Semantics of propositional logic

Question

I am learning about semantics in logic, starting with proportional logic (PL). However, I struggle to fully understand some of the concepts.

There seem to be multiple ways in which a wff (well-formed formulas) of PL can be assigned meaning. The literature often speaks of:

• Interpretations: for any atomic wff $\alpha$, an interpretation of $\alpha$ is an assignment of a truth value to $\alpha$.
• Valuations: for any wff $\alpha$ with an interpretation $\mathscr{I}(\alpha)$, a valuation of $\alpha$ is an assignment of truth value to $\alpha$ based on $\mathscr{I}(\alpha)$.

Now, assuming I got the above right, what confuses me is that some authors moreover speak of interpretations as assigning what I take to be extensions (or truth conditions) to wffs (e.g., Peter Smith seems to use "interpretation" in this way in his An Introduction to Formal Logic (2022)). This procedure seems to involve connecting $\alpha$ (where $\alpha$ is an atomic wff) with a condition such that $\alpha$ is true iff the world is such that that condition is met; otherwise, $\alpha$ is false. For example, if we assign "Plato is a philosopher" as an extension to some atomic wff $P$, $P$ is true iff Plato is a philosopher.

And finally, I also encountered talk of truth functions. It seems that each wff is moreover a truth function taking truth values as inputs and returning a unique truth value as output.

Now, finally my questions:

1. How do (what I labeled) "truth conditions" or "extension" fit within the overall semantics of PL? Is there an official name for this kind of meaning of a wff?
2. How does the truth-functional nature of a wff fit within the overall semantics of PL? Are wffs truth functions in addition to having an interpretation and a valuation, or does their truth functionality somehow follow from the previous?
3. Is the meaning of a wff determined by all of the above four concepts (interpretation, valuation, extension, or truth-functionality), or are some among them more fundamental than the others?

Answer 1

The distinction between interpretation and valuation is not very clear-cut; different authors use the terms differently. The word "interpretation" tends to have a wider scope, meaning a mapping from symbols to some sort of value, e.g. in first-order logic an interpretation assigns to the symbols individual entities, functions and relations, whereas "valuation" typically is used when the values are truth values. In propositional logic "interpretation" and "valuation" are often used to mean the same thing across different or even the same author, and one term may be used for both of the two concepts you distinguished. In any event, both refer at their core to the same idea: A mapping from symbols to some sort of semantic value, in the case of propositional logic a truth value.

The extension is the value of an interpretation applied to a formula. So in propositional logic, the extension of a wff is always 1 or 0. The term "extension" is used to distinguish from intension, where an extension is the concrete value of an expression in a specific situation, whereas the intension is an abstraction over possible extensions and comes closer to what one would think of as "sense"; two propositions may happen to have the same truth value (extension) under a given interpretation but may have different truth values under different circumstances and not be identical in meaning (have different intensions).

A truth condition is, well, a condition under which a formula is true, the "$= 1$" part of the valuation function. "Plato is a philosopher" is not an extension, it is a natural language description of what must be the case in order for a formula to have the extension 1.

A truth function is a function which takes truth values as input and gives a truth value as output. E.g. $f$ such that $f(0) = 1$ and $f(1) = 0$ is the truth function associated with the negation operator. Propositional logic is fully truth-functional, i.e. the semantic value of any complex expression is determined entirely by the truth values of the components and the operator that combines them. There are other logics with operators that are not truth-functional, where more goes into the truth value of an expression than just the truth values of its parts.

One can also generalize the notion "truth function" to mean the recursive computation of a truth value for any formula based on the truth functions of the operators together with a truth value assignment for the atoms, which then coincides with what you called "valuation" above. The semantics of propositional logic is given by this function definition.

The definition of a wff is a purely syntactical one and does not involve semantic notions such as interpretation or truth function. A wff is simply a string of symbols, formed according to the inductive rule schema "If $P$ is a wff then $\neg P$ is a wff" etc. The semantics then gets build on top of that, with the interpretation following the inductive scheme by which formulas are built up. A wff is not itself a truth value or a truth function; it has a truth value under a given interpretation, and how to compute this value from the form of the formula is defined by the truth function.