HP Grice, “Logic and Conversation,” Syntax and semantics 3: Speech arts (1975), Cole et al., pp.41-58.
Grice explicates what he takes to be necessary elements of successful conversation, insisting that whatever the difference between formal and natural languages, both adhere to the same elements and thus do not significantly diverge in meaning. For Grice, successful conversation necessitates adherence to an overarching “Cooperative Principle” (or, ‘CP’: communicate that which is required of the conversation such that the purpose of the conversation is achievable) [307R], plus – in certain instances at least – further maxims which fall under the following categories: Quantity (that which pertains to the amount of information communicated); Quality (that which pertains to the veracity of the information communicated); Relation (that which pertains to the relevance of the information communicated); and Manner (that which pertains to the way information is communicated). [308L-R] Central to Grice’s analysis are the notion of implicature and the assumption that communication is an essentially rational endeavor.
Grice begins by describing two opposing views on the respective roles of formal language (i.e., symbolic logic) and natural languages (e.g., English). The views diverge in light of the apparent difference in meaning between certain semantic units in formal language (the logical connectives and quantifiers) and their counterparts in natural language (e.g., ‘some’, the counterpart of the existential quantifier). [305L] Grice calls those who would emphasize the superiority of formal language formalists and those who argue that natural language has certain features which make it impossible to supplant informalists. [305-306] After sketching a generalized version of each position, Grice asserts that the apparent difference in meaning is largely illusory and can be traced to either side’s “inadequate attention to the nature and importance of the conditions governing conversation,” [306R] which he subsequently attempts to explicate.
Grice then gives a provisional list of the maxims associated with CP :
- Maxims associated with Quantity: ‘Communicate information that is as informative as required’; ‘Do not communicate more information than is necessary’.
- Maxims associated with Quality: ‘Do not communicate things you know to be false’; ‘Do not assert that which you have insufficient evidence for’.
- The maxim associated with Relation: ‘Be relevant’.
- Maxims associated with Manner: ‘Avoid obscurity’; ‘Avoid ambiguity’; ‘Be brief’; ‘Be orderly’. 
Grice thinks we properly assume that speakers will adhere to CP and related maxims not just because it is an empirical fact that they do, but because they represent norms that rational agents would adhere to. In other words, rather like economics prescribes certain utility-maximizing behaviors to agents on the assumption that they are rational, CP and related maxims are prescribed to speakers on the assumption that they are rational (and hence want to fulfill the purpose of communication in any given instance). [309L]
Besides building a rationality assumption (which Grice hopes – but does not
demonstrate – is what necessitates CP and related maxims) into successful communication, Grice’s analysis deploys the notion of implicature, or implying additional meanings above and beyond what is said. Grice defines the implicatum as that which is implied and subsequently delimits conventional and conversational implicatures (instances of implication): conventional implicatures are those which can arise solely as a result of the conventional meaning of the words of a sentence, whereas conversational implicatures result necessarily from inherent features of discourse, i.e., they are a function of adherence to CP and related norms (plus certain extralinguistic facts, e.g., context and background knowledge). [307R]
Grice provides a procedure for determining conversational implicature on the basis of CP and related norms: when a speaker says that p implicates q, such implication will be successful assuming: one, the speaker is adhering to CP (at the very least), two, the speaker genuinely thinks p must be so in order for his words to be in accord with CP, and three, the speaker thinks that listeners are aware of the latter point and that they know (or think) he is aware of it as well. 
After providing a catalog of examples meant to demonstrate how the calculus of conversational implicature in general ought to proceed [311-314], Grice expresses what he takes to be several properties of conversational implicature which result from his analysis: one, conversational implicature can be canceled in any given instance, since to provide such implicature necessitates adhering to CP, something no speaker is obligated to do; two, to the extent that the manner of expression plays no role in determining conversational implicature, there will be no alternative way of saying the same thing that does not also have the same implicature; three, the implacatum (implied meaning) of an expression is not a part of the meaning of the expression itself, since such meanings are conventional and implicata are by definition conversational implicatures (and thus determined by CP and related norm adherence), a class of non-conventional implicature; and four , it is possible for indeterminacy to result in instances where more than one explanation of what a speaker is implying adheres to the assumption that they are utilizing CP. [314R-315]