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:
- A and B
- 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.