Grátis

367 pág.

# semantics

- Denunciar

## Pré-visualização | Página 50 de 50

we have said, is not exactly equivalent to the logical connective &, whose meaning is deﬁned solely in terms of truth. It is interesting to ask about the meaning of the related English word but. Practice (1) Make a quick, intuitive judgement here. Do the English words and and but mean the same thing? Yes / No (2) Now we will work out a truth table for but. Remember that we are only concerned with the TRUTH in given situations of the sentences we cite. (a) In a situation where Henry died and Terry resigned are both true, is the sentence Henry died but Terry resigned true or false? T / F (b) Where Henry died is true, but Terry resigned is false, is Henry died but Terry resigned true or false? T / F (c) Where Henry died is false, but Terry resigned is true, is Henry died but Terry resigned true or false? T / F (d) Where Henry died and Terry resigned are both false, is Henry died but Terry resigned true or false? T / F (3) On the basis of your answers to questions (a)–(d) above, ﬁll in the values (T or F, as appropriate) in the right-hand column of the truth table for but below. p q p but q T T T F F T F F (4) Does the truth table that you have ﬁlled out diﬀer in any way from the truth table given for the logical connective &? Yes / No (5) Considered only from the point of view of their eﬀect on the truth of complex sentences containing them, do and and but diﬀer in meaning? Yes / No Feedback (1) No (2) (a) T (b) F (c) F (d) F (3) T, F, F, F (reading down the column) (4) No (5) No Comment The conclusion reached about and and but need not be perplexing, so long as one bears in mind that truth-conditional meaning, of the kind described in UNIT 14 Connectives: and and or 173 PART FOUR Logic 174 truth tables, may be only a part of the meaning, in a wider sense, of a word. From the point of view of truth alone, and and but make the same contribution to meaning; but they diﬀer in other aspects of meaning. In particular, the word but is preferred when the speaker wishes to indicate some kind of contrast between the two propositions involved. If the second proposition, but not the ﬁrst, gives an unexpected piece of information, for instance, the use of but, rather than and, is appropriate. The logical connective & captures the truth-conditional aspects of the meanings of both and and but. A truth table can also be given for V, corresponding approximately to English or (or either . . . or). We get you to construct one in practice below. Practice (1) In a situation where Henry died and Terry resigned are both true, is (Either) Henry died or Terry resigned true or false? T / F (2) Where Henry died is true, but Terry resigned is false, is (Either) Henry died or Terry resigned true or false? T / F (3) Where Henry died is false, but Terry resigned is true, is (Either) Henry died or Terry resigned true or false? T / F (4) Where Henry died and Terry resigned are both false, is (Either) Henry died or Terry resigned true or false? T / F (5) Using your answers to (1)–(4) above as a basis, ﬁll in the values (T or F, as appropriate) in the right-hand column of the table below. p q p V q T T T F F T F F Feedback (1) T (2) T (3) T (4) F (5) T, T, T, F (reading down the column) Comment The values T and F which appear in truth tables are the same values as those assigned to simple propositions in relation to the situations in the world which they describe (recall the previous unit). In the case of simple propositions, the values T and F ‘come from’ the world. In the case of complex propositions with connectives such as & and V, the combinations of the values of the component simple propositions are ‘looked up’ in the appropriate truth table, and the value of the whole complex proposition (either T or F) is arrived at. Thus in the case of complex propositions, their truth values ‘come from’ the truth values of their constituent simple propositions. UNIT 14 Connectives: and and or 175 Example Metaphorically, the truth value of a complex proposition is like the trunk of a tree whose roots reach down into the world. Truth values ﬂow from the world upwards through the roots, being aﬀected in various ways where the roots connect with each other, and eventually arriving at the trunk of the tree. The following practices will give you additional experience in working with complex propositions. Try to work through them completely before checking the answers in the feedback sections. ((j BEHIND e) & (r SMILE)) V (e STAND) T V ((j BEHIND e) & (r SMILE)) T & (j BEHIND e) T r SMILE T e STAND F Practice Using the world shown in the picture above as the source of truth values for simple propositions, and calculating the values of complex propositions from truth tables, ﬁll in the values T and F, as appropriate, in the boxes in the diagrams below. PART FOUR Logic 176 (1) ((e STAND) & (s PULL j)) V (e BEHIND j) V (e STAND) & (s PULLj) & e STAND s PULL j e BEHIND j ↑ ↑ ↑ (2) (In this example, the boxes are on the left-hand side of their propositions, simply to make the diagram neater.) (j BETWEEN r e) & ((e SIT) V (j STAND)) & (e SIT) V (j STAND) V j BETWEEN r e e SIT j STAND ↑ ↑ ↑ Feedback F T F T (1) F T F (2) T T T Practice Use the map overleaf as a ‘world’ to determine the truth values of the formulae (some simple, some complex) listed. We have used unabbreviated logical names, for convenience. Assume that the predicates have their usual English meanings. Boldface is not used in this practice. (1) mumbai IN India T / F (2) karachi IN iran T / F (3) tibet BORDER afghanistan T / F (4) (tashkent IN iran) & (kabul IN tibet) T / F (5) (karachi IN pakistan) V (iran BORDER india) T / F (6) (karachi SOUTH tashkent) & ((tashkent EAST tehran) & (nepal BORDER tibet)) T / F (7) ((chennai IN india) V (tehran IN tibet)) V (mumbai IN iran) T / F (8) ((chennai IN iran) V (tibet WEST iran)) & (kabul IN afghanistan) T / F Feedback (1) T (2) F (3) T (4) F (5) T (6) T (7) T (8) F Comment Truth values of complex propositions are derived from the truth values of their constituent simple propositions. This is an example of what has been called the compositionality of meaning. Definition The thesis of COMPOSITIONALITY of meaning is that the meaning of any expression is a function of the meanings of the parts of which it is composed. Comment The truth tables we have given for & and V are in fact functions of the kind mentioned in this deﬁnition. Summary The logical connectives & (corresponding to English and and but) and V (roughly English or) are used to form complex propositional formulae by connecting simple propositional formulae. Rules of inference can be given involving these connectives, and they can be deﬁned by means of truth tables. Unit 14 Study Guide and Exercises Directions After you have read Unit 14 you should be able to tackle the following questions to test your understanding of the main ideas raised in the unit. UNIT 14 Connectives: and and or 177 I R A N PA KI ST A N I N D I A Tehran Karachi Mumbai Chennai N E P A L Kabul Tashkent UZBEKISTAN A FG HA NI ST AN TIBET PART FOUR Logic 178 1 You should understand these terms and concepts from this unit: logical conjunction (with &) logical disjunction (with V) rules of inference commutativity of conjunction commutativity of disjunction compositionality of meaning truth tables 2 Translate the following into logical notation using either & or V (or both as required). a Fred arrived and Mary left b Either Fred is a Republican or Mary is a Democrat c Max and Erma are happy d Bill visited Jim and Mary e Max saw Mary or Fred f Jim briefed Jane and he saw Mary or he met Fred (disambiguate by means of diﬀerent bracketing) g Either Mary briefed Fred and she saw Jane or she saw David (is this ambiguous?) h Mary briefed Fred and either she saw Jane or she saw David (is this ambiguous?) 3 What can be connected with the logical connectives