Propositional calculus

Last updated

Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic . It deals with propositions (which can be true or false) and relations between propositions, including the construction of arguments based on them. Compound propositions are formed by connecting propositions by logical connectives. Propositions that contain no logical connectives are called atomic propositions.

Contents

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

Explanation

Logical connectives are found in natural languages. In English for example, some examples are "and" (conjunction), "or" (disjunction), "not" (negation) and "if" (but only when used to denote material conditional).

The following is an example of a very simple inference within the scope of propositional logic:

Premise 1: If it's raining then it's cloudy.
Premise 2: It's raining.
Conclusion: It's cloudy.

Both premises and the conclusion are propositions. The premises are taken for granted, and with the application of modus ponens (an inference rule), the conclusion follows.

As propositional logic is not concerned with the structure of propositions beyond the point where they can't be decomposed any more by logical connectives, this inference can be restated replacing those atomic statements with statement letters, which are interpreted as variables representing statements:

Premise 1:
Premise 2:
Conclusion:

The same can be stated succinctly in the following way:

When P is interpreted as "It's raining" and Q as "it's cloudy" the above symbolic expressions can be seen to correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference of this form, which will be valid on the same basis this inference is.

Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of axioms and inference rules allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem.

When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as , and [1] ) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.

In classical truth-functional propositional logic, formulas are interpreted as having precisely one of two possible truth values, the truth value of true or the truth value of false. [2] The principle of bivalence and the law of excluded middle are upheld. Truth-functional propositional logic defined as such and systems isomorphic to it are considered to be zeroth-order logic . However, alternative propositional logics are also possible. For more, see Other logical calculi below.

History

Although propositional logic (which is interchangeable with propositional calculus) had been hinted by earlier philosophers, it was developed into a formal logic (Stoic logic) by Chrysippus in the 3rd century BC [3] and expanded by his successor Stoics. The logic was focused on propositions. This advancement was different from the traditional syllogistic logic, which was focused on terms. However, most of the original writings were lost [4] and the propositional logic developed by the Stoics was no longer understood later in antiquity.[ citation needed ] Consequently, the system was essentially reinvented by Peter Abelard in the 12th century. [5]

Propositional logic was eventually refined using symbolic logic. The 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community. Consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan—completely independent of Leibniz. [6]

Just as propositional logic can be considered an advancement from the earlier syllogistic logic, Gottlob Frege's predicate logic can be also considered an advancement from the earlier propositional logic. One author describes predicate logic as combining "the distinctive features of syllogistic logic and propositional logic." [7] Consequently, predicate logic ushered in a new era in logic's history; however, advances in propositional logic were still made after Frege, including natural deduction, truth trees and truth tables. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz. Truth trees were invented by Evert Willem Beth. [8] The invention of truth tables, however, is of uncertain attribution.

Within works by Frege [9] and Bertrand Russell, [10] are ideas influential to the invention of truth tables. The actual tabular structure (being formatted as a table), itself, is generally credited to either Ludwig Wittgenstein or Emil Post (or both, independently). [9] Besides Frege and Russell, others credited with having ideas preceding truth tables include Philo, Boole, Charles Sanders Peirce, [11] and Ernst Schröder. Others credited with the tabular structure include Jan Łukasiewicz, Ernst Schröder, Alfred North Whitehead, William Stanley Jevons, John Venn, and Clarence Irving Lewis. [10] Ultimately, some have concluded, like John Shosky, that "It is far from clear that any one person should be given the title of 'inventor' of truth-tables.". [10]

Terminology

In general terms, a calculus is a formal system that consists of a set of syntactic expressions ( well-formed formulas ), a distinguished subset of these expressions (axioms), plus a set of formal rules that define a specific binary relation, intended to be interpreted as logical equivalence, on the space of expressions.

When the formal system is intended to be a logical system, the expressions are meant to be interpreted as statements, and the rules, known to be inference rules, are typically intended to be truth-preserving. In this setting, the rules, which may include axioms, can then be used to derive ("infer") formulas representing true statements—from given formulas representing true statements.

The set of axioms may be empty, a nonempty finite set, or a countably infinite set (see axiom schema). A formal grammar recursively defines the expressions and well-formed formulas of the language. In addition a semantics may be given which defines truth and valuations (or interpretations).

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 is any atomic formula, or any formula that can be built up from atomic formulas by means of operator symbols according to the rules of the grammar.

Mathematicians sometimes distinguish between propositional constants, propositional variables, and schemata. Propositional constants represent some particular proposition, while propositional variables range over the set of all atomic propositions. Schemata, however, range over all propositions. It is common to represent propositional constants by A, B, and C, propositional variables by P, Q, and R, [1] and schematic letters are often Greek letters, most often φ, ψ, and χ.

Basic concepts

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent, but differ in the details of:

  1. their language (i.e., the particular collection of primitive symbols and operator symbols),
  2. the set of axioms, or distinguished formulas, and
  3. the set of inference rules.

Any given proposition may be represented with a letter called a 'propositional constant', analogous to representing a number by a letter in mathematics (e.g., a = 5). All propositions require exactly one of two truth-values: true or false. For example, let P be the proposition that it is raining outside. This will be true (P) if it is raining outside, and false otherwise (¬P).

The conjunction of P and Q is true in case 1, and is false otherwise. Where P is the proposition that it is raining outside and Q is the proposition that a cold-front is over Kansas, PQ is true when it is raining outside and there is a cold-front over Kansas. If it is not raining outside, then P ∧ Q is false; and if there is no cold-front over Kansas, then PQ is also false.

It is very helpful to look at the truth tables for these different operators, as well as the method of analytic tableaux.

Closure under operations

Propositional logic is closed under truth-functional connectives. That is to say, for any proposition φ, ¬φ is also a proposition. Likewise, for any propositions φ and ψ, φψ is a proposition, and similarly for disjunction, conditional, and biconditional. This implies that, for instance, φψ is a proposition, and so it can be conjoined with another proposition. In order to represent this, we need to use parentheses to indicate which proposition is conjoined with which. For instance, PQR is not a well-formed formula, because we do not know if we are conjoining PQ with R or if we are conjoining P with QR. Thus we must write either (PQ) ∧ R to represent the former, or P ∧ (QR) to represent the latter. By evaluating the truth conditions, we see that both expressions have the same truth conditions (will be true in the same cases), and moreover that any proposition formed by arbitrary conjunctions will have the same truth conditions, regardless of the location of the parentheses. This means that conjunction is associative, however, one should not assume that parentheses never serve a purpose. For instance, the sentence P ∧ (QR) does not have the same truth conditions of (PQ) ∨ R, so they are different sentences distinguished only by the parentheses. One can verify this by the truth-table method referenced above.

Note: For any arbitrary number of propositional constants, we can form a finite number of cases which list their possible truth-values. A simple way to generate this is by truth-tables, in which one writes P, Q, ..., Z, for any list of k propositional constants—that is to say, any list of propositional constants with k entries. Below this list, one writes 2k rows, and below P one fills in the first half of the rows with true (or T) and the second half with false (or F). Below Q one fills in one-quarter of the rows with T, then one-quarter with F, then one-quarter with T and the last quarter with F. The next column alternates between true and false for each eighth of the rows, then sixteenths, and so on, until the last propositional constant varies between T and F for each row. This will give a complete listing of cases or truth-value assignments possible for those propositional constants.

Argument

The propositional calculus then defines an argument to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following list of propositions:

This is a list of three propositions, each line is a proposition, and the last follows from the rest. The first two lines are called premises, and the last line the conclusion. We say that any proposition C follows from any set of propositions , if C must be true whenever every member of the set is true. In the argument above, for any P and Q, whenever PQ and P are true, necessarily Q is true. Notice that, when P is true, we cannot consider cases 3 and 4 (from the truth table). When PQ is true, we cannot consider case 2. This leaves only case 1, in which Q is also true. Thus Q is implied by the premises.

This generalizes schematically. Thus, where φ and ψ may be any propositions at all,

Other argument forms are convenient, but not necessary. Given a complete set of axioms (see below for one such set), modus ponens is sufficient to prove all other argument forms in propositional logic, thus they may be considered to be a derivative. Note, this is not true of the extension of propositional logic to other logics like first-order logic. First-order logic requires at least one additional rule of inference in order to obtain completeness.

The significance of argument in formal logic is that one may obtain new truths from established truths. In the first example above, given the two premises, the truth of Q is not yet known or stated. After the argument is made, Q is deduced. In this way, we define a deduction system to be a set of all propositions that may be deduced from another set of propositions. For instance, given the set of propositions , we can define a deduction system, Γ, which is the set of all propositions which follow from A. Reiteration is always assumed, so . Also, from the first element of A, last element, as well as modus ponens, R is a consequence, and so . Because we have not included sufficiently complete axioms, though, nothing else may be deduced. Thus, even though most deduction systems studied in propositional logic are able to deduce , this one is too weak to prove such a proposition.

Generic description of a propositional calculus

A propositional calculus is a formal system , where:

The language of , also known as its set of formulas, well-formed formulas , is inductively defined by the following rules:

  1. Base: Any element of the alpha set is a formula of .
  2. If are formulas and is in , then is a formula.
  3. Closed: Nothing else is a formula of .

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

Example 1. Simple axiom system

Let , where , , , are defined as follows:

Example 2. Natural deduction system

Let , where , , , are defined as follows:

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.

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

In describing the transformation rules, we may introduce a metalanguage symbol . It is basically a convenient shorthand for saying "infer that". The format is , in which Γ is a (possibly empty) set of formulas called premises, and ψ is a formula called conclusion. The transformation rule means that if every proposition in Γ is a theorem (or has the same truth value as the axioms), then ψ is also a theorem. Note that considering the following rule Conjunction introduction, we will know whenever Γ has more than one formula, we can always safely reduce it into one formula using conjunction. So for short, from that time on we may represent Γ as one formula instead of a set. Another omission for convenience is when Γ is an empty set, in which case Γ may not appear.

Negation introduction
From and , infer .
That is, .
Negation elimination
From , infer .
That is, .
Double negation elimination
From , infer p.
That is, .
Conjunction introduction
From p and q, infer .
That is, .
Conjunction elimination
From , infer p.
From , infer q.
That is, and .
Disjunction introduction
From p, infer .
From q, infer .
That is, and .
Disjunction elimination
From and and , infer r.
That is, .
Biconditional introduction
From and , infer .
That is, .
Biconditional elimination
From , infer .
From , infer .
That is, and .
Modus ponens (conditional elimination)
From p and , infer q.
That is, .
Conditional proof (conditional introduction)
From [accepting p allows a proof of q], infer .
That is, .

Basic and derived argument forms

NameSequentDescription
Modus Ponens If p then q; p; therefore q
Modus Tollens If p then q; not q; therefore not p
Hypothetical Syllogism If p then q; if q then r; therefore, if p then r
Disjunctive Syllogism Either p or q, or both; not p; therefore, q
Constructive Dilemma If p then q; and if r then s; but p or r; therefore q or s
Destructive Dilemma If p then q; and if r then s; but not q or not s; therefore not p or not r
Bidirectional DilemmaIf p then q; and if r then s; but p or not s; therefore q or not r
Simplification p and q are true; therefore p is true
Conjunction p and q are true separately; therefore they are true conjointly
Addition p is true; therefore the disjunction (p or q) is true
Composition 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)The negation of (p and q) is equiv. to (not p or not q)
De Morgan's Theorem (2)The negation of (p or q) is equiv. to (not p and not q)
Commutation (1)(p or q) is equiv. to (q or p)
Commutation (2)(p and q) is equiv. to (q and p)
Commutation (3)(p is equiv. to q) is equiv. to (q is equiv. to p)
Association (1)p or (q or r) is equiv. to (p or q) or r
Association (2)p and (q and r) is equiv. to (p and q) and r
Distribution (1)p and (q or r) is equiv. to (p and q) or (p and r)
Distribution (2)p or (q and r) is equiv. to (p or q) and (p or r)
Double Negation and p is equivalent to the negation of not p
Transposition If p then q is equiv. to if not q then not p
Material Implication If p then q is equiv. to not p or q
Material Equivalence (1)(piffq) is equiv. to (if p is true then q is true) and (if q is true then p is true)
Material Equivalence (2)(piffq) is equiv. to either (p and q are true) or (both p and q are false)
Material Equivalence (3)(piffq) is equiv to., both (p or not q is true) and (not p or q is true)
Exportation [13] 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 If p then (if q then r) is equivalent to if p and q then r
Tautology (1)p is true is equiv. to p is true or p is true
Tautology (2)p is true is equiv. to p is true and p is true
Tertium non datur (Law of Excluded Middle) p or not p is true
Law of Non-Contradiction p and not p is false, is a true statement

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

Example of a proof in natural deduction system

Example of a proof
NumberFormulaReason
1premise
2From ( 1 ) by disjunction introduction
3From ( 1 ) and ( 2 ) by conjunction introduction
4From ( 3 ) by conjunction elimination
5Summary of ( 1 ) through ( 4 )
6From ( 5 ) by conditional proof

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

Example of a proof in a classical propositional calculus system

We now prove the same theorem in the axiomatic system by Jan Łukasiewicz described above, which is an example of a classical propositional calculus systems, or a Hilbert-style deductive system for propositional calculus.

The axioms are:

(A1)
(A2)
(A3)

And the proof is as follows:

  1.     (instance of (A1))
  2.     (instance of (A2))
  3.     (from (1) and (2) by modus ponens)
  4.     (instance of (A1))
  5.     (from (4) and (3) by modus ponens)

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. Note that the proofs for the soundness and completeness of the propositional logic are not themselves proofs in propositional logic ; these are theorems in ZFC used as a metatheory to prove properties of propositional logic.

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 formulas can then be formalized by defining for which "state of affairs" they are considered to be true, which is what is done by the following definition.

We define when such a truth assignment A satisfies a certain well-formed formula with the following rules:

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

Finally we define syntactical entailment such that φ is syntactically entailed by S if and only if 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 well-formed formulas Ssyntactically entails the well-formed formula φ then Ssemantically entails φ.

Completeness: If the set of well-formed formulas Ssemantically entails the well-formed formula φ then Ssyntactically 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.

  1. Basis. Show: If A is a member of G, then G implies A.
  2. Basis. Show: If A is an axiom, then G implies A.
  3. Inductive step (induction on n, the length of the proof):
    1. Assume for arbitrary G and A that if G proves A in n or fewer steps, then G implies A.
    2. For each possible application of a rule of inference at step n + 1, leading to a new theorem B, show that G implies B.

Notice that Basis Step II can be omitted for natural deduction systems because they have no axioms. When used, Step II involves showing that each of the axioms is a (semantic) logical truth.

The Basis steps demonstrate that the simplest provable sentences from G are also implied by G, for any G. (The proof 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. If we show that there is a model where A does not hold despite G being true, then obviously G does not imply A. The idea is to build such a model out of our very assumption that G does not prove A.

  1. G does not prove A. (Assumption)
  2. 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.
    1. Place an ordering (with order type ω) on all the sentences in the language (e.g., shortest first, and equally long ones in extended alphabetical ordering), and number them (E1, E2, ...)
    2. Define a series Gn of sets (G0, G1, ...) inductively:
      1. If proves A, then
      2. If does not prove A, then
    3. Define G as the union of all the Gn. (That is, G is the set of all the sentences that are in any Gn.)
    4. It can be easily shown that
      1. G contains (is a superset of) G (by (b.i));
      2. G does not prove A (because the proof would contain only finitely many sentences and when the last of them is introduced in some Gn, that Gn would prove A contrary to the definition of Gn); and
      3. 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)
  3. If G is a Maximal Set with respect to A, then it is truth-like. This means that it contains C if and only if it does not contain ¬C; If it contains C and contains "If C then B" then it also contains B; and so forth. In order to show this, one has to show the axiomatic system is strong enough for the following:
    • For any formulas C and D, if it proves both C and ¬C, then it proves D. From this it follows, that a Maximal Set with respect to A cannot prove both C and ¬C, as otherwise it would prove A.
    • For any formulas C and D, if it proves both CD and ¬CD, then it proves D. This is used, together with the deduction theorem, to show that for any formula, either it or its negation is in G: Let B be a formula not in G; then G with the addition of B proves A. Thus from the deduction theorem it follows that G proves BA. But suppose ¬B were also not in G, then by the same logic G also proves ¬BA; but then G proves A, which we have already shown to be false.
    • For any formulas C and D, if it proves C and D, then it proves CD.
    • For any formulas C and D, if it proves C and ¬D, then it proves ¬(CD).
    • For any formulas C and D, if it proves ¬C, then it proves CD.
    If additional logical operation (such as conjunction and/or disjunction) are part of the vocabulary as well, then there are additional requirement on the axiomatic system (e.g. that if it proves C and D, it would also prove their conjunction).
  4. 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. Note that the requirement that it is truth-like is needed to guarantee that the laws of semantic composition in the language will be satisfied by this truth assignment.
  5. A G-canonical valuation will make our original set G all true, and make A false.
  6. If there is a valuation on which G are true and A is false, then G does not (semantically) imply A.

Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:

The first five are used for the satisfaction of the five conditions in stage III above, and the last three for proving the deduction theorem.

Example

As an example, it can be shown that as any other tautology, the three axioms of the classical propositional calculus system described earlier can be proven in any system that satisfies the above, namely that has modus ponens as an inference rule, and proves the above eight theorems (including substitutions thereof). Out of the eight theorems, the last two are two of the three axioms; the third axiom, , can be proven as well, as we now show.

For the proof we may use the hypothetical syllogism theorem (in the form relevant for this axiomatic system), since it only relies on the two axioms that are already in the above set of eight theorems. The proof then is as follows:

  1.     (instance of the 7th theorem)
  2.     (instance of the 7th theorem)
  3.     (from (1) and (2) by modus ponens)
  4.     (instance of the hypothetical syllogism theorem)
  5.     (instance of the 5th theorem)
  6.     (from (5) and (4) by modus ponens)
  7.     (instance of the 2nd theorem)
  8.     (instance of the 7th theorem)
  9.     (from (7) and (8) by modus ponens)
  10.     (instance of the 8th theorem)
  11.     (from (9) and (10) by modus ponens)
  12.     (from (3) and (11) by modus ponens)
  13.     (instance of the 8th theorem)
  14.     (from (12) and (13) by modus ponens)
  15.     (from (6) and (14) by modus ponens)

Verifying completeness for the classical propositional calculus system

We now verify that the classical propositional calculus system described earlier can indeed prove the required eight theorems mentioned above. We use several lemmas proven here:

(DN1) - Double negation (one direction)
(DN2) - Double negation (another direction)
(HS1) - one form of Hypothetical syllogism
(HS2) - another form of Hypothetical syllogism
(TR1) - Transposition
(TR2) - another form of transposition.
(L1)
(L3)

We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.

  • p → (¬p → q) - proof:
    1.     (instance of (A1))
    2.     (instance of (TR1))
    3.     (from (1) and (2) using the hypothetical syllogism metatheorem)
    4.     (instance of (DN1))
    5.     (instance of (HS1))
    6.     (from (4) and (5) using modus ponens)
    7.     (from (3) and (6) using the hypothetical syllogism metatheorem)
  • (p → q) → ((¬p → q) → q) - proof:
    1.     (instance of (HS1))
    2.     (instance of (L3))
    3.     (instance of (HS1))
    4.     (from (2) and (3) by modus ponens)
    5.     (from (1) and (4) using the hypothetical syllogism metatheorem)
    6.     (instance of (TR2))
    7.     (instance of (HS2))
    8.     (from (6) and (7) using modus ponens)
    9.     (from (5) and (8) using the hypothetical syllogism metatheorem)
  • p → (q → (p → q)) - proof:
    1.     (instance of (A1))
    2.     (instance of (A1))
    3.     (from (1) and (2) using modus ponens)
  • p → (¬q → ¬(p → q)) - proof:
    1.     (instance of (L1))
    2.     (instance of (TR1))
    3.     (from (1) and (2) using the hypothetical syllogism metatheorem)
  • ¬p → (p → q) - proof:
    1.     (instance of (A1))
    2.     (instance of (A3))
    3.     (from (1) and (2) using the hypothetical syllogism metatheorem)
  • p → p - proof given in the proof example above
  • p → (q → p) - axiom (A1)
  • (p → (q → r)) → ((p → q) → (p → r)) - axiom (A2)

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.

Interpretation of a truth-functional propositional calculus

An interpretation of a truth-functional propositional calculus is an assignment to each propositional symbol of of one or the other (but not both) of the truth values truth (T) and falsity (F), and an assignment to the connective symbols of of their usual truth-functional meanings. An interpretation of a truth-functional propositional calculus may also be expressed in terms of truth tables. [14]

For distinct propositional symbols there are distinct possible interpretations. For any particular symbol , for example, there are possible interpretations:

  1. is assigned T, or
  2. is assigned F.

For the pair , there are possible interpretations:

  1. both are assigned T,
  2. both are assigned F,
  3. is assigned T and is assigned F, or
  4. is assigned F and is assigned T. [14]

Since has , that is, denumerably many propositional symbols, there are , and therefore uncountably many distinct possible interpretations of . [14]

Interpretation of a sentence of truth-functional propositional logic

If φ and ψ are formulas of and is an interpretation of then the following definitions apply:

Some consequences of these definitions:

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 well-formed formulas themselves would not contain any Greek letters, but only capital Roman letters, connective operators, and parentheses.) Then the axioms are as follows:

Axioms
NameAxiom SchemaDescription
THEN-1Add hypothesis χ, implication introduction
THEN-2Distribute hypothesis over implication
AND-1Eliminate conjunction
AND-2 
AND-3Introduce conjunction
OR-1Introduce disjunction
OR-2 
OR-3Eliminate disjunction
NOT-1Introduce negation
NOT-2Eliminate negation
NOT-3Excluded middle, classical logic
IFF-1Eliminate equivalence
IFF-2 
IFF-3Introduce equivalence

Inference rule

The inference rule is modus ponens:

.

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

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

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

The converse of DT is also valid:

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

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

If
then
1:
2:
and from (1) and (2) can be deduced
3:
by means of modus ponens, Q.E.D.

The converse of DT has powerful implications: it can be used to convert an axiom into an inference rule. For example, the axiom AND-1,

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

which is conjunction elimination, one of the ten inference rules used in the first version (in this article) of the propositional calculus.

Example of a proof

The following is an example of a (syntactical) demonstration, involving only axioms THEN-1 and THEN-2 :

Prove: (Reflexivity of implication).

Proof:

  1. Axiom THEN-2 with
  2. Axiom THEN-1 with
  3. From (1) and (2) by modus ponens.
  4. Axiom THEN-1 with
  5. From (3) and (4) by modus ponens.

Equivalence to equational logics

The preceding alternative calculus is an example of a Hilbert-style deduction system. In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens. Equational logic as standardly used informally in high school algebra is a different kind of calculus from Hilbert systems. Its theorems are equations and its inference rules express the properties of equality, namely that it is a congruence on terms that admits substitution.

Classical propositional calculus as described above is equivalent to Boolean algebra, while intuitionistic propositional calculus is equivalent to Heyting algebra. The equivalence is shown by translation in each direction of the theorems of the respective systems. Theorems of classical or intuitionistic propositional calculus are translated as equations of Boolean or Heyting algebra respectively. Conversely theorems of Boolean or Heyting algebra are translated as theorems of classical or intuitionistic calculus respectively, for which is a standard abbreviation. In the case of Boolean algebra can also be translated as , but this translation is incorrect intuitionistically.

In both Boolean and Heyting algebra, inequality can be used in place of equality. The equality is expressible as a pair of inequalities and . Conversely the inequality is expressible as the equality , or as . The significance of inequality for Hilbert-style systems is that it corresponds to the latter's deduction or entailment symbol . An entailment

is translated in the inequality version of the algebraic framework as

Conversely the algebraic inequality is translated as the entailment

.

The difference between implication and inequality or entailment or is that the former is internal to the logic while the latter is external. Internal implication between two terms is another term of the same kind. Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the metalanguage. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not.

Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category. In this interpretation the cut rule of the sequent calculus corresponds to composition in the category. Boolean and Heyting algebras enter this picture as special categories having at most one morphism per homset, i.e., one proof per entailment, corresponding to the idea that existence of proofs is all that matters: any proof will do and there is no point in distinguishing them.

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. Many species of graphs arise as parse graphs in the syntactic analysis of the corresponding families of text structures. The exigencies of practical computation on formal languages frequently demand that text strings be converted into pointer structure renditions of parse graphs, simply as a matter of checking whether strings are well-formed formulas 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 current use. It can be extended in several ways. (Aristotelian "syllogistic" calculus, which is largely supplanted in modern logic, is in some ways simpler – but in other ways more complex – than propositional calculus.) 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.

First-order logic (a.k.a. first-order predicate logic) results when the "atomic sentences" of propositional logic are broken up into terms, variables, predicates, and quantifiers, all keeping the rules of propositional logic with some new ones introduced. (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. Second-order logic and other higher-order logics are formal extensions of first-order logic. Thus, it makes sense to refer to propositional logic as "zeroth-order logic", when comparing it with these logics.

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". The translation between modal logics and algebraic logics concerns classical and intuitionistic logics but with the introduction of a unary operator on Boolean or Heyting algebras, different from the Boolean operations, interpreting the possibility modality, and in the case of Heyting algebra a second operator interpreting necessity (for Boolean algebra this is redundant since necessity is the De Morgan dual of possibility). The first operator preserves 0 and disjunction while the second preserves 1 and conjunction.

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. When the values form a Boolean algebra (which may have more than two or even infinitely many values), many-valued logic reduces to classical logic; many-valued logics are therefore only of independent interest when the values form an algebra that is not Boolean.

Solvers

Finding solutions to propositional logic formulas is an NP-complete problem. However, practical methods exist (e.g., DPLL algorithm, 1962; Chaff algorithm, 2001) that are very fast for many useful cases. Recent work has extended the SAT solver algorithms to work with propositions containing arithmetic expressions; these are the SMT solvers.

See also

Higher logical levels

Related Research Articles

An axiom, postulate or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'

In classical logic, disjunctive syllogism is a valid argument form which is a syllogism having a disjunctive statement for one of its premises.

First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a man", one can have expressions in the form "there exists x such that x is Socrates and x is a man", where "there exists" is a quantifier, while x is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

In propositional logic, modus ponens, also known as modus ponendo ponens or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore Q must also be true."

In propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.

In classical deductive logic, a consistent theory is one that does not entail a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences and the set of closed sentences provable from under some formal deductive system. The set of axioms is consistent when for no formula .

Contradiction Logical incompatibility between two or more propositions

In traditional logic, a contradiction consists of a logical incompatibility or incongruity between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic.

In classical logic, hypothetical syllogism is a valid argument form which is a syllogism having a conditional statement for one or both of its premises.

A rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion. For example, the rule of inference called modus ponens takes two premises, one in the form "If p then q" and another in the form "p", and returns the conclusion "q". The rule is valid with respect to the semantics of classical logic, in the sense that if the premises are true, then so is the conclusion.

In mathematical logic, sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology instead of an unconditional tautology. Each conditional tautology is inferred from other conditional tautologies on earlier lines in a formal argument according to rules and procedures of inference, giving a better approximation to the natural style of deduction used by mathematicians than David Hilbert's earlier style of formal logic where every line was an unconditional tautology. There may be more subtle distinctions to be made; for example, there may be non-logical axioms upon which all propositions are implicitly dependent. Then sequents signify conditional theorems in a first-order language rather than conditional tautologies.

In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs — to prove an implication A → B, assume A as an hypothesis and then proceed to derive B — in systems that do not have an explicit inference rule for this. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction.

In mathematical logic, Löb's theorem states that in Peano arithmetic (PA), for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is provable in PA. More formally, if Prov(P) means that the formula P is provable, then

In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that:

Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applications in many fields, including philosophy, theoretical computer science, artificial intelligence, economics and linguistics. While philosophers since Aristotle have discussed modal logic, and Medieval philosophers such as Avicenna, Ockham, and Duns Scotus developed many of their observations, it was C. I. Lewis who created the first symbolic and systematic approach to the topic, in 1912. It continued to mature as a field, reaching its modern form in 1963 with the work of Kripke.

In mathematical logic, the implicational propositional calculus is a version of classical propositional calculus which uses only one connective, called implication or conditional. In formulas, this binary operation is indicated by "implies", "if ..., then ...", "→", "", etc..

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert–Ackermann system, is a type of system of formal deduction attributed to Gottlob Frege and David Hilbert. These deductive systems are most often studied for first-order logic, but are of interest for other logics as well.

Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs and later applied to more general complex behaviors arising in linguistics, philosophy, AI, and other fields.

References

  1. 1 2 3 4 5 6 7 "Comprehensive List of Logic Symbols". Math Vault. 6 April 2020. Retrieved 20 August 2020.
  2. "Propositional Logic | Brilliant Math & Science Wiki". brilliant.org. Retrieved 20 August 2020.
  3. Bobzien, Susanne (1 January 2016). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy via Stanford Encyclopedia of Philosophy.
  4. "Propositional Logic | Internet Encyclopedia of Philosophy" . Retrieved 20 August 2020.
  5. Marenbon, John (2007). Medieval philosophy: an historical and philosophical introduction . Routledge. p.  137.
  6. Peckhaus, Volker (1 January 2014). Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy via Stanford Encyclopedia of Philosophy.
  7. Hurley, Patrick (2007). A Concise Introduction to Logic 10th edition. Wadsworth Publishing. p. 392.
  8. Beth, Evert W.; "Semantic entailment and formal derivability", series: Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, Nieuwe Reeks, vol. 18, no. 13, Noord-Hollandsche Uitg. Mij., Amsterdam, 1955, pp. 309–42. Reprinted in Jaakko Intikka (ed.) The Philosophy of Mathematics, Oxford University Press, 1969
  9. 1 2 Truth in Frege
  10. 1 2 3 "Russell: the Journal of Bertrand Russell Studies".
  11. Anellis, Irving H. (2012). "Peirce's Truth-functional Analysis and the Origin of the Truth Table". History and Philosophy of Logic. 33: 87–97. doi:10.1080/01445340.2011.621702.
  12. Wernick, William (1942) "Complete Sets of Logical Functions," Transactions of the American Mathematical Society51, pp. 117132.
  13. Toida, Shunichi (2 August 2009). "Proof of Implications". CS381 Discrete Structures/Discrete Mathematics Web Course Material. Department of Computer Science, Old Dominion University . Retrieved 10 March 2010.
  14. 1 2 3 4 5 6 7 8 9 10 11 Hunter, Geoffrey (1971). Metalogic: An Introduction to the Metatheory of Standard First-Order Logic. University of California Pres. ISBN   0-520-02356-0.

Further reading