Unlike modal logics, which are extensions of classical (propositional and predicate) logic, three-valued logics are its alternatives, as they abandon one of its basic principles – that of bivalence:
| Principle of bivalence: | Every statement is true or false. |
In the context of discussing three-valued logic, we must be careful not to consider the expressions “false” and “not true” to be synonymous (as we are accustomed to because of the principle of bivalence). “True” (T) and “false” (F) are the standard truth values. The sentence “p is false” says of p that it has the value “false” (F), while “p is not true” denies of p that it has the value “true” (T) but the latter does not imply that it has the value “false” (F) (it may have a third truth value.) The above principle states that each statement has one of the two standard truth values (T or F) and thus denies the existence of a third truth value. On the contrary, in three-valued logic it is assumed that some statements are neither true nor false but have a third truth value – hence its name.
If, as is natural, the “or” in the formulation of the principle of bivalence is interpreted as an exclusive “or” (each statement is true or false but not both), the principle becomes equivalent to Aristotle’s laws of excluded middle and non-contradiction. If, on the other hand, the “or” is interpreted as inclusive, the principle becomes equivalent only to the law of excluded middle. To see why, let us first formulate these two Aristotle’s laws.
| Law of non-contradiction: | A statement (α) and its negation (¬α) cannot be both true. |
| Law of excluded middle: | For each statement (α), either it or its negation (¬α) is true. |
The principle of bivalence is a consequence of these two laws due to the following. If, in accordance with the law of excluded middle, α is true, then this confirms the principle. Generally, because of the meaning of negation („not“), if a statement has the value T or the value F, its negation has the value F or the value T, respectively, and vice versa. A statement can be neither true nor false only if its negation is neither true nor false, and vice versa. So, in the other possible case that is in accordance with the law of excluded middle, when the true statement is ¬α, α will be false, which again confirms the principle of bivalence. It turns out that if the law of excluded middle is valid, then α (an arbitrary statement) is true or false (or both). The exclusive sense of “or” comes from the law of non-contradiction: if α is both true and false, then (because α is false) ¬α will be true, which means that α and ¬α are both true (contrary to the law of non-contradiction). The converse, the fact that the two laws of Aristotle are entailed by the principle of bivalence (or that only the law of excluded middle is entailed if “or” is interpreted as inclusive), is easy to see – α and ¬α are statements, so (according to the principle) each of them must have the value T or F and, since the value of ¬α is the opposite to that of α, one will be true and the other false.
Therefore, by affirming the two laws, Aristotle indirectly affirms the principle of bivalence. Despite of that, he is also the first to question that principle and thus the law of excluded middle (without questioning the law of non-contradiction). In On interpretation (Aristotle, 1991) he says that some statements about the future are neither true nor false. Such, according to him, is the sentence “A sea-battle will be fought tomorrow” on the day of its utterance. The reason he gives is that if it were true or false today, the future would seem to be determined – it would be determined today whether there will be a sea-battle tomorrow. If the statement is true, whatever we do, there will be a battle tomorrow; if it is false, there will be none. But because the future depends on our actions, it does not seem to be determined. Thus, (contrary to the principle of bivalence and the law of excluded middle) Aristotle concludes that at the time of its utterance this sentence is neither true nor false.
Aristotle’s argument is not irrefutable. In a sense, the statement about tomorrow’s sea-battle could be true or false today and yet (again in a sense) the future could be not determined by the following reason. Let us mean by “possible world” a certain overall development of the situation in the world in time. On the day of its utterance, the statement is true in every possible world in which there is a sea-battle the next day, and it is false in every possible world in which there is not. However, which among the infinitely many possible worlds will be realized and will become actual depends (of course not entirely) on our actions. By acting in a certain way, we help to realize a certain possible world. In it (and in each of the other unrealized possible worlds) the future is determined, and the statement is true or false in the day of the utterance. In the whole reality in which we live, however, the future is not determined because in it it is not determined which possible world will be realized. The future is determined only in the individual possible worlds, understood as certain overall developments of the situation in time, but it is not determined in the reality that includes all such possible developments of the situation. While acting, we live in this whole reality.
By the way, in the passage of On Interpretation, Aristotle denies the principle of bivalence, while at the same time attempting to preserve the law of excluded middle. The attempt does not seem convincing, however. He says that although the sentence “Tomorrow there will be a sea-battle” is neither true nor false today, the sentence “Tomorrow there will be or there will be no sea-battle” is true (and even necessarily so) today because, no matter how things turn out, tomorrow there will either be or there will be no sea-battle. This is an attempt to preserve the law for excluded middle because the latter can also be formulated as follows: “α or non-α (α∨¬α) is always true”. However, the overall position is not consistent because (due to the meaning of “or”) if a disjunction is true, at least one of the disjunction must be true and therefore (as either α or non-α is true) α must be true or false. If the principle of bivalence is denied, the law of excluded middle must be denied also, and vice versa; one cannot reject one and preserve the other.
Inspired by the passage of Aristotle in question, in the early 20th century the Polish logician Lukasiewicz created the first three-valued logic (Lukasiewicz, 1920). The principle of bivalence does not apply in it, so in addition to T and F the propositional letters can have a third truth value, which we will denote by “#”. In accordance with Aristotle’s understanding of tomorrow’s sea-battle, it signifies “not yet determined” (whether it is true or false).
In Lukasiewicz’s three-valued logic, logical connectives have alternative truth tables, in which the third truth value is present. These tables are almost identical to the truth tables used later by the American logician Kleene (Kleene, 1938). The difference is only one row in the conditional table and (as a result) one row in the biconditional table. Below are given the tables of Kleene. We will see later what the differences are between Kleene’s and Lukasiewicz’s tables of conditional and biconditional and what the reason is for them.
The truth table of negation is as follows:
| α | ¬α |
| T | F |
| # | # |
| F | T |
When α has the value T or F the table does not differ from the standard table in propositional logic. When the truth value of α is undetermined (#), the value of ¬α is also undetermined (#). If it is still uncertain whether there will be a naval battle tomorrow, it will be uncertain whether there will not be a naval battle tomorrow.
The tables for disjunction, conjunction and conditional are as follows:
| α | β | α ∨ β | α ∧ β | α → β |
| T | T | T | T | T |
| T | # | T | # | # |
| T | F | T | F | F |
| # | T | T | # | T |
| # | # | # | # | # |
| # | F | # | F | # |
| F | T | T | F | T |
| F | # | # | F | T |
| F | F | F | F | T |
The first two columns contain all combinations of three truth values for two sentences. The combinations are nine (each value for the first sentence is combined with each for the second, 3x3 = 9). When α and β have the value T or the value F (first, third, seventh, and ninth row), the truth values of disjunction, conjunction and conditional are the same as in propositional logic. The interesting cases are when α or β has the value #.
When α has the value T and β has the value # (second row), the disjunction between them will be true because the truth of one of the disjuncts is sufficient to guarantee the truth of the whole disjunction (in truth-value analysis T∨β was equivalent to T). Interpreting # in the spirit of Aristotle as (still) undetermined (whether true or false), then, no matter how β will be determined, the disjunction will be true because of the truth of α. For the same values of α and β, the conjunction between them will have the value # because when α is true, α∧β has the same truth value as β (in truth-value analysis T∧β was equivalent to β), so the truth value of β will be undetermined. For the same values of α and β, the conditional between them will have the value # because when α is true, the truth value of α→β is the same as that of β (in truth-value analysis T→β was equivalent to β).
When α has the value # and β has the value T (fourth row), the truth values of the disjunction and the conjunction between them will be the same as in the previous case – the values of α and β are swapped and the order of the members of a disjunction or conjunction does not affect their truth values. However, the conditional between α and β will be true in this case because its consequent β is true and this is sufficient to ensure the truth of α→β regardless of the truth value of α (in the truth-value analysis α→T was equivalent to T).
When the truth values of both α and β are undetermined (fifth row), the truth values of the disjunction, conjunction, and conditional between them are also undetermined. The truth values of α and β can be determined in any possible combination of T and F in the future, which allows for the three compound sentences to be determined as true as well as false, i.e. they are undetermined now. This is the case where the tables of Kleene and Lukasiewicz differ (the difference in the biconditional table is a consequence of that difference). According to Lukasiewicz’s table, when both sentences have the value #, the value of α→β is not undetermined, but truth. The reason for this choice is not semantic, i.e. it is not related to the meaning of “if…, then…”. Below we will see what it is related to.
When α has the value # and β has the value F (sixth row), the disjunction between them has the value # because when one of the disjuncts is false, the truth value of the disjunction is the same as the value of the other disjunct (in truth-value analysis α∨F was equivalent to α). The conjunction between α and β in this case is false because when one of the conjuncts is false, the conjunction is false regardless of the value of the other conjunct (in the truth-value analysis α∧F was equivalent to F). The biconditional between α and β has the value # because when the consequent of a biconditional is false, the truth value of the biconditional is the opposite of the truth value of the antecedent (in truth-value analysis α→F was equivalent to ¬α) and the latter is undetermined (above we saw that when α has the value #, ¬α also has value #, and vice versa).
When α is false and β is undetermined (next to the last row), the truth values of the disjunction and conjunction between them are the same as in the previous case we considered – the values of α and β are switched. The biconditional in this case is true because a biconditional is true when its antecedent is false, regardless of the truth value of its consequent (in truth-value analysis F→α was equivalent to T).
Once we have the truth tables of conjunction and conditional, we can derive the truth table of biconditional by constructing the table of (α→β)∧(β→α). The last scheme and α↔β should be logically equivalent, so their tables (the last columns) should be the same. In general, constructing a truth table of an arbitrary formula in three-valued logic does not differ from constructing an ordinary truth table, except that, as there are more truth values, the tables have more rows.
| α ↔ β | ||||
| α | β | α→β | β→α | (α→β)∧(β→α) |
| T | T | T | T | T |
| T | # | # | T | # |
| T | F | F | T | F |
| # | T | T | # | # |
| # | # | # T | # T | # T |
| # | F | # | T | # |
| F | T | T | F | F |
| F | # | T | # | # |
| F | Н | T | T | T |
In the fifth row, which corresponds to the case when both α and β have the value #, in the column under α→β, β→α and (α→β)∧(β→α) there are two values. The first is according to Kleene’s three-valued logic, the second according to Lukasiewicz’s.
In both classical and three-valued propositional logic (whether in the version of Kleene or Lukasiewicz), conjunction and disjunction can be expressed through each other with the help of negation. The scheme α∧β is logically equivalent to ¬(¬α∨¬β) and α∨β to ¬(¬α∧¬β)1. If we construct the three-valued tables of ¬(¬α∨¬β) and ¬(¬α∧¬β), we will see that they are the same as the elementary three-valued tables of α∧β and α∨β, respectively.
Apart from Aristotle’s argument concerning tomorrow’s naval battle, other reasons have been suggested for rejecting the principle of bivalence and adopting three-valued logic. One such reason is related to singular terms that do not denote anything (such as “Pegasus” or “the present king of France”). In 3.6 Identity we presented the classical solution to the logical problem with these terms – Russell’s theory of definite descriptions. In this theory, each (atomic) sentence that contains a singular term denoting a non-existing thing is considered false because it is interpreted as a conjunction of sentences one of which states that such a thing exists. For example, the sentence “The present king of France is bald” turns out to be false because, according to Russell’s analysis, it contains the sentence “There is a present king of France”. Not everyone agrees with this analysis. Among the positions of those who disagree, the most well-known is that of Strawson (Strawson, 1950). According to him, Russell presents the usage of definite descriptions in a distorted way, maintaining that statements involving them contain statements asserting existence and uniqueness. Strawson’s contention is that, for example, in the statement “The present king of France is bald”, the existence and the uniqueness of the king of France are not asserted but presupposed. As presuppositions, they are not parts of the statement but are necessary conditions for its proper (meaningful) use. Since in this case these necessary conditions are not fulfilled (France is a republic and has no king), the statement is neither true nor false, but meaningless. By „a presupposition of a statement“ Strawson means a sentence (or a state of affairs) that, unlike the statement of which it is a presupposition, is not affirmed, but whose truth is a necessary condition for the meaningfulness of the statement. Statements containing definite descriptions have at least two such presuppositions – an existential one (“There is a present king of France”) and a presupposition of uniqueness (“There are no more than one present kings of France”). Formally, a presupposition of a statement should be defined as follows:
| The sentence (state of affairs) p is a presupposition of the sentence q if and only if when p is false, q is neither true nor false (but meaningless). |
Thus, the third truth value (#) in Strawson’s theory goes to statements with false presuppositions.
Abandonment of the principle of bivalence and, accordingly, adoption of three-valued logic is also proposed as a solution to various paradoxes. Such is, for example, the sorites paradox. It is caused by some everyday language general terms (such as “bald” or “heap”), the meaning of which is not strictly defined and therefore in borderline cases it is difficult to say whether something falls or does not fall under them. It is clear that a person with a lot of hair on his head (say 100,000 hairs) is not bald and that a lot of grains of sand in one place (again say 100,000) is a heap; as it is clear that a person with one hair on his head is bald and that one grain is not a heap. However, if we start to tear hairs one by one from the head of a man who is nor bald or (preferably) remove grains from a heap, sooner or later we will reach a point where it will be difficult to say whether the person is bald and whether the grains still form a heap.
Sorites paradox2 is very old. Formally, the logical problem with it is the following. These two premises seem true:
| 100,000 grains of sand (in one place) are a heap. |
| If we remove a grain from a heap, it will continue to be a heap. |
Clearly form them follows the sentence “99,999 grains of sand are a heap”, which therefore also has to be true. From the last sentence and second premise follows the sentence “99,998 grains of sand are a heap”, etc. Continuing, we will finally derive the truth of the sentence “One grain of sand is a pile” (and similarly the truth of “A man without a single hair on his head is not bald”).
The abandonment of the principle of bivalence (accordingly, the adoption of three-valued logic) has been suggested as a solution to the sorites paradox and other paradoxes because of the following. The idea is the problem sentences, here the sentences corresponding to situations where it cannot be said whether something falls under the term or not, to be declared neither true nor false. Assume hypothetically that such a borderline case is the sentence “1000 grains of sand are a heap”. If it is neither true nor untrue, (because it is not true) from it and the second premise above, we can no longer deduce the truth of “999 grains of sand are a heap”. So, the sequence of inferences is interrupted.
Indeed, it remains unclear where the boundary is at which sentences cease to be true and begin to be neither true nor false. If we say that this can always be determined arbitrarily, as we define it to be 1000 in our example, then why not simply define arbitrarily that 1000 or more grains are a heap and less than 1000 are not? Then the second premise (“A heap minus a grain is a heap”) ceases to be true because 1000 grains are a heap and 999 are not, which again breaks the chain of inferences and solves the paradox. But anyway, what is important for us is why the abandonment of the principle of bivalence and the usage of three-valued logic could be suggested as a solution to some paradoxes.
The suggested reasons for the abandonment of the principle of bivalence that we have considered so far were of two different kinds. For Aristotle, the sentences he considers neither true nor false (“There will be a sea-battle tomorrow”) are correct and meaningful. Their truth value is undetermined now, but they will become true or false at some time in the future. (The same is true, buy the way, of the proposed use of three-valued logic in quantum mechanics (Reichenbach, 1944). For example, in the imaginary situation of Schrödinger’s cat, at some time it is not objectively determined whether the sentence “The cat is alive” is true or false, but at the time of opening the box the sentence becomes true or false.) On the contrary, for Strawson the statements he considers neither true nor false (“The present king of France is bald”) are semantically incorrect senseless utterances. They violate certain rules of meaning, which is why they are not and will never be true or false. The same is true of the suggested solutions to certain paradoxes by regarding some sentences (“1000 grains are a heap”) as neither true nor false. These sentences are not and will never be true or false because in the borderline cases the predicate in question (“heap”, “bald”) is inapplicable. Thus, we have two fundamentally different interpretations of the third truth value #: 1) as undetermined and 2) as meaningless.
The three-valued logics of Kleene and Lukasiewicz, whose truth tables we have considered, is based on the first interpretation. The second interpretation requires changes in the tables because when a meaningful sentence is connected through a logical connective with a meaningless sentence, the meaninglessness of the latter, so to speak, “infects” the resulting compound sentence and it also becomes meaningless. For example, although “Socrates is a philosopher” is a meaningful and true sentence, and for a disjunction to be true, it is enough at least one of its members to be true, we would rather consider the sentence in the form of disjunction “Socrates is a philosopher or rsld f kdsfy” not to be true, but to be meaningless.
The truth tables below are associated with Bochvar’s name (Бочвар, 1938) and are based on the second interpretation of # – as meaningless, not as (still) undetermined (whether true or false). (The table of negation is the same as in the three-valued logics of Kleene and Lukasiewic.)
| α | β | α ∨ β | α ∧ β | α → β |
| T | T | T | T | T |
| T | # | # | # | # |
| T | F | T | F | F |
| # | T | # | # | # |
| # | # | # | # | # |
| # | F | # | # | # |
| F | T | T | F | T |
| F | # | # | # | # |
| F | F | F | F | T |
These tables for the logical connectives are simpler. When α and β have the values T or F, the truth values of the disjunction, conjunction and conditional between them are the same as in classical propositional logic. When α or β (or both) have the value # (meaningless), it “infects” the compound sentence and it also has the value #.
Unlike classical (propositional and predicate) logic, three-valued logic and modal logic have problems that cast a shadow over their logical status. Let us look at some of the problems in three-valued logic.
In three-valued logic, there are no tautologies. Even the formal versions of the laws of non-contradiction (¬(α∧¬α)) and excluded middle (α∨¬α) are not tautologies. When all propositional letters in a formula have the value #, the whole formula also has the value #, no matter how these letters are further connected by logical connectives. This is so because the negation of a formula that has the value # also has the value #, and the conjunction, disjunction, conditional, and biconditional between formulas that have the value # also have the value #. So, whatever its logical structure, every formula will have the value # when its propositional letters have that value. Therefore, no formula can be a tautology (always true). As an illustration, below are given the truth tables of the formal versions of the law of non-contradiction and the law of excluded middle:
| α | ¬α | α ∧ ¬α | ¬(α ∧ ¬α) | α ∨ ¬α |
| T | F | F | T | T |
| # | # | # | # | # |
| F | T | F | T | T |
In fact, of the three-valued logics considered, tautologies are missing in the logics of Kleene and Bochvar, but not in the logic of Lukasiewic. In the latter, due to the assumption that a conditional has the value T when its antecedent and consequent have the value #, some formulas containing “→” or “↔” become tautologies. The problem is that the only reason for this assumption is the existence of tautologies. It is unjustified from a semantic point of view (from the point of view of the meaning of “if–then”, “and”, “not”, etc.). Why, for example, should the statement “If there will be a sea-battle tomorrow, then there will be a sea-battle tomorrow” (“p→p”) be true at the time of the utterance and the statement “There will be a sea-battle tomorrow or there will be no sea-battle tomorrow” (“p∨¬p”) be undetermined, provided that what they say is equally trivial? Because of this arbitrary assumption about conditional’s truth values, in Lukasiewic’s logic conditional and biconditional are indefinable through the other logical connectives although the mutual definability of logical connectives (“if–then”, “and”, “or”, “not” etc.) is contained in their intuitive meanings. The idea of three-valued logic is to reject the principle of bivalence, not to change the meanings of the logical words. In the logics of Kleene and Bochvar, the logical connectives are mutually definable just as they are in propositional logic, but there are no tautologies in them. So, it seems that the more consistent position is to accept as a fact that the nature of three-valued logic is such that it excludes tautologies.
We mentioned above that although Aristotle abandoned the principle of bivalence with respect to sentences about future events, he could not accept the fact that then the formal version of the law of excluded middle (α∨¬α) ceases to be a tautology (this is a fact including in Lukasiewic’s three-valued logic). In the same place where he criticizes the principle of bivalence, Aristotle defends the logical validity of the scheme in question, saying that the sentence “There will be a sea-battle tomorrow or there will be no sea-battle tomorrow” is not only true now, but true with necessity (that it is a tautology in modern terms). The problem is that if α∨¬α is always true, the meaning of “or” implies that one of the two (α or ¬α) is always true and at the same time it is assumed that there are cases when both are not such.
The lack of tautologies is a more serious problem for the variant of three-valued logic, in which the third truth value is interpreted as (still) undetermined (the variant of Kleene and Lukasiewic). Although the truth values of “There will be a sea-battle tomorrow” (α) and its negation (¬α) have not yet been determined, when they are determined, no matter how they are determined, one will have the value T, which will make the disjunction α∨¬α true. So, α∨¬α can only be determined as true. Why should we now (as opposed to tomorrow) consider its truth value to be undetermined? In the same way, the fact that α and ¬α are undetermined with respect to their truth values does not mean that the contradiction that occurs when they are connected with conjunction is also undetermined in truth value – obviously we do not have to wait until tomorrow to realize that fighting and not fighting a sea battle cannot be a fact (neither now nor tomorrow).
The lack of tautologies is less problematic for the second interpretation of the third truth value (as meaningless rather than as undetermined), because we could agree that sentences such as “Borogoves are mimsy or not mimsy” and “Borogoves are mimsy and not mimsy” are meaningless (i.e., that they have the value # rather than T and F, respectively). However, there are other problems with this interpretation of the third truth value.
Let us return to Strawson’s theory of presuppositions and the statement “The king of France is bald”. This statement has two presuppositions – of existence and uniqueness – but let us assume for simplicity that there is just one – that of existence. The problem we will point out does not depend on the number of presuppositions. Consider the following statement
| If there is a king of France, then the king of France is bald. |
The sentence “There is a king of France” is a presupposition of the statement “The king of France is bald” but it is not a presupposition of the above statement because it is an explicit part of the latter. Therefore, the whole if-then-statement does not have a false presupposition that requires it to have the value #. Then, since the antecedent of the conditional (“There is a king of France”) is false, the whole statement must be true. However, it has the value # in Bochvar’s three-valued logic, as the consequent (“The king of France is bald”) has a false presupposition (“There is a king of France”), which makes its truth value #, and the latter infects the entire conditional. Here, the three-valued logic of Kleene would be more adequate because it would give the whole statement the value T, which seems correct, but we have already seen that Bochvar’s logic is the one that corresponds to the way the third truth value is understood in Strawson’s theory – false presuppositions make statements meaningless, not (yet) undetermined (in truth value).
There are also problems with three-valued predicate logic. The use of three-valued logic for the sorites paradox falls within the realm of the latter. To show one of them, let us first see how universal and existential sentences get their truth values in three-valued logic. In principle, a universal sentence (one whose symbolic representation begins with a universal quantifier) resembles a conjunction, and an existential sentence resembles disjunction. If, for example, F symbolizes the predicate “…is a student” and the universe of discourse consists of three people in a room denoted by a, b and c, then the sentence “All are students” (∀xFx) will be equivalent to the conjunction “a, b and c are students” (Fa ∧ Fb ∧ Fc). Accordingly, the sentence “Someone is a student” (∃xFx) will be equivalent to the disjunction “a, b or c is a student” (Fa ∨ Fb ∨ Fc). In fact, if the universes of discourse were always finite, we could do without quantifiers, because instead of universal or existential sentences we could always use conjunctions or disjunctions. However, if the universe of discourse consists of infinitely many things (as in arithmetic, for example, where there are infinitely many numbers), the universal and existential sentences correspond to conjunctions and disjunctions with infinitely many members, and infinite sentences cannot exist. Anyway, this connection between the universal quantifier and conjunction, and the existential quantifier and disjunction, shows how the truth values of ∀xFx and ∃xFx are determined in three-valued logic. For both three-valued logics (Kleene’s and Bochvar’s), ∀xFx will be true only if F is true of all things in the universe of discourse, because for both logics a conjunction is true only if all its members are true. ∀xFx will be false for Kleene’s logic (considering the way conjunctions get their truth values there), if F is false of at least one thing of the universe of discourse D, and will have the value # if F is false of nothing in D but there is at least one thing of which it has the value #. For Bochvar’s logic, ∀xFx will be false only if F is false of each thing in D. If there is even one thing of which F has the value #, ∀xFx will have the value #. Considering how a disjunction obtains its truth values in both three-valued logics, we see that for both ∃xFx will be false only if F is false of each thing in D. In Kleene’s logic, ∃xFx will be true if F is true of at least one thing, whether or not there are things of which F has the value #. If there are no things in D of which F is true but there is at least one thing of which F has the value #, ∃xFx will have the value #. For Bochvar’s logic, ∃xFx will be true only if F is true of at least one thing and there are no things of which F has the value #; otherwise (if F is true of some things and has the value # of others) ∃xFx will have the value #.
Sentences such as “Every heap is a heap” and “Every bald man is bald” are obviously true, but they have the value # for both Kleene’s and Bochvar’s three-valued logic. To see why, let us symbolize “heap” with F, “bald” with G and “man” with H. Then the first sentence is symbolized by
| (1) | ∀x(Fx → Fx) |
and the second one with
| (2) | ∀x[(Gx ∧ Hx) → Gx] |
If a is a collection of grains of which it cannot be said whether it is a heap or not (i.e., if it belongs to the boundary cases for this term) and b is a person of whom it is equally unclear whether is bald or not, then Fa→Fa and (Gb∧Hb)→Gb will have the value # for both three-valued logics, because the antecedent and the consequent of both conditionals have the value #. We have seen above that for both Kleene’s and Bochvar’s logics, for a universal sentence to be true, the predicate coinciding with the scope of the universal quantifier (Fx→Fx in (1) and (Gx∧Hx)→Gx in (2)) must be true of everything. In this case it is not so, as a and b are things of which the predicates have the value #, not T. For Bochvar’s logic, the latter automatically means (see previous paragraph) that (1) and (2) have the value #. Considering that there are no things of which the predicates have the value F, we see that the two sentences have the value # also for Kleene’s logic.
| (1) For each formula, construct two truth tables – one for Kleene’s three-valued logic and one for Bochvar’s. Actually, construct a single table but write two values in the cells where the two tables differ. |
| 1) | p → ¬p |
| 2) | (p→¬q) ∧ p |
| 3) | ¬p ∨ (q∧¬q) |
| 4) | (p∧q) → (¬p∨¬q) |
| 5) | (p∨q) ↔ (¬p→q) |
| 6) | p → (q→(¬q→p)) |