Propositional calculus

The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. (Compare this to the predicate calculus which is quantificational and whose atomic formulas are propositional functions, and modal logic which may be non-truth-functional.)

A calculus is a logical system which is used to prove valid formulas (i.e. its theorems) and arguments. It is a set of axioms (which may be an empty or countably infinite set) or axiom schemata, and inference rules for deriving valid inferences. A formal grammar (or syntax) 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 (i.e. theorems).

In the propositional calculus the language consists of propositional variables (or placeholders) and sentential operators (or connectives). A wff is any atomic formula or a formula built up from sentential operators.

In what follows we will outline a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language (i.e. which operators and variables are part of the language); (2) which (if any) axioms they have; (3) which inference rules are employed.

Grammar

The language consists of:

The capital letters of the alphabet standing as propositional variables. These are atomic formulas.
Symbols denoting the following connectives (or logical operators): Ź, &and, &or, &rarr, &harr. (We may do with fewer operators (and thus symbols) by having some abbreviate others — e.g. P → Q is equivalent to ¬ P ∨ Q.)
The left and right parentheses: (, ).

The set of wffs is recursively defined by the following rules:

Basis: Letters of the alphabet (usually capitalized such as A, B, etc.) are wffs.

Inductive clause I: If φ is a wff, then Ź φ is a wff.

Inductive clause II If φ and ψ are wffs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are wffs.

Closure clause: Nothing else is a wff.

Repeated applications of these three rules permit the generation of complex wffs. For example:

By rule 1, A is a wff.
By rule 2, Ź A is a wff.
By rule 1, B is a wff.
By rule 3, ( Ź A ∨ B ) is a wff.
Calculus

For simplicity, we will use a natural deduction system, which has no axioms; or, equivalently, which has an empty axiom set.

Derivations using our calculus will be laid out in the form of a list of numbered lines, with a single wff and a justification on each line. Any premises will be at the top, with a "p" for their justification. The conclusion will be on the last line. A derivation will be considered complete if every line follows from previous ones by correct application of a rule.
(For a contrasting approach, see proof-trees).

Axioms

Our axiom set is the empty set.

Inference rules

Our propositional calculus has ten inference rules. These rules allow us to derive other true formulas given a set of formulas that are assumed to be true. The first eight simply state that we can infer certain wffs from other wffs. The last two rules however use hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a certain other formula. Since the first eight rules don't do this they are usually described as non-hypothetical rules, and the last two as hypothetical rules.

Double negative elimination -
From the wff Ź Ź φ, we may infer φ

Conjunction introduction -
From any wff φ and any wff ψ, we may infer ( φ ∧ ψ ).

Conjunction elimination -
From any wff ( φ ∧ ψ ), we may infer φ and ψ

Disjunction introduction -
From any wff φ, we may infer (φ ∨ ψ) and (ψ ∨ φ), where ψ is any wff.

Disjunction elimination -
From wffs of the form ( φ ∨ ψ ), ( φ → χ ), and ( ψ → χ ), we may infer χ.

Biconditional introduction -
From wffs of the form ( φ → ψ ) and ( ψ → φ ), we may infer ( φ ↔ ψ ).

Biconditional elimination -
From the wff ( φ ↔ ψ ), we may infer ( φ → ψ ) and ( ψ → φ ).

Modus ponens -
From wffs of the form φ and ( φ → ψ ), we may infer ψ.

Conditional proof -
If ψ can be derived while assuming the hypothesis φ, we may infer ( φ → ψ ).

Reductio ad absurdum -
If we can derive both ψ and Ź ψ while assuming the hypothesis φ, we may infer Ź φ.

Example of a proof

The following is an example of a (syntactical) demonstration:
Prove:
Proof:

Interpret as "Assuming A, infer A". Read as "Assuming nothing, infer that A implies A," or "It is a tautology that A implies A," or "It is always true that A implies A."

Soundness and completeness of the rules

The crucial properties of this set of rules is that they are sound and complete. Informally this means that the rules are correct and that no other rules are required. These claims can be made more formal as follows.

We define a truth assignment as a function that maps propositional variables to true or false. Informally such a truth assignment can be understood as the description of a possible state of affairs (or possible worlds) where certain statements are true and others are not. The semantics of formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.

We define when such a truth assignment A satisfies a certain wff with the following rules:

A satisfies the propositional variable P iff A(P) = true

A satisfies Ź φ iff A does not satisfy φ

A satisfies (φ ∧ ψ) iff A satisfies both φ and ψ

A satisfies (φ ∨ ψ) iff A satisfies at least one of either φ or ψ

A satisfies (φ → ψ) iff it is not the case that A satisfies φ but not ψ

A satisfies (φ ↔ ψ) iff A satisfies both φ and ψ or satisfies neither one of them

With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulas. Informally this is true if in all worlds that are possible given the set of formulas S the formula φ also holds. This leads to the following formal definition: We say that a set S of wffs semantically entails (or implies) a certain wff φ if all truth assignments that satisfy all the formulas in S also satisfy φ.

Finally we define syntactical entailment such that φ is syntactically entailed by S iff we can derive it with the inference rules that were presented above in a finite number of steps. This allows us formulate exactly what it means for the set of inference rules to be sound and complete:

Soundness -
If the set of wffs S syntactically entails wff φ then S semantically entails φ

Completeness -
If the set of wffs S semantically entails wff φ then S syntactically entails φ
For the above set of rules this is indeed the case.

Sketch of a soundness proof

(For most logical systems, this is the comparatively "simple" direction of proof)

Notational conventions: Let "G" be a variable ranging over sets of sentences. Let "A", "B", and "C" range over sentences. For "G syntactically entails A" we write "G proves A". For "G semantically entails A" we write "G implies A".

We want to show: (A)(G)(If G proves A then G implies A)

We note that "G proves A" has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If G proves A then . . ." So our proof proceeds by induction.

I. Basis. Show: If A is a member of G then G implies A
[II. Basis. Show: If A is an axiom, then G implies A]
III. Inductive step: (a) Assume for arbitrary G and A that if G proves A then G implies A. (If necessary, assume this for arbitrary B, C, etc. as well)
::(b) For each possible application of a rule of inference to A, leading to a new sentence B, show that G implies B.

(N.B. Basis Step II can be omitted for the above calculus, which is a natural deduction system and so has no axioms. Basically, it involves showing that each of the axioms is a (semantic) logical truth.)

The Basis step(s) demonstrate(s) that the simplest provable sentences from G are also implied by G, for any G. (The is simple, since the semantic fact that a set implies any of its members, is also trivial.) The Inductive step will systematically cover all the further sentences that might be provable--by considering each case where we might reach a logical conclusion using an inference rule--and shows that if a new sentence is provable, it is also logically implied. (For example, we might have a rule telling us that from "A" we can derive "A or B". In III.(a) We assume that if A is provable it is implied. We also know that if A is provable then "A or B" is provable. We have to show that then "A or B" too is implied. We do so by appeal to the semantic definition and the assumption we just made. A is provable from G, we assume. So it is also implied by G. So any semantic valuation making all of G true makes A true. But any valuation making A true makes "A or B" true, by the defined semantics for "or". So any valuation which makes all of G true makes "A or B" true. So "A or B" is implied.) Generally, the Inductive step will consist of a lengthy but simple case-by-case analysis of all the rules of inference, showing that each "preserves" semantic implication.

By the definition of provability, there are no sentences provable other than by being a member of G, an axiom, or following by a rule; so if all of those are semantically implied, the deduction calculus is sound.

Sketch of completeness proof

(This is usually the much harder direction of proof.)

We adopt the same notational conventions as above.

We want to show: If G implies A, then G proves A. We proceed by contraposition: We show instead that If G does not prove A then G does not imply A.

I. G does not prove A. (Assumption)
II. If G does not prove A, then we can construct an (infinite) "Maximal Set", G*, which is a superset of G and which also does not prove A.
*(a)Place an "ordering" on all the sentences in the language. (e.g., alphabetical ordering), and number them E1, E2, . . .
*(b)Define a series Gn of sets (G0, G1 . . . )inductively, as follows. (i)G0=G. (ii) If {Gk, E(k+1)} proves A, then G(k+1)=Gk. (iii) If {Gk, E(k+1)} does not prove A, then G(k+1)={Gk, E(k+1)}
*(c)Define G* as the union of all the Gn. (That is, G* is the set of all the sentences that are in any Gn).
*(d) It can be easily shown that (i) G* contains (is a superset of) G (by (b.i)); (ii) G* does not prove A (because if it proves A then some sentence was added to some Gn which caused it to prove A; but this was ruled out by definition); and (iii) G* is a "Maximal Set" (with respect to A): If any more sentences whatever were added to G*, it would prove A. (Because if it were possible to add any more sentences, they should have been added when they were encountered during the construction of the Gn, again by definition)
III. If G* is a Maximal Set (wrt A), then it is "truth-like". This means that it contains the sentence "A" only if it does not contain the sentence not-A; If it contains "A" and contains "If A then B" then it also contains "B"; and so forth.
IV. If G* is truth-like there is a "G*-Canonical" valuation of the language: one that makes every sentence in G* true and everything outside G* false while still obeying the laws of semantic composition in the language.
V. A G*-canonical valuation will make our original set G all true, and make A false.
VI. If there is a valuation on which G are true and A is false, then G does not (semantically) imply A.

Q.E.D

Alternative calculus

It is possible to define another version of propositional calculus, which defines most of the syntax of the logical operators by means of axioms, and which uses only one inference rule.

Axioms

Let φ, χ and ψ stand for well-formed formulas. (The wffs themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are

THEN-1: φ → (χ → φ)

THEN-2: (φ → (χ → ψ)) → ((φ → χ) → (φ → ψ))

AND-1: φ ∧ χ → φ

AND-2: φ ∧ χ → χ

AND-3: φ → (χ → (φ ∧ χ))

OR-1: φ → φ ∨ χ

OR-2: χ → φ ∨ χ

OR-3: (φ → ψ) → ((χ → ψ) → (φ ∨ χ → ψ))

NOT-1: (φ → χ) → ((φ → Ź χ) → Ź φ)

NOT-2: φ → (Ź φ → χ)

NOT-3: φ ∨ Ź φ

Axiom THEN-2 may be considered to be a "distributive property of implication with respect to implication." Axioms AND-1 and AND-2 correspond to "conjunction elimination". The relation between AND-1 and AND-2 reflects the commutativity of the conjunction operator. Axiom AND-3 corresponds to "conjunction introduction." Axioms OR-1 and OR-2 correspond to "disjunction introduction." The relation between OR-1 and OR-2 reflects the commutativity of the disjunction operator. Axiom NOT-1 corresponds to "reductio ad absurdum." Axiom NOT-2 says that "anything can be deduced from a contradiction." Axiom NOT-3 is called "tertium non datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulas: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. Intuitionistic logicians do not accept the axiom NOT-3.

Inference rule

The inference rule is modus ponens:

.
If the double-arrow equivalence operator is also used, then the following "natural" inference rules may be added:

IFF-1:

IFF-2:

Meta-inference rule

Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstyle and the conclusion to the right of the turnstyle. Then the deduction theorem can be stated as follows:
: If the sequence
::
: has been demonstrated, then it is also possible to demonstrate the sequence
::.

This deduction theorem (DT) is not itself formulated with propositional calculus: it is not a theorem of propositional calculus, but a theorem about propositional calculus. In this sense, it is a meta-theorem, comparable to theorems about the soundness or completeness of propositional calculus.

On the other hand, DT is so useful for simplifying the syntactical proof process that it can be considered and used as another inference rule, accompanying modus ponens. In this sense, DT corresponds to the natural conditional proof inference rule which is part of the first version of propositional calculus introduced in this article.

The converse of DT is also valid:
: If the sequence
::
: has been demonstrated, then it is also possible to demonstrate the sequence
::
in fact, the validity of the converse of DT is almost trivial compared to that of DT:
: If
::
: then
:: 1:
:: 2:
: and from (1) and (2) can be deduced
:: 3:
: by means of modus ponens, Q.E.D.

The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, the axiom AND-1,
:
can be transformed by means of the converse of the deduction theorem into the inference rule
:
which is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus.

Example of a proof

The following is an example of a (syntactical) demonstration, involving only axioms THEN-1 and THEN-2:
Prove: A → A (Reflexivity of implication).
Proof:
:1. (A → ((A → A) → A) → ((A → (A → A)) → (A → A))
::Axiom THEN-2 with φ = A, χ = A → A, ψ = A
:2. A → ((A → A) → A)
::Axiom THEN-1 with φ = A, χ = A → A
:3. (A → (A → A)) → (A → A)
::From (1) and (2) by modus ponens.
:4. A → (A → A)
::Axiom THEN-1 with φ = A, χ = A
:5. A → A
::From (3) and (4) by modus ponens.

Other logical calculi

Propositional calculus is about the simplest kind of logical calculus in any current use. (Aristotelian "syllogisitic" calculus, which is largely supplanted in modern logic, is in some ways simpler--but in other ways more complex--than propositional calculus.) It can be extended in several ways.

The most immediate way to develop a more complex logical calculus is to introduce rules that are sensitive to more fine-grained details of the sentences being used. When the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, they yield first-order logic, or first-order predicate logic, which keeps all the rules of propositional logic and adds some new ones. (For example, from "All dogs are mammals" we may infer "If Rover is a dog then Rover is a mammal.)

With the tools of first-order logic it is possible to formulate a number of theories, either with explicit axioms or by rules of inference, that can themselves be treated as logical calculi. Arithmetic is the best known of these; others include set theory and mereology.

Modal logic also offers a variety of inferences that cannot be captured in propositional calculus. For example, from "Necessarily p" we may infer that p. From p we may infer "It is possible that p".

Many-valued logics are those allowing sentences to have values other than true and false. (For example, neither and both are standard "extra values"; "continuum logic" allows each sentence to have any of an infinite number of "degrees of truth" between true and false.) These logics often require calculational devices quite distinct from propositional calculus.