Beyond Propositional Logic

A logical system is defined using three key parameters:

1. Syntax: A formal language
2. Semantics: A way to interpret formulas in the formal language (i.e., a definition of truth for formulas)
3. Consequence: A definition of valid inference rules

The parameters of propositional logic are:

1. Syntax: Recall the rules to construct formulas of propositional logic. Any atomic proposition is a formula, and if $Z$ and $Y$ are formulas, then so are $\neg X$, $X\wedge Y$, $X\vee Y$, and $X\rightarrow Y$.
2. Semantics: Truth of formulas is defined using truth value functions (see the section on truth of formulas of propositional logic). There are two key aspects of the definition of truth in propositional logic:
   • truth-functionality: the truth-value of a formula only depends on the truth values of its components
   • bivalence: formulas are either true or false, with nothing in-between
3. Consequence: Recall the definition of validity from the section on validity for propositional logic: $X_1,\ldots, X_n\models Y$ when every truth value function that assigns the truth value $\mathsf{T}$ to each of the formulas $X_1,\ldots, X_n$ also assigns the truth value $\mathsf{T}$ to $Y$.

Different logical systems are defined by changing one or more of the above parameters. For example, many-valued propositional logics drop the assumption of bivalence and allows formulas to be assigned more than two truth values. Another example is modal logic that extends the language of propositional logic with a new symbol that is intended to represent modals. A modal is a phrase that modifies the truth of a statement, such as in the sentence: "It must be raining". The semantics for modal logic is not truth-functional.

The next two sections are a brief introduction to a logical system that can represent arguments containing quantifiers.