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Encyclopedia > Propositional logic

Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. Unlike predicate logic or syllogistic logic, the internal structure of a clause or sentence has no effect on validity in propositional logic. Instead, validity is determined by the relationship of the clauses to each other in sentences compounded with words like 'and', 'or', and 'if ... then ...'. So the smallest possible expressions in a propositional logic (its atomic formulas) are clauses or variables that stand for clauses. Such variables are called propositional variables. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... Proposition is a term used in logic to describe the content of assertions. ... ... In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ...


In general terms, a propositional or sentential calculus is a formal system that consists of a set of syntactic expressions (well-formed formulae, 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. Most systems of propositional logic also have a formal semantics that specifies how the truth or falsity of a compound sentence is determined by the truth values of its clauses and connecting words. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ... In mathematics, the concept of binary relation, sometimes called dyadic relation, is exemplified by such ideas as is greater than and is equal to in arithmetic, or is congruent to in geometry, or is an element of or is a subset of in set theory. ...


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 intepreted 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. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ... In computer metadata, Semantic Equivalence is a declaration that two data elements from different vocabularies contain data that has similar meaning. ... In logic, statements p and q are logically equivalent if they have the same logical content. ... An axiom is a sentence or proposition that is taken for granted as true, and serves as a starting point for deducing other truths. ...


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. In computer science a formal grammar is an abstract structure that describes a formal language precisely, i. ... In the main, semantics (from the Greek and in greek letters σημαντικός or in latin letters semantikós, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...


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. In logical calculus, logical operators or logical connectors serve to connect statements into more complicated compound statements. ... In formal logic, logical connectives, also known as logical connectors and sometimes logical constants, serve to connect statements into more complicated compound statements. ...


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.

Contents


Grammar

The language consists of:

  1. The capital letters of the alphabet, standing as propositional variables. These are atomic formulae. Conventionally, either the Latin alphabet (A, B, C) or the Greek alphabet (χ, φ, ψ) is used, but the two are not mixed.
  2. Symbols denoting the following connectives (or logical operators): ¬, , , , . (We may do with fewer operators (and thus symbols) by having some abbreviate others — e.g. PQ is equivalent to ¬PQ.) Many authors prefer the tilde, ~, to represent the logical not. The ampersand, &, is also a common symbol for the logical conjunction.
  3. The left and right parentheses: (, ).

The set of well-formed formulas (wffs) is recursively defined by the following rules: In mathematical logic, a propositional variable (also called a sentential variable) is a variable which can either be true or false. ... In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. ... In formal logic, logical connectives, also known as logical connectors and sometimes logical constants, serve to connect statements into more complicated compound statements. ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ... In logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... The tilde (~) is a grapheme with several uses. ... The roman ampersand at left is stylised, but the italic one at right reveals its origin in the Latin word An ampersand (&, &, &), also commonly called an and sign, is a logogram representing the conjunction and. The symbol is a ligature of the letters in et, which is Latin for and... In logic, WFF is an abbreviation for well-formed formula. ... See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...

  1. Basis: Letters of the alphabet (usually capitalized such as A, B, φ, χ, etc.) are wffs.
  2. Inductive clause I: If φ is a wff, then ¬φ is a wff.
  3. Inductive clause II: If φ and ψ are wffs, then (φ ∧ ψ), (φ ∨ ψ), (φ → ψ), and (φ ↔ ψ) are wffs.
  4. Closure clause: Nothing else is a wff.

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

  1. By rule 1, A is a wff.
  2. By rule 2, ¬A is a wff.
  3. By rule 1, B is a wff.
  4. By rule 3, (¬AB) 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. In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...


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 wffs of the form (φ ∨ ψ), (φ → χ), and (ψ → χ), we may infer χ.
Biconditional introduction
From the 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 the 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 ¬ψ while assuming the hypothesis φ, we may infer ¬φ.
Basic argument forms of the calculus
Name Sequent Description
Modus Ponens ((pq) ∧ p) ├ q if p then q; p; therefore q
Modus Tollens ((pq) ∧ ¬q) ├ ¬p if p then q; not q; therefore not p
Hypothetical Syllogism ((pq) ∧ (qr)) ├ (pr) if p then q; if q then r; therefore, if p then r
Disjunctive Syllogism ((pq) ∧ ¬p) ├ q Either p or q; not p; therefore, q
Constructive Dilemma ((pq) ∧ (rs) ∧ (pr)) ├ (qs) If p then q; and if r then s; but either p or r; therefore either q or s
Destructive Dilemma ((pq) ∧ (rs) ∧ (¬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 (pq) ├ p p and q are true; therefore p is true
Conjunction p, q ├ (pq) p and q are true separately; therefore they are true conjointly
Addition p ├ (pq) p is true; therefore the disjunction (p or q) is true
Composition ((pq) ∧ (pr)) ├ (p → (qr)) 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) ¬(pq) ├ (¬p ∨ ¬q) The negation of (p and q) is equiv. to (not p or not q)
De Morgan's Theorem (2) ¬(pq) ├ (¬p ∧ ¬q) The negation of (p or q) is equiv. to (not p and not q)
Commutation (1) (pq) ├ (qp) (p or q) is equiv. to (q or p)
Commutation (2) (pq) ├ (qp) (p and q) is equiv. to (q and p)
Association (1) (p ∨ (qr)) ├ ((pq) ∨ r) p or (q or r) is equiv. to (p or q) or r
Association (2) (p ∧ (qr)) ├ ((pq) ∧ r) p and (q and r) is equiv. to (p and q) and r
Distribution (1) (p ∧ (qr)) ├ ((pq) ∨ (pr)) p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2) (p ∨ (qr)) ├ ((pq) ∧ (pr)) 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 (pq) ├ (¬q → ¬p) If p then q is equiv. to if not q then not p
Material Implication (pq) ├ (¬pq) If p then q is equiv. to either not p or q
Material Equivalence (1) (pq) ├ ((pq) ∧ (qp)) (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) (pq) ├ ((pq) ∨ (¬q ∧ ¬p)) (p is equiv. to q) means, either (p and q are true) or ( both p and q are false)
Exportation ((pq) → r) ├ (p → (qr)) 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 → (qr)) ├ ((pq) → r)
Tautology p ├ (pp) 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

In logic and the propositional calculus, double negative elimination is a rule that states that double negatives can be removed from a proposition without changing its meaning: means the same as: Formally: ¬ ¬ A ∴ A The rule of double negative introduction states the converse, that double negatives can be added without... Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true. ... In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ... Disjunction introduction is the logic principle that, if A is true, then its true that either A or B is true. ... In propositional calculus disjunction elimination is the inference that, if A or B is true, and A entails C, and B entails C, then we may justifiably infer C. The reasoning is simple: since at least one of the statements A and B is true, and since either of them... Biconditional introduction is the inference that, if B follows from A, and A follows from B, then A if and only if B. For example: if Im breathing, then Im alive; also, if Im alive, then Im breathing. ... Biconditional elimination allows one to infer a conditional from a biconditional: if ( A ↔ B ) is true, then one may infer one direction of the biconditional, either ( A → B ) or ( B → A ). For example, if its true that Im breathing if and only if Im alive, then it... In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις άτοπον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must...

Example of a proof

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

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

Interpret AA as "Assuming A, infer A". Read ├ AA 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. (This article discusses the soundess notion of informal logic. ... In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...


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. Partial plot of a function f. ... State of affairs has some technical usages in philosophy, as well as being a phrase in everyday speech in English. ... In philosophy and logic, the concept of possible worlds is used to express modal claims. ...


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. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...


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) Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...


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.) In mathematical logic, natural deduction is an approach to proof theory that attempts to provide a formal model of logical reasoning as it naturally occurs. ...


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. Case analysis is one of the most general and applicable methods of analytical thinking, depending only on the division of a problem, decision or situation into a sufficient number of separate cases. ...


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. In traditional logic, contraposition is a form of immediate inference in which from a given categorical proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality (affirmation or negation). ...

  • 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. Q.E.D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, which was to be demonstrated). This is a translation of the Greek (hoper edei deixai) which was used by many early mathematicians including Euclid and Archimedes. ...


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 "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.
To meet Wikipedias quality standards, this article or section may require cleanup. ... Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...


Inference rule

The inference rule is modus ponens: In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: P → Q P ⊢ Q where ⊢ represents the logical assertion. ...

  • .

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 turnstile and the conclusion to the right of the turnstile. Then the deduction theorem can be stated as follows: In mathematical logic, the deduction theorem states that if a formula F is deducible from E then the implication E → F is demonstrable (i. ...

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. Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...


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. In logic, conjunction elimination is the inference that, if the conjunction A and B is true, then A is true, and B is true. ...


Example of a proof

The following is an example of a (syntactical) demonstration, involving only axioms THEN-1 and THEN-2:
Prove: AA (Reflexivity of implication).
Proof:

1. (A → ((AA) → A)) → ((A → (AA)) → (AA))
Axiom THEN-2 with φ = A, χ = AA, ψ = A
2. A → ((AA) → A)
Axiom THEN-1 with φ = A, χ = AA
3. (A → (AA)) → (AA)
From (1) and (2) by modus ponens.
4. A → (AA)
Axiom THEN-1 with φ = A, χ = A
5. AA
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, 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.) There is no really adequate definition of singular term. ... In computer science and mathematics, a variable (sometimes called a pronumeral) is a symbol denoting a quantity or symbolic representation. ... In mathematics, a predicate is a relation. ... In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ... It has been suggested that Predicate calculus be merged into this article or section. ...


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. Arithmetic is the current mathematics collaboration of the week! Please help improve it to featured article standard. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ...


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". A modal logic is any logic for handling modalities: concepts like possibility, impossibility, and necessity. ...


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. Multi-valued logics are logical calculi in which there are more than two possible truth values. ...


See also

Logical levels

Zeroth-order logic is a term in popular use among practitioners for the subject matter otherwise known as boolean functions, monadic predicate logic, propositional calculus, or sentential calculus. ... First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ... In mathematical logic, second-order logic is an extension of either propositional logic or first-order logic which contains variables in predicate positions (rather than only in term positions, as in first-order logic), and quantifiers binding them. ... In mathematics, higher-order logic is distinguished from first-order logic in a number of ways. ...

Building blocks

Exclusive disjunction (usual symbol XOR occasionally EOR) is a logical operator that results in true if one of the operands, but not both of them, is true. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... OR logic gate. ... XNOR Logic Gate Symbol Logical equality is a logical operator that corresponds to equality in boolean algebra and to the logical biconditional in propositional calculus. ... In propositional calculus, or logical calculus in mathematics, the material conditional or the implies operator is a binary truth-functional logical operator yielding the form If a then c, where a and c are statement variables (to be replaced by any meaningful indicative sentence of the language). ... NAND Logic Gate The Sheffer stroke, |, is the negation of the conjunction operator. ... NOR Logic Gate The logical NOR or joint denial is a boolean logic operator which produces a result that is the inverse of logical or. ... Negation (i. ...

Related topics

Ampheck, from Greek double-edged, is a term coined by Charles Sanders Peirce for either one of the pair of logically dual operators, variously referred to as Peirce arrows, Sheffer strokes, or NAND and NNOR. Either of these logical operators is a sole sufficient operator for deriving or generating all... Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ... Algebra of sets Ampheck Boole, George Boolean algebra Boolean domain Boolean function Boolean logic Boolean implicant Boolean prime ideal theorem Boolean-valued function Boolean-valued model Boolean satisfiability problem Booles syllogistic Canonical form (Boolean algebra) Characteristic function Compactness theorem Complete Boolean algebra De Morgan, Augustus De Morgans laws... A boolean domain B is a generic 2-element set, say, B = {0, 1}, whose elements are interpreted as logical values, typically 0 = false and 1 = true. ... In mathematics, a boolean function is usually a function F(b1, b2, ..., bn) of a number n of boolean variables bi from the two-element boolean algebra B = {0, 1}, such that F also takes values in B. A function on an arbitrary set X taking values in B is... Boolean logic is a complete system for logical operations. ... A boolean-valued function is a function of the type , where is an arbitrary set, where is a generic 2-element set, typically , and where the latter is frequently interpreted for logical applications as . ... John F. Sowas Conceptual Graphs allow the graphical statement of logic propositions, or predicates. ... A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: P or Q Not P Therefore, Q In logical operator notation: ¬ where represents the logical assertion. ... An entitative graph is an element of the graphical syntax for logic that Charles Sanders Peirce developed under the name of qualitative logic in the 1880s, taking the coverage of the formalism only as far as the propositional or sentential aspects of logic are concerned. ... An existential graph is a type of diagrammatic or visual notation for logical expressions, invented by Charles Sanders Peirce, who wrote his first paper on graphical logic in 1882 and continued to develop the method until his death in 1914. ... In mathematical logic Freges propositional calculus was the first axiomatization of propositional calculus. ... GEB cover Gödel, Escher, Bach: an Eternal Golden Braid (commonly GEB) is a Pulitzer Prize-winning book by Douglas Hofstadter, published in 1979 by Basic Books. ... Douglas Richard Hofstadter (born February 15, 1945) is an American academic. ... The phrase Laws of Form refers to either of two things: The book, hereinafter abbreviated LoF by G. Spencer-Brown. ... A logical graph is a special type of graph-theoretic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic. ... In logic and mathematics, a logical value, also called a truth value, is a value indicating to what extent a proposition is true. ... In logic and mathematics, the minimal negation operator is a multigrade operator where each is a k-ary boolean function defined in such a way that if and only if exactly one of the arguments is 0. ... In logic and mathematics, a multigrade operator is a parametric operator with parameter k in the set N of non-negative integers. ... In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ... In logic and mathematics, a parametric operator with parameter in the parametric set is a indexed family of operators with index in the index set . ... Peirces law in logic is named after the philosopher and logician Charles Sanders Peirce. ... In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

External links


  Results from FactBites:
 
Logical Fallacy: Fallacy of Propositional Logic (303 words)
Propositional logic is a system which deals with the logical relations that hold between propositions taken as a whole, and those compound propositions which are constructed from simpler ones with truth-functional connectives.
Moreover, the connective "and" which joins them is truth-functional, that is, the truth-value of the compound proposition is a function of the truth-values of its components.
Propositional logic studies the logical relations which hold between propositions as a result of truth-functional combinations, for instance, the example conjunction implies "today is Sunday".
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