Formal Logic/Sentential Logic/Goals

← Start of Sentential Logic

↑ Sentential Logic

The Sentential Language →

Goals

Sentential logic

Sentential logic attempts to capture certain logical features of natural languages. In particular, it covers truth-functional connections for sentences. Its formal language specifically recognizes the sentential connections

it is not the case that _____

[both] _____ and _____

[either] _____ or _____

[either] _____ or _____, but not both

if _____, then _____

_____ if and only if _____

The blanks are to be filled with true or false sentences. The truth-value (truth or falsity) of the entire sentence can be computed from the truth values of the blank-filling components.

The listed sentential connections do not include all possible truth-functions. However, all possible truth functions can be built up using them.

Issues

Already we have tacitly taken a position in ongoing controversy. Some questions already raised by the seemingly innocuous beginning above are listed.

• Should we admit into our logic only sentences that are true or false? Multi-valued logics admit a greater range of sentences.

• Are the connections listed above truly truth functional? Should we admit connections that are not truth functional sentences into our logic?

• What should logic take as its truth-bearers (objects that are true or false)? The two leading contenders today are sentences and propositions.

• Sentences. These consist of a string of words and perhaps punctuation. The sentence 'The cat is on the mat' consists of six elements: 'the', 'cat', 'is', 'on', another 'the', and 'mat'.

• Propositions. These are the meanings of sentences. They are what is expressed by a sentence or what someone says when he utters a sentence. The proposition that the cat is on the mat consists of three elements: a cat, a mat, and the on-ness relation.

Elsewhere in Wikibooks and Wikipedia, you will see the name 'Propositional Logic' (or rather 'Propositional Calculus', see below) and the treatment of propositions much more often than you will see the name 'Sentential Logic' and the treatment of sentences. Our choice here represents the contributor's view as to which position is more popular among current logicians and what you are most likely to see in standard textbooks on the subject. Considerations as to whether the popular view is actually correct are not taken up here.

Some authors will use talk about statements instead of sentences. Most (but not all) such authors you are likely to encounter take statements to be a subset of sentences, namely those sentences that are either true or false. This use of 'statement' does not represent a third position in the controversy, but rather places such authors in the sentences camp. (However, other—particularly older—uses of 'statement' may well place its authors in a third camp.)

Sometimes you will see 'calculus' rather than 'logic' such as in 'Sentential Calculus' or 'Propositional Calculus' as opposed to 'Sentential Logic' or 'Propositional Logic'. While the choice between 'sentential' and 'propositional' is substantive and philosophical, the choice between 'logic' and 'calculus' is merely stylistic.