Previously, we talked a little bit about what logics are and we set up our language. Now, we’re going to discuss the interpretation and semantics for this language. In a later post, we’ll set up the system using the language.
Interpretation is how we thing of the language; it’s what it means. The same exact language can have different interpretations and you can have a formal system with an uninterpreted language. An example of differing interpretations for the same language (in the context of a system as well, actually) is that of quantum mechanics. The two main interpretations are the Copenhagen Interpretation and the Many Worlds interpretation. They are two ways of making sense of the data of QM experiments and they use the same math. They’re just different ideas of what the math “means”.
The interpretation of our is pretty straightforward. From the common name of the system (which we’ll get to later) with this interpretation and semantic, “Propositional Calculus”, we can pretty much guess. It’s the math of ideas. It is how we get from one proposition to another proposition. The symbols we introduced in the previous post will be given names corresponding to bits of natural language.
The semantic is how we use the symbols to approximate natural language. We define the way the connectives work such that they behave like the bit of natural language from which they get their names. The connectives are operators and are defined by something called the assignment function. I could bore you with the exact definition of the assignment function, or I could show it to you in tables. Yeah, I’ll go with that.
So, let’s start with the “horseshoe”. It’s technically called the “conditional” or “material conditional”. This connective corresponds with the bit of English “If a, then b”. In that example, “a” is called the “antecedent” and “b” is called the “consequent”. Now, let’s get to that table.
Now let’s do conjunction. This corresponds to “and”, and it is easy to see why via the table. Each of the wffs connected to the conjunction is called a “conjunct”.
Having done conjunction, let’s move on to disjunction. This corresponds to one type of “or”. This is an inclusive or (think “and/or”) rather than an xor. Each of the wffs connected to the the disjunction is called a “disjunct”.
There is a connected related to the conditional called the “biconditional”. It corresponds to “if and only if” and is often abbreviated in text as “iff”. We’ll see in the next post why “if and only if” is a very good description. This connective is define, as you guessed, by yet another table:
Now we end the post with the easiest assignment table. This one is for negation. It corresponds to “not”.