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CONNECTIVES: AND AND OR

By Agus Khairi - Thursday, March 3, 2016
Agus Khairi
NIM. 157835401
Language and Literature Program
Post Graduate Program
The State University of Surabaya

A.      INTRODUCTION
In communication of human being there are lot of utterances are uttered. Utterances do not flow spontaneously but construct in human mind first. In this process some propositions construct in simple and complex form. Simple proposition is single proposition while complex proposition is proposition that consist of single proposition combination.
Combining simple proposition into complex proposition can be done by using connective. Connective functioned as connection between two simple propositions that has truth value. By combining two simple propositions into single proposition make the proposition shorter than before combining them. 
  
B.       DISCUSSION
1.         Logical Connective
1.1    Logic
The simplest definition of logic can be found in most of the dictionaries around the world. One of the dictionary-based definitions of logic according to Cambridge Dictionaryis “a particular way of thinking, especially one which is reasonable and based on good judgment”. This definition, since it is based on the dictionary, will not satisfy us to understand the logic especially when we discuss the term logic in semantics point of view. Therefore, we need to find more comprehend definition of what logic is. According to Hurford, Heasley, and Smith (2007, p. 150) logic deals with meanings in a language system, not with actual behavior. Furthermore, Hurford et al. (2007, p. 143) argues that logicgives rules for calculation which can be applied to get a rational being from goals and assumptions to action.


1.2    Connective
In English there are some words are used to connect or combine propositions. Those words are called connectives, is that word and and or. Hurford et al. (2007, p. 165) argue that “connectives provide a way of joining simple propositions to form complex proposition.” Complex proposition consist of ideas as the result of joining simple propositions. In other word logical connective is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the sense of the compound sentence produced depends only on the original sentences.
In terms of propositional logic, “connectives use to express the combination of propositions” (Frawley, 1992, p. 26). Logical connectives are symbolized by V ('or'),- ('not'), &  ('and'),~ ('if, then'). Those symbols are used in propositional notation that express complex proposition. Exactly connectives such as and, or, and but do not have clear lexical meaning. These kinds of words are called function words because they specify grammatical relations and have little or no semantic content (Fromkin, Rodman, & Hyams, 2014, p. 35).
Various English words and word pairs express logical connectives, and some of them are synonymous. Examples (with the name of the relationship in parentheses) are:
·      "and" (conjunction)
·      "or" (disjunction)
·      "either...or" (exclusive disjunction)
·      "implies" (implication)
·      "if...then" (implication)
·      "if and only if" (equivalence)
·      "only if" (implication)
·      "just in case" (equivalence)
·      "but" (conjunction)
·      "however" (conjunction)
·      "not both" (NAND)
"neither...nor" (NOR)
1.3    Conjunction
Conjunction is one of the items in grammar, a body rules specifying how meaning are created in English. Conjunction is a word that links words, phrases, and clauses. It can be said that conjunctions were “linking“ or “joining“ words which joined together various things. According to Leech & Svartvik in Leung (2005, p. 4) “conjunctions  play  an  important  role  in  discourse  as  they  are  used  as  coordination  to conjoin “different grammatical units: clauses, clause elements, words”. Moreover, Leech & Svartvik in Leung (2005, p. 4) “conjunctions are the most common way of coordination and the most frequently used  and  central  conjunctions  are and, or, and but.” The function of conjunctions are to link parts of clauses (clause elements): noun phrases, verb phrases, complements, and adverbial.  

2.         Logical Connective and
Connective and symbolized with & in notation proposition. It is called conjunction.
a.    Function of Connective and
In Hurford et al. (2007, p. 65) stated that “Any number of individual well formed formulae can be placed in a sequence with the symbol & between each adjacent pair in the sequence and the result is The result is a complex well formed formula.
Here are the examples:
Robert is fat
Deny is thin
r FAT
d THIN
Those are simple proposition and can be combined with conjunction as below:
r FAT & d THIN
for more than two propositions can be formed as below:
Robert meet Deny
Ane call Deny
Jack text Deny
r MEET d
a CALL d
j TEXT d

(r MEET d) & (a CALL d) & (j TEXT d)
Parentheses use to make the structure of formula clear and avoid ambiguity. Parentheses are absolutely unnecessary in this case, we can avoid it depend on necessary to make the formula clear. The most important rule that should be considered is the & only used to join the whole propositions not for predicates and names (Hurford et al., 2007, p. 165).
b.   Joining Compound Propositions
 Propositions in English are not only in simple form but sometimes in compound propositions. Compound proposition is formed by combine two propositions in one complex proposition. For instance, proposition Andrew and Jay are diligent can be formed as a DILIGENT & j DILIGENT. It cannot be formed as a & j DILIGENT because the rules is only to combine proposition not predicates or names.
c.    Commutativity of Conjunction
According to Hurford et al. (2007, p. 167) “commutativity is conjunction of  two propositions in a given order one can infer the conjunction of the same two propositions in the opposite order.” By this commutativity we can infer one simple proposition with other simple proposition that formed one complex proposition and formed new complex proposition that equivalent with the previous proposition. Here is the example:
 Ages invite Marya and Ages invite Rose
a INVITE m & a INVITE r
this proposition equivalent with:
a INVITE r & a INVITE m

From the example above can be formed the formula or the rule of commutativity conjunction as follow:
(Premiss)
(Conclusion)
p and q are variable ranging over propositions, that is, p and q stand for any propositions one may think of. 
In this rule premiss and conclusion can be completely change whereas (q & p) as premiss and (p & q) as conclusion. The diagram can be seen as below:
  
(Premiss)
(Conclusion)

This complete interchangeability of premiss and conclusion amounts to their logical equivalence. The relationship between formulae and propositions can be seen in a way somewhat parallel to the relationship between names and their referents. We shall say that equivalent formulae actually stand for the same proposition, just as dierent (though equivalent) names stand for the same individual (Hurford et al., 2007, p. 167)

3.         Logical Connective or
Connective or is symbolized with V (Latin: vel meaning or). This is called disjunction.
a.    Function of Connective or
Logical connective V corresponding to English or is functioned to join the simple proposition into complex propositions with the rule “any number of well formed formulae can be placed in a sequence with the symbol V between each adjacent pair in the sequence: the result is a complex well formed formula.” (Hurford et al., 2007, p. 169) 
Here are the examples:
Dorothy saw Bill
Dorothy saw Alan
d SEE b
d SEE a

(d SEE b) V (d SEE a)
Dorothy saw Bill or Dorothy saw Alan.
b.   Joining Compound Propositions
In case of compound proposition as Dorothy saw Bill or Alan can be formed as (d SEE b) V (d SEE a) Dorothy saw Bill or Dorothy saw Alan. it must be remembered that only proposition can be joined by connective V, it cannot join the predicates or names.   
c.    Commutativity of Disjunction
The rule of commutativity of disjunction is same as the rule commutativity conjunction. The difference is in commutativity of disjunction involve V. By this commutativity we can infer one simple proposition with other simple proposition that formed one complex proposition and formed new complex proposition that equivalent with the previous proposition. Here is the example:
Dorothy or Bernard saw Rose
d SEE r V b SEE r
this formula uquivalent with:
 b SEE r V d SEE r
The diagram can be seen as below:
  
(Premiss)
(Conclusion)

p and q are variable ranging over propositions, that is, p and q stand for any propositions one may think of. 
In this rule premiss and conclusion can be completely change whereas (q V p) as premiss and (p V q) as conclusion. The diagram can be seen as below:
  
(Premiss)
(Conclusion)

4.         Resolving Ambiguity
Ambiguity can be resolved by placing the word either in one of two positions.
Example:
Alice went to Birmingham and She met Cyril or She called on David
The sentence grammatically is ambiguous.
The two meanings can be expressed by using brackets to group together the propositions differently in each logical formula.
a GO b & (a MEET c V a CALL-ON d)
or
(a GO b & a MEET C) V a CALL-ON d
Here are the ways to resolve it:
Either Alice went to Birmingham and She met Cyril or She called on David.
(a GO b & a MEET c) V a CALL-ON d
Alice went to Birmingham and either She met Cyril or She called on David.
a GO b & (a MEET c V a CALLED-ON d)
5.         Inference
Here are the rules of of inference stated by (Hurford et al., 2007, p. 172):
Rules of inference, which can be seen as contributing to a description of the meanings of connectives such as & and V, can be interpreted in terms of truth. A rule of inference states, in e ect, that a situation in which the premiss (or premisses) is (are) true is also a situation in which the conclusion is true. But rules of inference do not explicitly operate with the terms ‘true’ and ‘false’, and do not, either explicitly or implicitly, state any relationships between propositions which happen to be false.
A full account of the contribution that a connective such as & or V makes to the truth or falsehood of a complex proposition can be given in the form of a truth table.

6.         Truth Values Table
a.    Truth Table for “&”
p
q
p & q
T
T
T
T
F
F
F
T
F
F
F
F

The table above explains about truth table for & where p and q are variables standing for any propositions. The rows in the left-hand column list all possible combinations of the values T (for true) and F (for false) that can be assigned to a pair of propositions. The corresponding values in the right-hand column are the values of the formula p & q for those combinations of values.(Hurford et al., 2007, p. 173)
From the table can be concluded that the value p & q of will be False if one of the value that is p and q is False. If both of them are True so the value of p & q will be True and if both of them are False so the value of p & q are False.  
b.   Truth Table for “V”
p
q
p V q
T
T
T
T
F
T
F
T
T
F
F
F

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 (Hurford et al., 2007, p. 174)
The table can be explained as follow, if one of the value p and q are true or false, the value of p V q are True. If both of them are True the value of p V q are True and will be False if both of them are False.
    
c.    Truth Value of Complex Propositions
As said before that proposition is not only simple form but also in complex form. Even though complex proposition, it can be made the truth value based on the proposition. Of course the truth value of complex proposition is different with simple proposition. In Hurford et al. (2007, p. 175) explained as below:
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 aected in various ways where the roots connect with each other, and eventually arriving at the trunk of the tree.

Base on the explanation above can be elaborate how truth values of complex proposition are formed as given example below:

Taken from (Hurford et al., 2007, p. 175)
  
C.      CONCLUSION
Logical connectives are used to joining the propositions into complex proposition. The connective & corresponding English and called conjunction and connective V corresponding English or called disjunction. These connectives are used to combine proposition and used to make formula of compound proposition into well formed formula. The results of joining propositions by using logical connective are complex propositions which have truth value based on each proposition formed them. 

D.      REFERENCE
______________________. (Ed.) (2008) Cambridge Advance Learner's Dictionary. Cambridge: Cambridge University Press.
Frawley, W. (1992). Linguistic Semantics. New York: Routkedge.
Fromkin, V., Rodman, R., & Hyams, N. (2014). An Introduction to Language. United State Of America: Cengage Learning.
Hurford, J. R., Heasley, B., & Smith, M. B. (2007). Semantics: A Course Book. Cambridge: Cambridge University Press.

Leung, C. (2005). A comparison of the use of major English conjunctions by American and Hong Kong university students (Using the HKUST corpus, HKBU corpus and the ICLE corpus of American English). (Bachelor), Lunds Universitet.