2.1.1. Classical Propositional Logic
The alphabet of classical propositional logic, , consists of (i) a countable set of propositional constants, (ii) the truth constants “⊤” (“truth”) and “⊥” (“falsehood”), (iii) the primitive logical connectives “¬” (“negation”) and “” (“implication”), and (iv) the punctuation symbols “(“and”)”. The class of formulas is built from elements of the alphabet of in the usual inductive fashion, whereby the propositional constants and truth constants constitute the atomic formulas. Formulas which are non-atomic are referred to as composite formulas.
Besides the primitive connectives ¬ and , we also make use of the standard connectives “” (“disjunction”), “” (“conjunction”), and “” (“equivalence”), defined in the usual way: , , and .
In what follows, we will use the letters “P”, “Q”, “R”, … (possibly appended with subscripts and/or with primes) or words from everyday English to refer to propositional constants, and use the letters “A”, “B”, “C”, …(again possibly appended with subscripts and/or with primes) to refer to arbitrary formulas (distinct such letters need not represent distinct formulas).
A (two-valued) interpretation is a mapping I assigning each propositional constant from an element from the set , whose elements are referred to as truth values, where represents truth and represents falsity. The truth value of a composite formula A under an interpretation I, denoted by , is defined in terms of the usual truth-table conditions of classical propositional logic. Accordingly, a formula A is true under I iff , and false under I if . If A is true under I, then I is said to be a model of A, and if A is false under I, then I is a countermodel of A. If I is a countermodel of A, then we also say that I refutes A. We call A satisfiable (in ) if it has some model, and falsifiable (in ), or refutable (in ), if it has some countermodel. Moreover, A is unsatisfiable (in ) if it has no model. Finally, A is a tautology, symbolically , if it is true in every interpretation, and refutable (in ), symbolically , otherwise.
A set of formulas is also referred to as a theory. An interpretation I is a model of a theory T if I is a model of all elements of T, otherwise I is a countermodel of T. If a theory T has a model, then T is satisfiable, and if T has a countermodel, then T is falsifiable. A theory is unsatisfiable if it has no model.
A formula A is a valid consequence of a theory T (in ), or T entails A (in ), in symbols , iff A is true in any model of T. Two formulas, A and B, are (logically) equivalent (in ) iff . In general, two theories are (logically) equivalent iff they have the same models.
As customary, we will write expressions like “” as “”, and similarly for finite sets of form instead of a singleton set .
We denote by
the usual derivability operator of
with respect to some fixed sound and complete Hilbert-type system. The deductive closure operator
is given by:
is a theory. A theory T
is deductively closed
. As well known, the operator
enjoys the following properties (for any theory T
implies . (“Monotonicity”.)
If A is not derivable from T, then we indicate this by writing . Later on, we will define proof systems axiomatising formulas that are not derivable from a given theory. Such axiom systems are accordingly also referred to as complementary calculi as they axiomatise the complement of the provable formulas of a logic.
We say that a theory T is consistent iff there is a formula A such that . Clearly, T is consistent iff it is satisfiable. Moreover, a formula A is consistent with T iff .
2.1.2. Łukasiewicz’s Three-Valued Logic
We now turn to the three-valued logic of Łukasiewicz [11
] for the propositional case, henceforth denoted by
. Our presentation follows the one given by Radzikowska [9
The alphabet of consists of the alphabet of along with the additional truth constant ⊔ (“undetermined”). Again, we assume as a countable set of propositional constants. The class of formulas of is built similarly to the formulas of , except that ⊔ is counted as an additional atomic formula.
A difference to the syntax of the logic
concerns the defined connectives; while conjunction,
, and material equivalence,
, are defined as in propositional logic, disjunction in
is defined differently:
Furthermore, there are also additional unary defined operators, viz.
the connective “
” (“weak negation”), given by
the unary operators “
” (“certainty operator”) and “
” (“possibility operator”), defined by
which, according to Łukasiewicz [11
], were first formalised in 1921 by Tarski; and
the operator “
”, given by
Intuitively, expresses that A is certain, whilst means that A is possible. These operators will be used subsequently to distinguish between certain knowledge and defeasible conclusions. Furthermore, expresses that A is contingent or modally indifferent.
A (three-valued) interpretation is a mapping m assigning to each propositional constant from an element from . Here, besides the truth values and , the symbol represents a truth value standing for “undetermined” or “indeterminacy”. As usual, is the truth value of P under m, where now P is true under an interpretation m if , false under m if , and has undetermined truth value if .
The truth value, , of an arbitrary formula A under an interpretation m is given subject to the following conditions:
If , then .
If , then .
If , then .
If A is an atomic formula, then .
, for some formula B
, for some formulas C
is determined according to the truth tables given in Figure 1
(there, the corresponding truth conditions for the defined connectives are also given).
If , then A is true under m, if , then A is undetermined under m, and if , then A is false under m. If A is true under m, then m is a model of A. If A is true in every interpretation, then A is valid (in ), written .
Clearly, the classically valid principle of tertium non datur, i.e., the law of excluded middle, , as well as the corresponding law of non-contradiction, , are not valid in . However, their three-valued pendants, viz., the principle of quartum non datur, i.e., the law of excluded fourth, , and the corresponding extended non-contradiction principle, , are valid in .
In classical logic, two formulas are logically equivalent if and only if, they have the same models, where logical equivalence between formulas A
is defined by the condition that
holds. However, such a relation between logical equivalence and equality of models does not hold in general in the three-valued logic case. Indeed, following Radzikowska [9
], let us define that two formulas A
are strongly equivalent
. That is, A
are strongly equivalent iff, for any three-valued interpretation m
. Furthermore, let us call A
and B equivalent
, iff A
have the same models. Clearly, strong equivalence implies equivalence, but in general not vice versa. For instance, P
, for an atom P
, are equivalent but not strongly equivalent. In addition, strong equivalence is an equivalence relation (i.e., reflexive, symmetric, and transitive) and enjoys a substitution principle, similar to the one of classical logic, i.e., if a formula
contains a subformula A
is the result of substituting at least one occurrence of A
by a formula B
Let us also note some strong equivalences which hold in :
, for and .
, for .
The notion of a theory in is defined as in , i.e., a theory is a set of formulas. Likewise, the notion of a model or of a countermodel of a theory, and of a theory being satisfiable, falsifiable (or refutable), or unsatisfiable are defined in mutatis mutandis as in . A theory T is said to entail a formula A (in ), or A is a valid consequence of T (in ), symbolically , iff every model (in ) of T is also a model (in ) of A.
Sound and complete Hilbert-style axiomatisations of the logic
can be readily found in the literature [47
]; the first one was introduced by Wajsberg in 1931 [49
]. We write
has a derivation (in some fixed Hilbert-style calculus) from T
. As well, the deductive closure operator
is given by
is a theory. The notions of a theory being deductively closed
and of being consistent
, as well as of a formula being consistent with
a theory, are defined similarly as in
. Moreover, the properties of inflationaryness, idempotency, and monotonicity hold for
, and consistency of a theory T
is equivalent to the satisfiability of T
While in we have the well-known properties that (i) iff is inconsistent and (ii) iff (the “only if” part of the latter is generally referred to as the deduction theorem), for a theory T and formulas A and B, in sight variations thereof hold:
Let T be a theory, and A and B formulas.
iff is inconsistent (in ).
Note that, as a consequence, the consistency of a formula A with a theory T implies the consistency of the theory , but it does not necessarily imply the consistency of . For instance, is consistent with , for an atomic formula P, so is consistent, but is not.
Furthermore, although in it always holds that implies , it is the converse direction (i.e., the classical version of the deduction theorem) that fails in general.