentence is a 0-place sentence functor.
When determining what sentences satisfy what sentence functors, the logician is interested in their truth value (rather than the actual meaning or sense). This information can be summarised in the form of a truth table.
A truth table stipulates all combinations of truth for a given set of sentences and what the truth value of a sentence functor is for each combination.
Consider the sentence functor phi and psi. The truth table for this sentence functor is drawn up as follows:
|P||Q||P and Q|
Notice that the letters P and Q are used rather than phi and psi. Sentence variables cannot bear truth values. P and Q instantiate actual sentences, the truth values of which are considered below these letters in the table: 'T' stands for when that sentence is true and 'F' stands for when it is false. Note also that "P and Q" is a sentence, not a 2-place sentence functor (sentence functors with at least one sentence variable cannot bear truth values). To know what the value of the declarative sentence yielded by replacing the sentence variables of the sentence functor with sentences of various truth values, the row (known as a structure) containing the desired truth values is selected and the letter in that row below the complex sentence taken.
In the above example, the sentence "P and Q" is true when P is true and Q is true but false for any other values of P and Q (so when P is true but Q is false, "P and Q" is false).
Consider the truth table of the sentence functor Hume knew that phi.
|P||Hume knew that P|
The structure where P is false is false for "Hume knew that P", for it is not possible to know something that is false. However, the structure where P is true has the symbol "-" in it (this will be referred to as a blank). This symbol does not mean that the sentence is neither true nor false in this structure. It means that there are true sentences that satisfy this functor and there are sentences that do not satisfy this functor. For example, Hume knew that his (Hume's) first name was David. However, he did not know that Russell was (or rather would be from Hume's perspective) a 20th century philosopher.