Not to be confused with Propositional analysis.
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.
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.
Logical connectives are found in natural languages.
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.
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:
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.
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.
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.
However, alternative propositional logics are also possible.
For more, see Other logical calculi below.
Main article: History of logic
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 and expanded by his successor Stoics.
The logic was focused on propositions.
However, most of the original writings were lost and the propositional logic developed by the Stoics was no longer understood later in antiquity.
Consequently, the system was essentially reinvented by Peter Abelard in the 12th century.
Propositional logic was eventually refined using symbolic logic.
Although his work was the first of its kind, it was unknown to the larger logical community.
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."
Truth trees were invented by Evert Willem Beth.
The invention of truth tables, however, is of uncertain attribution.
Within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables.
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.
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).
The language of a propositional calculus consists of
- a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and
- 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, and schematic letters are often Greek letters, most often φ, ψ, and χ.
The following outlines a standard propositional calculus.
Many different formulations exist which are all more or less equivalent, but differ in the details of:
- their language (i.e., the particular collection of primitive symbols and operator symbols),
- the set of axioms, or distinguished formulas, and
- 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).
- We then define truth-functional operators, beginning with negation. ¬P represents the negation of P, which can be thought of as the denial of P. In the example above, ¬P expresses that it is not raining outside, or by a more standard reading: "It is not the case that it is raining outside." When P is true, ¬P is false; and when P is false, ¬P is true. As a result, ¬ ¬P always has the same truth-value as P.
- Conjunction is a truth-functional connective which forms a proposition out of two simpler propositions, for example, P and Q. The conjunction of P and Q is written P ∧ Q, and expresses that each are true. We read P ∧ Q as "P and Q". For any two propositions, there are four possible assignments of truth values:
- P is true and Q is true
- P is true and Q is false
- P is false and Q is true
- P is false and Q is false
- 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, P ∧ Q 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 P ∧ Q is also false.
- Disjunction resembles conjunction in that it forms a proposition out of two simpler propositions. We write it P ∨ Q, and it is read "P or Q". It expresses that either P or Q is true. Thus, in the cases listed above, the disjunction of P with Q is true in all cases—except case 4. Using the example above, the disjunction expresses that it is either raining outside, or there is a cold front over Kansas. (Note, this use of disjunction is supposed to resemble the use of the English word "or". However, it is most like the English inclusive "or", which can be used to express the truth of at least one of two propositions. It is not like the English exclusive "or", which expresses the truth of exactly one of two propositions. In other words, the exclusive "or" is false when both P and Q are true (case 1). An example of the exclusive or is: You may have a bagel or a pastry, but not both. Often in natural language, given the appropriate context, the addendum "but not both" is omitted—but implied. In mathematics, however, "or" is always inclusive or; if exclusive or is meant it will be specified, possibly by "xor".)
- Material conditional also joins two simpler propositions, and we write P → Q, which is read "if P then Q". The proposition to the left of the arrow is called the antecedent, and the proposition to the right is called the consequent. (There is no such designation for conjunction or disjunction, since they are commutative operations.) It expresses that Q is true whenever P is true. Thus P → Q is true in every case above except case 2, because this is the only case when P is true but Q is not. Using the example, if P then Q expresses that if it is raining outside, then there is a cold-front over Kansas. The material conditional is often confused with physical causation. The material conditional, however, only relates two propositions by their truth-values—which is not the relation of cause and effect. It is contentious in the literature whether the material implication represents logical causation.
- Biconditional joins two simpler propositions, and we write P ↔ Q, which is read "P if and only if Q". It expresses that P and Q have the same truth-value, and in cases 1 and 4. 'P is true if and only if Q' is true, and is false otherwise.
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, φ ∧ ψ is a proposition, and similarly for disjunction, conditional, and biconditional.
This implies that, for instance, φ ∧ ψ 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, P ∧ Q ∧ R is not a well-formed formula, because we do not know if we are conjoining P ∧ Q with R or if we are conjoining P with Q ∧ R. Thus we must write either (P ∧ Q) ∧ R to represent the former, or P ∧ (Q ∧ R) 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 ∧ (Q ∨ R) does not have the same truth conditions of (P ∧ Q) ∨ 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 2 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.
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 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.
Generic description of a propositional calculus
Example 1. Simple axiom system
Example 2. Natural deduction system
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.
Basic and derived argument forms
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
- To be shown that A → A.
- One possible proof of this (which, though valid, happens to contain more steps than are necessary) may be arranged as follows:
Example of a proof in a classical propositional calculus system
The axioms are:
And the proof is as follows:
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.
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:
- A satisfies the propositional variable P if and only if A(P) = true
- A satisfies ¬φ if and only if A does not satisfy φ
- A satisfies (φ ∧ ψ) if and only if A satisfies both φ and ψ
- A satisfies (φ ∨ ψ) if and only if A satisfies at least one of either φ or ψ
- A satisfies (φ → ψ) if and only if it is not the case that A satisfies φ but not ψ
- A satisfies (φ ↔ ψ) if and only if A satisfies both φ and ψ or satisfies neither one of them
With this definition we can now formalize what it means for a formula φ to be implied by a certain set S of formulas.
Informally this is true if in all worlds that are possible given the set of formulas S the formula φ also holds.
This leads to the following formal definition: We say that a set S of 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 S syntactically entails the well-formed formula φ then S semantically entails φ.
Completeness: If the set of well-formed formulas S semantically entails the well-formed formula φ then S syntactically entails φ.
For the above set of rules this is indeed the case.
Sketch of a soundness proof
(For most logical systems, this is the comparatively "simple" direction of proof)
Notational conventions: Let G be a variable ranging over sets of sentences.
Let A, B and C range over sentences.
For "G syntactically entails A" we write "G proves A".
For "G semantically entails A" we write "G implies A".
We want to show: (A)(G) (if G proves A, then G implies A).
We note that "G proves A" has an inductive definition, and that gives us the immediate resources for demonstrating claims of the form "If G proves A, then ...".
So our proof proceeds by induction.
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.
Thus every system that has modus ponens as an inference rule, and proves the following theorems (including substitutions thereof) is complete:
- p → (¬p → q)
- (p → q) → ((¬p → q) → q)
- p → (q → (p → q))
- p → (¬q → ¬(p → q))
- ¬p → (p → q)
- p → p
- p → (q → p)
- (p → (q → r)) → ((p → q) → (p → r))
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.
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:
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:
We also use the method of the hypothetical syllogism metatheorem as a shorthand for several proof steps.
Another outline for a completeness proof
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
Interpretation of a sentence of truth-functional propositional logic
Main article: Interpretation (logic)
Some consequences of these definitions:
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.
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:
- Axiom THEN-2 may be considered to be a "distributive property of implication with respect to implication."
- Axioms AND-1 and AND-2 correspond to "conjunction elimination". The relation between AND-1 and AND-2 reflects the commutativity of the conjunction operator.
- Axiom AND-3 corresponds to "conjunction introduction."
- Axioms OR-1 and OR-2 correspond to "disjunction introduction." The relation between OR-1 and OR-2 reflects the commutativity of the disjunction operator.
- Axiom NOT-1 corresponds to "reductio ad absurdum."
- Axiom NOT-2 says that "anything can be deduced from a contradiction."
- Axiom NOT-3 is called "tertium non datur" (Latin: "a third is not given") and reflects the semantic valuation of propositional formulas: a formula can have a truth-value of either true or false. There is no third truth-value, at least not in classical logic. Intuitionistic logicians do not accept the axiom NOT-3.
The inference rule is modus ponens:
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:
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:
in fact, the validity of the converse of DT is almost trivial compared to that of DT:
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
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.
is translated in the inequality version of the algebraic framework as
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.
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.
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.
Finding solutions to propositional logic formulas is an NP-complete problem.
Higher logical levels
Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Propositional calculus.