Truth-functional connectives - functions of what exactly?

Question

In a recent piece of work I described sentences containing a specific natural language connective as ᴛʀᴜᴛʜ ғᴜɴᴄᴛɪᴏɴᴀʟ. More specifically, I wrote something along the lines of [wording changed to make the sentence less jargonistic]:

The truth value of such a sentence is determined by the truth values of its constituent clauses taken individually. In other words, it is assumed that such sentences are truth-functional.

One of my supervisors - from a very different field - wrote in a note attached to the phrase 'determined by the truth values of its constituent clauses taken individually' : "What about the connective itself?"

In a necessarily very brief conversation with them later they said something which led me to ask myself:

• Is the truth value of this kind of so-called truth-functional sentence merely a function of the truth values of the individual clauses of the sentence, or is it in fact a function of both a) those individual values of the two clauses and b) the connective.

From the little that I understand, a function is something with has inputs which always map onto specific outputs. If the inputs are the same, the outputs are always the same.

Now I am doubting whether the truth value of sentences with truth-functional connectives are really a function of the values of the individual atomic 'sentences' themselves, or if they are a function of three inputs, the two values of those atomic sentences AND the connective.

The reason is that the values of the larger sentences comprised of the two smaller individual conjoined clauses yield different values depending on the connective involved. So, for example, if A is true and B is false, then:

1. A and B
2. A or B

... have different truth values. Example (1) is false and (2) is true. So it seems that the connective is one of the inputs into the function mapping such sentences onto true and false values.

So, therefore, my question is:

• Are the truth values of sentences containing truth-functional connectives a function of the truth values of the atomic sentences involved, or of the atomic sentences AND the connective?

If the answer is that they are functions just of the values of the two indiviual atomic sentences involved, then how can we say that, for example, A and B is not equivalent to A or B?

I'm a syntactician, not a logician or semanticist. So a clear and simple explanation for dummies like me would be appreciated.

Answer 1

You're overthinking this. If there's a function such that f(x,y)=z, then it's not strictly correct to say that the value of z is determined by the values of x and y, because you should also mention the function itself. Usually it's obvious that you mean something like: "in the context of f, the value of z is determined by x and y", which is why the supervisor's comment might seem a bit pedantic, but it's correct nonetheless.

Answer 2

I'm not sure if I my answer will clear things up, but Ill start with your core question:

Are the truth values of sentences containing truth-functional connectives a function of the truth values of the atomic sentences involved, or of the atomic sentences AND the connective?

I think the use of "function" in the question is going to be a source of confusion.

It's probably easier to understand the connective as the function that relates the truth values of atomic sentences.

An atomic sentence is something like:

   R = It is raining
   B = There are clouds in the sky

In logic "and" is a truth function. It relates the two atomic sentences such that if they are both true, then R and B is true. Otherwise it is false. In other words, we can see it as the function that takes R and B as inputs and does something with them.

Same thing with "or", inclusive or takes R and B as inputs and returns TRUE in all cases except where R is false and B is false.

Let's add one more concept from logic, "well-formed formula". A well-formed formula is one where all of the functions have the right set of inputs to work. In other words, if we use "and", we need two inputs. If we use "or", we need two inputs. If we use "not", we need one and only input.

Given this background, we can now turn to your sentence:

The truth value of such a sentence is determined by the truth values of its constituent clauses taken individually.

Now let's think about this sentence:

  It is raining outside and I am happy; the world is full of life; the sky is blue or I am part of the blue man group.

This sentence has constituent parts that each have at truth value:

It is raining outside and I am happy (at least where I am false -- not raining)
The world is full of life (true)
The sky is blue or I am part of the blue man group (true)

But then how do we relate the false, true, true that are separated by semi-colons? Answer: I have no idea, because this is not a well-formed formula.

So then I would suggest that your supervisor is onto something. In order for your sentence to be truth-functional, it must be the case that

1. Every constituent part has a truth value
2. All constituent parts are joined together into a whole using truth-functional connectors
3. Every level is well-formed (every connector has the right number of inputs and it all resolves out to one value on the top).

At least on my read from a distance, that's what your supervisor is trying to point to -- that the pieces of a larger truth-functional sentence all have to be ordered together and that's what the connectors do. It's a whole -- not an aggregate.

---

Confusing addendum: you can make the functions an input if you really want to. We could understand it like this:

 function logic_resolver(connector,var1,var2 ...)
 {
   if connector == "and" ...
   if connector == "or" ...
 }

and so on, with it handling the function choice as an input as well and running recursively.

Compare:

   A = TRUE
   B = TRUE
   C = FALSE
   A and B   --> TRUE
   A and C   --> FALSE
   TF("and",A,B)  --> TRUE
   TF("and",A,C)  --> FALSE
   (A and B) and C --> FALSE
   TF("and",TF("and",A,B),C) --> FALSE

This is kind of the idea behind reverse polish notation.

But for simpler tasks, I would just recommend seeing the truth-functional connectors themselves as functions.