T4: 6 of 10
Semantics for Extensional Logic
Semantics has to do with truth and meaning for sentences. We will wait until the next tutorial to give a formal semantics for our language. But some of the ideas are easy and worth previewing.
Truth Functionality
Recall our simple example:
1. Agnes will go to law school and Bob will go to law school.
We said that we'd symbolize this as:
Wa&Wb
The easy and obvious thing to say about the meaning of this English sentence, and of it's SL symbolization, is that both are true if, and only if, their components are true. So, for example, the English version is true just in case both "Agnes will go to law school" is true and "Bob will go to law school" is true. Obviously!
And the same can be said for our symbolization Wa&Wb'. This one is true if and only if Wa and 'Wb' are both true.
Here's the less trivial idea. Our logic is to be defined so that the truth value of a compound sentence (like Wa&Wb') is completely determined by the truth values of its component parts. We will say that '&' and all the rest of our SL connectives are truth functional because the truth values of sentences formed from any of our connectives is a function (i.e., is determined by) the truth values of the component parts. But the details of why this is so are in the next tutorial. (You may recall doing "truth tables" in a previous logic class. These tables just spell out the function involved...much like multiplication tables spell out a function.)
But it is worth noting that though all connectives of our symbolic language SL are truth functional, there are connectives of English which are not. The words "and then" typically are used in a way that is not truth functional. For example, take 1 above but change it by adding the word "then":
Agnes will go to law school and then Bob will go to law school.
This one is not truth functional because the truth value of the whole sentence depends on more than the truth of the two components "Agnes will go to law school" and "Bob will go to law school". Why? Because if this sentence is to be true, it needs for Agnes to go to law school before Bob, not just that both go.
Leibniz's Law
Suppose that Agnes will attend law school. So, Wa is true. If Agnes happens to be the eldest member of the Burke clan, then it follows that the eldest member of the Bruke clan will attend law school. Trivial! Here's the idea.
If a=b and a has some property P, then b also has the property P. Or more formally:
If a=b, then Pa=Pb
This is just one form of Leibniz's law. But it's enough for the moment. For more details see Chapter Nine of the Logic Café.
For now we should notice that Liebniz's law is a little like truth functionality. For example, the truth value of Wa is determined by the reference of 'a'. And the truth value of "the eldest member of the Burke clan will attend law school" is determined by the reference of "the eldest member of the Burke clan". The truth value doesn't depend on just how Agnes is described, but only on her.
Compositionality and Extensional Logic
The idea behind compositionality is roughly that the meaning of a whole sentence is determined by the meaning of the parts. This is the idea first made very explicit by Frege in the late 19th century. For us this compositionality is a generalization of truth functionality:
The reference or truth value of a compound linguistic structure is determined by the reference and truth value of its component parts.
So, to take an example that generalizes beyond truth functionality and Leibniz's law:
If "unmarried adult male" refers to the same group of people as "bachelor", then if "George is a bachelor" is true, then "George is an unmarried adult male" is also true.
A logic that is compositional in this way is also often called "extensional". There are however examples of non-extenstional (or "intentional") logics. For example, the logic of knowledge. Gail may know that Muhammad Ali was a famous boxer. But she may fail to know that Cassius Clay was a famous boxer, even though Cassius Clay is the same person as Muhammad Ali. Notice that this shows a failure of Liebniz's law.
Other intensional logics are described in the Flashboard above and in the next tutorial.