Disjunction: ViewIII.doc 1 or every sentences A and B, there is a sentence: A B, which is the disjunction of A and B. he sentences A and B are, respectively, the first disjunct and the second disjunct of. A B. Note: A similar terminology is used for conjunctions. A and B are, respectively, the first conjunct and the second conjunct of A B. Like conjunction, disjunction denotes the operation as well as the resulting sentence; we speak of A B as a disjunction. he semantics of disjunction is given through the truth-table of A B: A B A B In words: he disjunction gets just when at least one of he disjuncts gets (including the case where both get ). It gets just when both disjuncts get.
ViewIII.doc 2 he connectives correspond to ways of forming sentences in natural language: : negating a sentence by means of not, or it is not the case that. : combining sentences by means of and. : combining sentences by means of or, or either or. In the case of negation and conjunction, the truth-values of the compound reflect our natural language intuitions: John went to the movie John did not go the movie John is tall Mary is tall John is tall and Mary is tall In the case of disjunction there are two possible interpretations of the natural language counterpart: the particle or. Inclusive or : If both disjuncts are true the disjunction is true. o Jill is at home or Jill s husband is at home, [given as an answer to the question whether someone is at Jill s home]. Eclusive or : If both disjuncts are true the disjunction is false. o You will either pay the fine, or go to jail, [asserted by a judge as part of the verdict].
ViewIII.doc 3 Of the two, inclusive or is the basic logical connective; eclusive or enjoys no special status and does not merit a notation. Arguably, in many natural language cases, given as eamples of eclusive or, the eclusivity does not derive from the meaning of the logical particle, but from other background facts. herefore in such eamples inclusive or will do as well. If, we consider a connective,, that represents eclusive or [there is no such customary notation], then its truth-table is: A B A B he following equivalence holds: A B (A B) (A B) Intuitively, the left-hand side says: A or B, and not both, which is what eclusive or amounts to. ruth-table checking shows that the right-hand side has indeed the required truth-values: A B A B A B (A B) (A B) (A B)
his can be stated by saying that,, and. ViewIII.doc 4 is epressible in terms of Note that epressibility is a semantic notion. It means that we can translate compounds formed by into compounds formed by the other connectives, which have the same truth-value behavior. One can also easily verify that: A B (A B) ( A B ) Each of and can be epressed in terms of negation and the other connective: 1. A B ( A B) 2. A B ( A B) We can construct a system of sentential logic that uses only the connectives and. In this system we epress disjunction via the equivalence 1. We can still use, but we regard epressions involving as shorthand notations for longer epressions obtained by translating them via 1. hat is, we stipulate: A B = Df ( A B). Similarly, using 2., we can base the system solely on and. he present system uses both and as primitive connectives. he sentences A B and ( A B) are different but logically equivalent. Homework 2.5: Show that can be epressed in terms of,, and (i.e., construct a sentence, using only,, and whose column in the truth-table is the same as that of A B). In fact, one can epress using only and, without using (this is easier than it looks; think intuitively what the sentence can say).,
ViewIII.doc 5 Grouping convention for : he disjunction symbol binds more weakly than the symbols for conjunction and negation. Eample: A B C is read as: ( A) (( B) C) Ways of proving equivalence: o show: it suffices that one show any of the following: (i) If get, then gets, and if gets, then gets. (ii) If get, then gets, and if gets, then gets. (iii) If get, then gets, and if gets, then gets. (iv) If get, then gets, and if gets, then gets Some equivalence claims are easier to prove using one of these ways, rather than another. e.g., (iii) is most suited for proving: A B ( A B)
ViewIII.doc 6 Homework 2.6: ind all the pairs of the following sentences that are logically equivalent (in general). Write your results by filling in a table as in Homework 2.4. (he grouping convention applies.) 1. A B 2. (A B) 3. ( A B) (C C) 4. (A B) ( A B) 5. (B C) (B C) 6. (A B) C 7. (A B) C or each pair that is not logically equivalent in general, find a truth-value assignment to the sentential variables under which the sentences have different truth-values.
ViewIII.doc 7 Homework 2.7 ind all the tautologies and all contradictions among the following sentences. or sentences not listed as a tautology find a truth-value assignment under which the sentence gets. or sentences not listed as a contradiction find a truth-value assignment under which the sentence gets. Note that some sentences might be neither tautologies nor contradictions. [It is better not to construct full truth-tables, but to find directly whether a sentence can get and whether it can get.] 1. (A B) (A B) 2. A ( (A B) (C A)) 3. (A B) ( A B) 4. (A B) ( A B) 5. (A B) (B A) 6. (A B) (A C)