Propositional calculus

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A propositional calculus (or a sentential calculus) is a formal system that represents the materials and the principles of propositional logic (or sentential logic). Propositional logic is a domain of formal subject matter that is, up to somorphism, constituted by the structural relationships of mathematical objects called propositions.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas or wffs), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions.

When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of simplifying expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical axioms.

The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems.

The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and (2) a set of operator symbols, variously interpreted as logical operators or logical connectives. A well-formed formula (wff) is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols.

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distingushed formulas, and (3) the set of transformation rules that are available.

Abstraction and application

Although it is possible to construct an abstract formal calculus that has no immediate practical use and next to nothing in the way of obvious applications, the very name calculus indicates that this species of formal system owes its origin to the utility of its prototypical members in practical calculation. Generally speaking, any mathematical calculus is designed with the intention of representing a given domain of formal objects, and typically with the aim of facilitating the computations and inferences that need to be carried out in this representation. Thus some idea of the intended denotation, the formal objects that the formulas of the calculus are intended to denote, is given in advance of developing the calculus itself.

Viewed over the course of its historical development, a formal calculus for any given subject matter normally arises through a process of gradual abstraction, stepwise refinement, and trial-and-error synthesis from the array of informal notational systems that inform prior use, each of which covers the object domain only in part or from a particular angle.

Generic description of a propositional calculus

A propositional calculus is a formal system \(\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), whose formulas are constructed in the following manner:

  • The alpha set \(\Alpha\!\) is a finite set of elements called proposition symbols or propositional variables. Syntactically speaking, these are the most basic elements of the formal language \(\mathcal{L},\) otherwise referred to as atomic formulas or terminal elements. In the examples to follow, the elements of \(\Alpha\!\) are typically the letters p, q, r, and so on.
  • The omega set \(\Omega\!\) is a finite set of elements called operator symbols or logical connectives. The set \(\Omega\!\) is partitioned into disjoint subsets as follows:
\[\Omega = \Omega_0 \cup \Omega_1 \cup \ldots \cup \Omega_j \cup \ldots \cup \Omega_m.\!\]
In this partition, \(\Omega_j\!\) is the set of operator symbols of arity \(j.\!\)
In the more familiar propositional calculi, \(\Omega\!\) is typically partitioned as follows:
\[\Omega_1 = \{ \lnot \},\!\]
\[\Omega_2 \subseteq \{ \land, \lor, \rightarrow, \leftrightarrow \}.\!\]
A frequently adopted option treats the constant logical values as operators of arity zero, thus:
\[\Omega_0 = \{ 0, 1 \}.\!\]
Some writers use the tilde (~) instead of (¬) and some use the ampersand (&) instead of (∧). Notation varies even more for the set of logical values, with symbols like {false, true}, {F, T}, {0, 1}, and {\(\bot\), \(\top\)} all being seen in various contexts.
  • Depending on the precise formal grammar or the grammar formalism that is being used, syntactic auxiliaries like the left parenthesis, "(", and the right parentheses, ")", may be necessary to complete the construction of formulas.

The language of \(\mathcal{L}\), also known as its set of formulas, well-formed formulas or wffs, is inductively or recursively defined by the following rules:

  1. Base. Any element of the alpha set \(\Alpha\!\) is a formula of \(\mathcal{L}\).
  2. Step (a). If p is a formula, then ¬p is a formula.
  3. Step (b). If p and q are formulas, then (p ∧ q), (p ∨ q), (p → q), and (p ↔ q) are formulas.
  4. Close. Nothing else is a formula of \(\mathcal{L}\).

Repeated applications of these rules permits the construction of complex formulas. For example:

  1. By rule 1, p is a formula.
  2. By rule 2, ¬p is a formula.
  3. By rule 1, q is a formula.
  4. By rule 3, (¬p ∨ q) is a formula.
  • The zeta set \(\Zeta\!\) is a finite set of transformation rules that are called inference rules when they acquire logical applications.
  • The iota set \(\Iota\!\) is a finite set of initial points that are called axioms when they receive logical interpretations.

Example 1. Simple axiom system

Let \(\mathcal{L}_1 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), where \(\Alpha,\ \Omega,\ \Zeta,\ \Iota\) are defined as follows:

  • The alpha set \(\Alpha \!\) is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:
\[\Alpha = \{p, q, r, s, t, u \} \,.\]

Of the three connectives for conjunction, disjunction, and implication (∧, ∨, and →), one can be taken as primitive and the other two can be defined in terms of it and negation (¬). Indeed, all of the logical connectives can be defined in terms of a sole sufficient operator. The biconditional (↔) can of course be defined in terms of conjunction and implication, with a ↔ b defined as (a → b) ∧ (b → a).

Adopting negation and implication as the two primitive operations of a propositional calculus is tantamount to having the omega set \(\Omega = \Omega_1 \cup \Omega_2\) partition as follows:

\[\Omega_1 = \{ \lnot \} \,,\]
\[\Omega_2 = \{ \Rightarrow \} \,.\]

An axiom system discovered by Jan Lukasiewicz formulates a propositional calculus in this language as follows:

  • \(p \Rightarrow (q \Rightarrow p)\)
  • \((p \Rightarrow (q \Rightarrow r)) \Rightarrow ((p \Rightarrow q) \Rightarrow (p \Rightarrow r))\)
  • \((\neg p \Rightarrow \neg q) \Rightarrow (q \Rightarrow p)\)

The inference rule is modus ponens:

  • From p, (p ⇒ q), infer q.

Then we have the following definitions:

  • p ∨ q is defined as ¬p ⇒ q.
  • p ∧ q is defined as ¬(p ⇒ ¬q).

Example 2. Natural deduction system

Let \(\mathcal{L}_2 = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)\), where \(\Alpha,\ \Omega,\ \Zeta,\ \Iota\) are defined as follows:

  • The alpha set \(\Alpha \!\) is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:
\[\Alpha = \{p, q, r, s, t, u \} \,.\]
  • The omega set \(\Omega = \Omega_1 \cup \Omega_2\) partitions as follows:
\[\Omega_1 = \{ \lnot \} \,,\]
\[\Omega_2 = \{ \land, \lor, \rightarrow, \leftrightarrow \} \,.\]

In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called natural deduction system. The particular system presented here has no initial points, which means that its interpretation for logical applications derives its theorems from an empty axiom set.

  • The set of initial points is empty, that is, \(\Iota = \varnothing \,.\)
  • The set of transformation rules, \(\Zeta ,\!\), is described as follows:

Graphical calculi

Main article : Logical graph

It is possible to generalize the definition of a formal language from a set of finite sequences over a finite basis to include many other sets of mathematical structures, so long as they are built up by finitary means from finite materials. What's more, many of these families of formal structures are especially well-suited for use in logic.

For example, there are many families of graphs that are close enough analogues of formal languages that the concept of a calculus is quite easily and naturally extended to them. Indeed, many species of graphs arise as parse graphs in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are wffs or not. Once this is done, there are many advantages to be gained from developing the graphical analogue of the calculus on strings. The mapping from strings to parse graphs is called parsing and the inverse mapping from parse graphs to strings is achieved by an operation that is called traversing the graph.

References

  • Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
  • Chang, C.C., and Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam, Netherlands.
  • Kohavi, Zvi (1978), Switching and Finite Automata Theory, 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
  • Korfhage, Robert R. (1974), Discrete Computational Structures, Academic Press, New York, NY.
  • Lambek, J., and Scott, P.J. (1986), Introduction to Higher Order Categorical Logic, Cambridge University Press, Cambridge, UK.
  • Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company.

Resources

  • Klement, Kevin C. (2006), “Propositional Logic”, in James Fieser and Bradley Dowden (eds.), Internet Encyclopedia of Philosophy. Online.

Syllabus

Focal nodes

Peer nodes

Logical operators

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