This leads essentially to a logic without proper disjunction.
For example, they defined a disjunction as true if and only if exactly one disjunct is true (the modern “exclusive” disjunction).
We can think of disjunction as a means of entertaining different alternatives.
Now the main difference between conjunction and disjunction in a dialogue is the player (ρ−1(!)
Although each blindspot is inaccessible, a disjunction of blindspots is normally not a blindspot.
These languages have two words for interrogative disjunction and standard disjunction.
We may be able to specify the relevant term only in principle in cases where the disjunction is infinitely long.
But the modal operator must be taken to apply to the disjunction as a whole as in (3b) and not to each disjunct as in (3c).
Universal (existential) quantification is treated as equivalent to a possibly infinite conjunction (disjunction) of propositions.
Avicenna refers to the first type as the real (or genuine) disjunction (munfaṣila ḥaqīqiyya) and to the other two types as unreal (ġayr ḥaqīqī) disjunctions.
Purely hypothetical syllogisms are those in which the combination of the premises involve only hypotheticals (conditional-conditional; conditional-disjunction; disjunction-disjunction).
And so some find it difficult to make sense of general logical laws, e.g. the law that any two falsehoods form a false disjunction, since the disjunction may not get uttered or written (Quine 1969, 143).
In turn, the existence of an operation of disjunction is ensured if Modal Criterion and B3 are accepted—in particular under the view of facts as sets of worlds; in that case, union is a disjunction operation.
In his ‘Principles of Arithmetic Presented by a New Method’ (1889), Peano introduces propositional connectives in the modern sense (an implication, negation, conjunction, disjunction, and a biconditional) and propositional constants (a verum and a falsum).
If the antecedent is a disjunction of atomic propositions, or a disjunction of conjunctions of atomic propositions, then the consequent must be true when every possible intervention or set of interventions described by the antecedent is performed.
Independently, in Beziau 2004 it was observed that by putting together the sequent rules for classical conjunction and the rules for classical disjunction, the resulting sequent calculus will (unexpectedly) prove the distributivity between conjunction and disjunction.
The combinations considered in this case all involve at least an unreal disjunction, which may be paired with a real disjunction, the negation of a real disjunction or with another unreal disjunction, where the shared part may be affirmative or negative.
Furthermore, if conjunction and disjunction satisfy the distribution axiom mentioned in the previous section, they can be modelled straightforwardly too: a conjunction is true at a point just when both conjuncts are true at that point, and a disjunction is true at a point just when at least one disjunct is true there.
In the special case of disjunction at the sentence level, the Boolean operator boils down to the classical propositional operator on truth values (see section 7.1 for a recent account which identifies disjunction with the join operator in a Heyting algebra, which at the sentence level yields a non-classical (inquisitive) propositional operator).
Humberstone (2000a, 368) challenges rejectivism, by asking the rejectivists to “show how the claim for the conceptual priority of rejection over negation is any more plausible than the corresponding claim for the conceptual priority of alterjection over disjunction—or indeed, ambi-assertion over conjunction,” where alterjection (ambi-assertion) is the supposedly primitive speech act the linguistic embodiment of which is disjunction (conjunction).
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Humberstone 2000a 368 challenges rejectivism by asking the rejectivists to show how the claim for the conceptual priority of rejection over negation is any more plausible than the corresponding claim for the conceptual priority of alterjection over disjunction—or indeed ambi-assertion over conjunction where alterjection ambi-assertion is the supposedly primitive speech act the linguistic embodiment of which is disjunction conjunction