Propositional calculus

Propositional calculus is a calculus that represents the logical structure of truth-functional connectives (not, and, or, if…, then, etc), which combine some propositions to produce new propositions. It is often referred to as propositional logic.

For instance, consider the following argument:

If Jack is innocent, then Jack has an alibi and Jack is not a murderer.

Jack does not have an alibi.

Therefore, Jack is not innocent.

The truth-values, truth or falsity, of the sentences in this argument are exclusively dependent on whether each of the simple propositions in the sentences (i.e. "Jack is innocent", "Jack has an alibi", and "Jack is a murderer") is true or false. In other words, once the truth-values of the simple sentences are determined, the complex sentences in the arguments are determined only in terms of the meanings of the connectives, “if…then”, “not,” and “and”, which are examples of truth-functional connectives. Propositional logic, focusing on connectives of such kinds, clarifies what form a given argument $A$ (such as the one in question here) has, and studies how the correctness or incorrectness of $A$ depends on the truth-functional connectives that it contains.

The language of propositional calculus consists of propositional variables, truth-functional connectives, (most often, $\lnot ,\wedge ,\vee ,\rightarrow ,\leftrightarrow$) and parentheses. Formulas are built up from propositional variables by using truth-functional connectives and parentheses.

To propositional variables, the truth-values are assigned and, relative to the truth-value assignment, the truth-value of an arbitrary well-formed formula (for the definition, see the section Syntax) that contains the propositional variables is calculated based on truth-functional connectives in the well-formed formula.

A propositional calculus has a set of axioms (possibly empty) and rules of inference. There are various kinds of propositional calculus (we will see two kinds of systems below), for which the soundness and completeness can be proved. (for the definitions of soundness and completeness, see the correponding section below)

What Propositional Calculus Studies?

Some sentences have truth-values, truth or falsity, and some do not. The sentences of the latter kind are excluded from what propositional calculus studies. Thus, in propositional calculus, it is assumed that every sentence is either true or false. (This assumption is called the principle of bivalence.)

Among such sentences, the sentences that do not include sentential connectives such as and, or, etc. (e.g. “John is a bachelor.”) are called atomic sentences. More complex sentences (e.g. “John is a bachelor and Ben is married”) are built from atomic sentences and sentential connectives.

Some sentential connectives determine the truth-values of the complex sentences in which they occur, once the truth-values of atomic sentences that the complex sentences contain are determined. For instance, the truth-value of “John is a bachelor and Ben is married” is determined purely by the meaning of the connective “and” once the truth-values of the two atomic sentences “John is a bachelor” and “Ben is married” are determined. The connectives of such a kind is called truth-functional. (Notice that this does not apply to all the sentential connectives. Consider “Ben is happy because Ben is married.” The truth-value of this sentence is still undetermined even if both the atomic sentences in this sentence are true.) Truth-functional connectives are the connectives that propositional calculus studies. Examples of such connectives are "and", "or", "if…then..." (These connectives of a certain use only. Some uses of the connectives are not truth-functional. For instance, consider counterfactual statements), etc.

Syntax

The language of propositional calculus consists of 1. propositional variables, usually annotated by p, q, r,…, 2. truth-functional connectives, $\lnot ,\wedge ,\vee ,\rightarrow ,\leftrightarrow$, and 3. parentheses “(“ and “)”. Propositional variables represent atomic sentences and $\lnot ,\wedge ,\vee ,\rightarrow$ , and $\leftrightarrow$ are usually considered as “not”, “and”, “or”, “if…then...”, and “...if and only if...” respectively. $\lnot$ is called unary (meaning that it is attachable to one wff. For the definitnion of wffs, see below.) and the other four connectives are called binary (meaning that they combine two wffs). Parentheses are used to represent the punctuations in sentences.

Well-formed formulas (wffs) are recursively built in the following way.

• Propositional variables are wffs.
• If $\alpha$ is a wff, then $\lnot \alpha$ is a wff.
• If $\alpha$ and $\beta$ are wffs, then $(\alpha \star \beta )$ is a wff where $\star$ is a binary connective.

Conventionally the outermost set of parentheses is dropped. Also, the order of strength in which propositional connectives bind is stipulated as: $\leftrightarrow ,\rightarrow ,\ wedge$ and $\vee ,\lnot$. Therefore, taking these two conventions into account, say, the wff “$((\lnot p\vee q)\rightarrow r)$” built up by the above definition is written as “$\lnot p\vee q\rightarrow r$”.

The connective in a given wff $\phi$ that binds last is called the main connective of $\phi$. Thus, in the case of $\lnot p\vee q\rightarrow r$, the main connective is $\rightarrow$. Wffs with $\lnot ,\wedge ,\vee ,\rightarrow ,$ and $\leftrightarrow$ as their main connectives are called negation, conjunction, disjunction, conditional, and biconditional respectively.

An argument is a set of wffs and a distinguished wff. The wffs of the former kind are called premises and the distinguished wff is called the conclusion. The set of premises of a given argument can possibly be empty.

For instance, the set of sentences about Jack in the opening example is represented in the language of propositional logic as follows:

$p\rightarrow (q\wedge \lnot r)$

$\lnot q$

$\lnot p$

where p, q, and r represent “Jack is innocent,” “Jack has an alibi,” and “Jack is a murderer,” respectively. The first two wffs are the premises and the last wff is the conclusion of the argument.

Semantics

Every wff in propositional calculus gets either of the two truth-values, True and False (T and F). Relative to the assignment V of truth-values to propositional variables (a function from the set of propositional variables to {T, F}, the truth-values of other wffs are determined recursively as follows:

• p is true iff V(p)=T
• $\lnot \alpha$ is T iff $\alpha$ is F.
• $\alpha \wedge \beta$ is T iff $\alpha$ is T and $\beta$ is T.
• $\alpha \vee \beta$ is T iff $\alpha$is T or $\beta$ is T (in the inclusive sense of "or" i.e. including the case in which both are T)
• $\alpha \rightarrow \beta$ is T iff $\alpha$ is F or $\beta$ is T.
• $\alpha \leftrightarrow \beta$ is T iff $\alpha$ and $\beta$ coincide in their truth-values.

For instance, when p, q, and r get T, T and F respectively, $(p\vee \lnot q)\leftrightarrow (r\wedge q)$ gets F. For the left side of the biconditional is T because p is T and $\lnot q$ is F, and the right side is F because r is F and q is T.

A wff that gets T no matter what truth-value assignment is given is called a tautology. A set $\Gamma$ of wffs (maybe empty) implies a wff $\phi$ if and only if $\phi$ is T relative to every truth-value assignment V that assigns Ts to all the wffs in $\Gamma$. An argument, consisting of a set $\Gamma$ of wffs and a wff $\phi$, is said to be valid if $\Gamma$ implies $\phi$. (For instance, readers are invited to check that the argument about Jack is valid.)

If an argument, consisting of a premise set $\Gamma$ and a conclusion $\phi$, is valid, , we write “$\Gamma \models \phi$”, which often reads as “$Gamma$ implies $\phi$.” (For the left hand side of “$\models$,” the wffs in $\Gamma$ are written with commas between them, e.g. if $\Gamma$ is {p, q, r}, we write “p, q, r $\models \phi$.”)

Propositional Calculi

Proofs in a propositional calculus

A propositional calculus consists of a set of specified wff called axioms (the set can possibly be empty) and rules of inference. A proof of an argument is a sequence of wffs in which (1) each wff is a premise, an axiom, or a wff that is derived from previous wffs in the sequence by a rule of inference and (2) the last wff of the sequence is the conclusion of the argument. If an argument, consisting of a premise set $\Gamma$ and a conclusion $\phi$, has a proof, we write “$\Gamma \vdash \phi$”, which reads as “$\Gamma$ proves $\phi$.” (The convention for the left hand side of “$\vdash$” is the same as the one for “$\models$”.)

Particularly, if there is a proof for an argument with an empty set of premises, i.e. if the conclusion of the argument can be derived only from axioms based on the rules of inference, then the conclusion is called a theorem. Thus, if $\phi$ is a theorem, we can write “$\vdash \phi$,” which reads as “$\phi$ is a theorem.”

There are various propositional calculi. We present two of the most famous ones below.

Hilbert-Style Propositional Calculus

One famous deductive system takes the language of propositional calculus that consists of propositional variables, the connectives, $\rightarrow$ and $\lnot$, and parentheses. The other connectives are defined as follows:

• $\alpha \wedge \beta :=\lnot (\alpha \rightarrow \lnot \beta )$
• $\alpha \vee \beta :=\lnot \alpha \rightarrow \beta$
• $\alpha \leftrightarrow \beta :=\lnot ((\alpha \rightarrow \beta )\rightarrow \lnot (\beta \rightarrow \alpha ))$

The axioms have one of the following forms:

• A1 $\alpha \rightarrow (\beta \rightarrow \alpha )$
• A2 $(\alpha \rightarrow (\beta \rightarrow \gamma )\rightarrow ((\alpha \rightarrow \beta )\rightarrow (\alpha \rightarrow \gamma ))$
• A3 $(\lnot \alpha \rightarrow \lnot \beta )\rightarrow ((\lnot \alpha \rightarrow \beta )\rightarrow \alpha )$

The only rule of inference is [modus ponens], i.e. from $\alpha$ and $\alpha \rightarrow \beta$, derive $\beta$.

Here is an example of a proof in this system for $p,(r\rightarrow p)\rightarrow (r\rightarrow (p\rightarrow (p\rightarrow s))\vdash r\rightarrow s$:

Number	wff	Justification
1	A	premiss
2	A ∨ A	From (1) by disjunction introduction
3	(A ∨ A) ∧ A	From (1) and (2) by conjunction introduction
4	A	From (3) by conjunction elimination
5	A ├ A	Summary of (1) through (4)
6	├ A → A	From (5) by conditional proof

Natural Deduction

Another example takes the language of propositional calculus that consists of propositional variables, the connectives, \lnot, \wedge, \vee, \rightarrow, \leftrightarrow, and parentheses. The set of axioms is empty. However, it has the following rules of inference:

Here is an example of a proof in this system again for P, (R \rightarrow P) \rightarrow (R \rightarrow (P \rightarrow (P \rightarrow S)) \vdash R\rightarrow S.

Soundness and Completeness

A calculus is sound if, for all \Gamma and \phi, \Gamma \vdash \phi implies \Gamma \models \phi. A calculus is complete if, for all \Gamma and \phi, \Gamma \models \phi implies \Gamma \vdash \phi. There are various sound and complete propositional calculi, i.e. the calculi in which the notion of proof and that of validity correspond. The two calculi above are the examples of sound and complete propositional calculi.

Example 2. Natural deduction system

Let ${\mathcal {L}}_{2}={\mathcal {L}}\ (\mathrm {A} ,\ \Omega ,\ \mathrm {Z} ,\ \mathrm {I} )$, where $\mathrm {A} ,\ \Omega ,\ \mathrm {Z} ,\ \mathrm {I}$ are defined as follows:

• The alpha set $\mathrm {A} \!$ is a finite set of symbols that is large enough to supply the needs of a given discussion, for example:

$\mathrm {A} =\{p,q,r,s,t,u\}\,.$

• 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, $\mathrm {I} =\varnothing \,.$

• The set of transformation rules, $\mathrm {Z} ,\!$, is described as follows:

Our propositional calculus has ten inference rules. These rules allow us to derive other true formulae given a set of formulae that are assumed to be true. The first nine simply state that we can infer certain wffs from other wffs. The last rule however uses 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 formulae to see if we can infer a certain other formula. Since the first nine rules don't do this they are usually described as non-hypothetical rules, and the last one as a hypothetical rule.

• Reductio ad absurdum (negation introduction)

From (p→q), (p→ ¬q), infer ¬p.

• Double negative elimination

From ¬¬p, infer p.

• Conjunction introduction

From p and q, infer (p ∧ q).

• Conjunction elimination

From (p ∧ q), infer p;

From (p ∧ q), infer q.

• Disjunction introduction

From p, infer (p ∨ q);

From p, infer (q ∨ p).

• Disjunction elimination

From (p ∨ q), (p → r), (q → r), infer r.

• Biconditional introduction

From (p → q), (q → p), infer (p ↔ q).

• Biconditional elimination

From (p ↔ q), infer (p → q);

From (p ↔ q), infer (q → p).

• Modus ponens (conditional elimination)

From p, (p → q), infer q.

• Conditional proof (conditional introduction)

If accepting p allows a proof of q, infer (p → q).

Basic and Derived Argument Forms
Name	Sequent	Description
Modus Ponens	((p → q) ∧ p) ├ q	if p then q; p; therefore q
Modus Tollens	((p → q) ∧ ¬q) ├ ¬p	if p then q; not q; therefore not p
Hypothetical Syllogism	((p → q) ∧ (q → r)) ├ (p → r)	if p then q; if q then r; therefore, if p then r
Disjunctive Syllogism	((p ∨ q) ∧ ¬p) ├ q	Either p or q; not p; therefore, q
Constructive Dilemma	((p → q) ∧ (r → s) ∧ (p ∨ r)) ├ (q ∨ s)	If p then q; and if r then s; but either p or r; therefore either q or s
Destructive Dilemma	((p → q) ∧ (r → s) ∧ (¬q ∨ ¬s)) ├ (¬p ∨ ¬r)	If p then q; and if r then s; but either not q or not s; therefore either not p or not r
Simplification	(p ∧ q) ├ p	p and q are true; therefore p is true
Conjunction	p, q ├ (p ∧ q)	p and q are true separately; therefore they are true conjointly
Addition	p ├ (p ∨ q)	p is true; therefore the disjunction (p or q) is true
Composition	((p → q) ∧ (p → r)) ├ (p → (q ∧ r))	If p then q; and if p then r; therefore if p is true then q and r are true
De Morgan's Theorem (1)	¬(p ∧ q) ├ (¬p ∨ ¬q)	The negation of (p and q) is equiv. to (not p or not q)
De Morgan's Theorem (2)	¬(p ∨ q) ├ (¬p ∧ ¬q)	The negation of (p or q) is equiv. to (not p and not q)
Commutation (1)	(p ∨ q) ├ (q ∨ p)	(p or q) is equiv. to (q or p)
Commutation (2)	(p ∧ q) ├ (q ∧ p)	(p and q) is equiv. to (q and p)
Association (1)	(p ∨ (q ∨ r)) ├ ((p ∨ q) ∨ r)	p or (q or r) is equiv. to (p or q) or r
Association (2)	(p ∧ (q ∧ r)) ├ ((p ∧ q) ∧ r)	p and (q and r) is equiv. to (p and q) and r
Distribution (1)	(p ∧ (q ∨ r)) ├ ((p ∧ q) ∨ (p ∧ r))	p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2)	(p ∨ (q ∧ r)) ├ ((p ∨ q) ∧ (p ∨ r))	p or (q and r) is equiv. to (p or q) and (p or r)
Double Negation	p ├ ¬¬p	p is equivalent to the negation of not p
Transposition	(p → q) ├ (¬q → ¬p)	If p then q is equiv. to if not q then not p
Material Implication	(p → q) ├ (¬p ∨ q)	If p then q is equiv. to either not p or q
Material Equivalence (1)	(p ↔ q) ├ ((p → q) ∧ (q → p))	(p is equiv. to q) means, (if p is true then q is true) and (if q is true then p is true)
Material Equivalence (2)	(p ↔ q) ├ ((p ∧ q) ∨ (¬q ∧ ¬p))	(p is equiv. to q) means, either (p and q are true) or ( both p and q are false)
Exportation	((p ∧ q) → r) ├ (p → (q → r))	from (if p and q are true then r is true) we can prove (if q is true then r is true, if p is true)
Importation	(p → (q → r)) ├ ((p ∧ q) → r)
Tautology	p ├ (p ∨ p)	p is true is equiv. to p is true or p is true
Tertium non datur (Law of Excluded Middle)	├ (p ∨ ¬ p)	p or not p is true

Proofs in propositional calculus

One of the main uses of a propositional calculus, when interpreted for logical applications, is to determine relations of logical equivalence between propositional formulas. These relationships are determined by means of the available transformation rules, sequences of which are called derivations or proofs.

In the discussion to follow, a proof is presented as a sequences of numbered lines, with each line consisting of a single formula followed by a reason or justification for introducing that formula. Each premiss of the argument, that is, an assumption introduced as a hypothesis of the argument, is listed at the beginning of the sequence and is marked as a "premiss" in lieu of other justification. The conclusion is listed on the last line. A proof is complete if every line follows from previous ones by the correct application of a transformation rule. (For a contrasting approach, see proof-trees).

Example of a proof

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

Number	wff	Justification
1	A	premiss
2	A ∨ A	From (1) by disjunction introduction
3	(A ∨ A) ∧ A	From (1) and (2) by conjunction introduction
4	A	From (3) by conjunction elimination
5	A ├ A	Summary of (1) through (4)
6	├ A → A	From (5) by conditional proof

Interpret A ├ A as "Assuming A, infer A". Read ├ A → A 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 are 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 world) where certain statements are true and others are not. The semantics of formulae 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 ¬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 formulae. Informally this is true if in all worlds that are possible given the set of formulae 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 formulae 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 to 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 G_n of sets (G₀, G₁ ... ) inductively, as follows. (i)G₀ = G. (ii) If {G_k, E_(k+1)} proves A, then G_(k+1) = G_k. (iii) If {G_k, E_(k+1)} does not prove A, then G_(k+1) = {G_k, E_(k+1)}
  • (c)Define G* as the union of all the G_n. (That is, G* is the set of all the sentences that are in any G_n).
  • (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 G_n 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 G_n, again by definition)
• III. If G* is a Maximal Set (wrt A), then it is "truth-like". This means that it contains the sentence "C" only if it does not contain the sentence not-C; If it contains "C" and contains "If C 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.

Another outline for a completeness proof

If a formula is a tautology, then there is a truth table for it which shows that each valuation yields the value true for the formula. Consider such a valuation. By mathematical induction on the length of the subformulas, show that the truth or falsity of the subformula follows from the truth or falsity (as appropriate for the valuation) of each propositional variable in the subformula. Then combine the lines of the truth table together two at a time by using "(P is true implies S) implies ((P is false implies S) implies S)". Keep repeating this until all dependencies on propositional variables have been eliminated. The result is that we have proved the given tautology. Since every tautology is provable, the logic is complete.

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 wff's themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are

Axioms
Name	Axiom Schema	Description
THEN-1	φ → (χ → φ)	Add hypothesis χ, implication introduction
THEN-2	(φ → (χ → ψ)) → ((φ → χ) → (φ → ψ))	Distribute hypothesis φ over implication
AND-1	φ ∧ χ → φ	Eliminate conjunction
AND-2	φ ∧ χ → χ
AND-3	φ → (χ → (φ ∧ χ))	Introduce conjunction
OR-1	φ → φ ∨ χ	Introduce disjunction
OR-2	χ → φ ∨ χ
OR-3	(φ → ψ) → ((χ → ψ) → (φ ∨ χ → ψ))	Eliminate disjunction
NOT-1	(φ → χ) → ((φ → ¬χ) → ¬ φ)	Introduce negation
NOT-2	φ → (¬φ → χ)	Eliminate negation
NOT-3	φ ∨ ¬φ	Excluded middle, classical logic
IFF-1	(φ ↔ χ) → (φ → χ)	Eliminate equivalence
IFF-2	(φ ↔ χ) → (χ → φ)
IFF-3	(φ → χ) → ((χ → φ) → (φ ↔ χ))	Introduce equivalence

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 formulae: 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:

• $\phi ,\ \phi \rightarrow \chi \vdash \chi$.

Meta-inference rule

Let a demonstration be represented by a sequence, with hypotheses to the left of the turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows:

If the sequence

$\phi _{1},\ \phi _{2},\ ...,\ \phi _{n},\ \chi \vdash \psi$

has been demonstrated, then it is also possible to demonstrate the sequence

$\phi _{1},\ \phi _{2},\ ...,\ \phi _{n}\vdash \chi \rightarrow \psi$.

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

$\phi _{1},\ \phi _{2},\ ...,\ \phi _{n}\vdash \chi \rightarrow \psi$

has been demonstrated, then it is also possible to demonstrate the sequence

$\phi _{1},\ \phi _{2},\ ...,\ \phi _{n},\ \chi \vdash \psi$

in fact, the validity of the converse of DT is almost trivial compared to that of DT:

If

$\phi _{1},\ ...,\ \phi _{n}\vdash \chi \rightarrow \psi$

then

1: $\phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \chi \rightarrow \psi$

2: $\phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \chi$

and from (1) and (2) can be deduced

3: $\phi _{1},\ ...,\ \phi _{n},\ \chi \vdash \psi$

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,

$\vdash \phi \wedge \chi \rightarrow \phi$

can be transformed by means of the converse of the deduction theorem into the inference rule

$\phi \wedge \chi \vdash \phi$

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 → ((B → A) → A)) → ((A → (B → A)) → (A → A))

Axiom THEN-2 with φ = A, χ = B → A, ψ = A

2. A → ((B → A) → A)

Axiom THEN-1 with φ = A, χ = B → A

3. (A → (B → A)) → (A → A)

From (1) and (2) by modus ponens.

4. A → (B → A)

Axiom THEN-1 with φ = A, χ = B

5. A → A

From (3) and (4) by modus ponens.

Graphical calculi

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 stuctures. 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.

Other logical calculi

Propositional calculus is about the simplest kind of logical calculus in any current use. (Aristotelian "syllogistic" 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.

References

ISBN links support NWE through referral fees

• 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.

See also

Logical levels

• Zeroth order logic
• First order logic
• Second order logic
• Higher order logic

Building blocks

Exclusive disjunction Logical conjunction Logical disjunction Logical equality

Logical implication Logical NAND Logical NOR Negation

Related topics

Ampheck Boolean algebra Boolean algebra topics Boolean domain Boolean function Boolean logic Boolean-valued function Categorical logic Combinational logic

Combinatory logic Conceptual graph Disjunctive syllogism Entitative graph Existential graph Frege's propositional calculus Implicational propositional calculus Intuitionistic propositional calculus Laws of Form

Logical graph Logical value Minimal negation operator Multigrade operator Operation Parametric operator Peirce's law Symmetric difference Truth table

Related works

• Hofstadter, D.R., Gödel, Escher, Bach

External links

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

• Introduction to Mathematical Logic

• Elements of Propositional Calculus

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