Propositional Logic

A calculus is a Logical System which is used to prove valid formulae (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 Rule s for deriving valid inferences. A Formal Grammar (or ''syntax'') recursively defines the expressions and well-formed formulae (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 Connective s). 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 Variable s. These are Atomic Formula e. Conventionally, either the Latin alphabet (''A'', ''B'', ''C'') or the Greek alphabet (χ, φ, ψ) is used, but the two are not mixed.
# Symbols denoting the following , ∧ , ∨ , → , ↔ . (We may do with fewer operators (and thus symbols) by having some abbreviate others — e.g. ''P'' → ''Q'' is equivalent to ¬''P'' ∨ ''Q''.) Many authors prefer the Tilde , ~, to represent the logical not. The Ampersand , &, is also a common symbol for the logical conjunction.
# The left and right parentheses: (, ''')'''.

The set of Well-formed Formula s (wffs) is Recursive ly defined by the following rules:
# ''Basis:'' Letters of the alphabet (usually capitalized such as ''A'', ''B'', φ, χ, etc.) are wffs.
# ''Inductive clause I:'' If φ is a wff, then ¬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, ¬''A'' is a wff.
# By rule 1, ''B'' is a wff.
# By rule 3, (¬''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 formulae given a set of formulae 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 formulae 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'' ¬¬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 the ''wff''s of the form (φ ∨ ψ), (φ → χ), and (ψ → χ), we may infer χ.
; Biconditional Introduction : From the ''wff''s 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 the ''wff''s 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 ¬ψ while assuming the hypothesis φ, we may infer ¬φ.

Example of a proof

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

Prove: ''A'' → ''A''

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 System s, 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

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)

(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 formulae. (The wff's 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 → ¬χ) → ¬ φ)

NOT-2: φ → (¬φ → χ)

NOT-3: φ ∨ ¬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 "ians do not accept the axiom NOT-3.

Inference rule

The inference rule is Modus Ponens :

$\phi, \ \phi
ightarrow \chi dash \chi$ .

IFF-1: $\phi \leftrightarrow \chi dash \chi
ightarrow \phi$

IFF-2: $\phi
ightarrow \chi, \ \chi
ightarrow \phi dash \phi \leftrightarrow \chi$

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 dash \psi

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

\phi_1, \ \phi_2, \ ..., \ \phi_n dash \chi 
ightarrow \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 dash \chi 
ightarrow \psi

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

\phi_1, \ \phi_2, \ ... , \ \phi_n, \ \chi dash \psi

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

\phi_1, \ ... , \ \phi_n dash \chi 
ightarrow \psi

: ''then''
:: 1:

\phi_1, \ ... , \ \phi_n, \ \chi dash \chi 
ightarrow \psi

:: 2:

\phi_1, \ ... , \ \phi_n, \ \chi dash \chi

: ''and from (1) and (2) can be deduced''
:: 3:

\phi_1, \ ... , \ \phi_n, \ \chi dash \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,
:

dash \phi \wedge \chi 
ightarrow \phi

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

\phi \wedge \chi dash \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'' → ((''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 "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 , Variable s, Predicate s, and Quantifier s, 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 Logic s 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.

SEE ALSO

Boolean Algebra Topics

Boolean Logic

First-order Logic

Frege's Propositional Calculus

'' Gödel, Escher, Bach ''

Hofstadter, D.R.

Logical Graph

Zeroth-order Logic

EXTERNAL LINKS

Article on ''Propositional logic'' at the Internet Encyclopedia Of Philosophy

Introduction to Mathematical Logic

Elements of Propositional Calculus

	CATEGORIES ABOUT PROPOSITIONAL CALCULUS

	sentential logic

	systems of formal logic

	logical calculi

	boolean algebra

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Propositional Logic