10  Overview

The semantics of a language consists of rules governing the meaning of formulas in it. In SL and FOL, the meaning of a sentence is its truth value. In classical languages such as SL and FOL, there are just two truth values (i.e., just two possible meanings):

10.1 Logical and non-logical symbols

All symbols in SL and FOL are either logical or non-logical symbols.

  • A logical symbol is a symbol whose meanings are fixed
  • A non-logical symbol is a symbol whose meanings are not fixed

10.2 Interpretation and truth-functionality

The meaning of a sentence in SL or FOL is governed by two things:

  • an interpretation, or assignment of a meaning (=truth value) to all non-logical symbols, and
  • the truth-functional semantics of all logical symbols in it.

Our semantics for each of our languages will depend on our definition of two functions:

  • the interpretation function \(i\), which assigns a meaning to all relevant non-logical symbols, and
  • the truth function \(v\), which determines the meaning of all formulas.

10.3 The interpretation function \(i\)

The interpretation function assigns a meaning to all non-logical symbols in a sentence or set of sentences. We can describe it in two ways:

  • as a function type, that is, by specifying a domain (the inputs it takes) and codomain (the outputs it returns), or
  • as a function application, that is, by showing how it maps a particular input to a particular output.

Interpretation, moreover, works slightly differently in SL and in FOL. See:

for particulars.

10.4 The truth function \(v\)

The truth function returns, for every formula \(\phi\) on an interpretation \(i\), a truth value:

\[v_i: \phi \mapsto \{1,0\}\]

In application:

\[ v_i(\phi) = 1 \text{ or } v_i(\phi) = 0\]

Whether \(v_i\) returns a 1 or 0 depends both on the relevant interpretation and on the particular structure of \(\phi\). In addition to the sections on interpretation, see:

10.5 Meaning = interpretation + truth function

The basic gist of our semantics is that the meaning of formulas in a language is premised on

  • the interpretation of its non-logical parts, and
  • the truth-functional semantics of its logical parts.

Take a sentence, apply an interpretation, feed it to the truth function, get a truth value. The process as a whole is called truth evaluation. For structural and technical insight into the whole, see

for particulars.