Sentential Logic

# 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 _____
_____ and _____
Either _____ or _____
_____ or _____ (or both)
if _____, then _____
_____ if and only if _____

The blanks are to be filled with statements that can be true or false. For example, "it is raining today" or "it will snow tomorrow". Whether the final sentence is true or false is entirely determined on whether the filled statements are true or false. For example, if it is raining today, but it will not snow tomorrow, then it is true to say that "Either it is raining today or it will snow tomorrow". On the other hand, it is false to say "it is raining today and it will snow tomorrow", since it won't snow tomorrow.

"Whether a statement is true or false" is called the truth value in logician slang. Thus "Either it is raining today or it is not raining today" has a truth value of true and "it is raining today and it is not raining today" has truth value of false.

Note that the above listed sentential connections do not include all possible truth value combinations. For example, there is no connection that is true when both sub-statements are true, both sub-statements are false or the first sub-statement is true while the other is false, and that is false else. However, you can combine the above connections together to build any truth combination of any number of sub-statements.

## 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.

# The Sentential Language

This page informally describes our sentential language which we name ${\mathcal {L_{S}}}\,\!$ . A more formal description will be given in Formal Syntax and Formal Semantics

## Language components

### Sentence letters

Sentences in ${\mathcal {L_{S}}}\,\!$ are represented as sentence letters, which are single letters such as $\mathrm {P} ,\ \mathrm {Q} ,\ \mathrm {R} ,$ and so on. Some texts restrict these to lower case letters, and others restrict them to capital letters. We will use capital letters.

Intuitively, we can think of sentence letters as English sentences that are either true or false. Thus, $\mathrm {P} \,\!$ may translate as 'The Earth is a planet' (which is true), or 'The moon is made of green cheese' (which is false). But $\mathrm {P} \,\!$ may not translate as 'Great ideas sleep furiously' because such a sentence is neither true nor false. Translations between English and ${\mathcal {L_{S}}}\,\!$ work best if they are restricted to timelessly true or false present tense sentences in the indicative mood. You will see in the translation section below that we do not always follow that advice, wherein we present sentences whose truth or falsity is not timeless.

### Sentential connectives

Sentential connectives are special symbols in Sentential Logic that represent truth functional relations. They are used to build larger sentences from smaller sentences. The truth or falsity of the larger sentence can then be computed from the truth or falsity of the smaller ones.

${\mbox{Conjunction:}}\ \land \,\!$ • Translates to English as 'and'.
• $\mathrm {P} \land \mathrm {Q} \,\!$ is called a conjunction and $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are its conjuncts.
• $\mathrm {P} \land \mathrm {Q} \,\!$ is true if both $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true—and is false otherwise.
• Some authors use an & (ampersand), (heavy dot) or juxtaposition. In the last case, an author would write
$\mathrm {PQ} \,\!$ $\mathrm {P} \land \mathrm {Q} \ .\,\!$ ${\mbox{Disjunction:}}\ \lor \,\!$ • Translates to English as 'or'.
• $\mathrm {P} \lor \mathrm {Q} \,\!$ is called a disjunction and $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are its disjuncts.
• $\mathrm {P} \lor \mathrm {Q} \,\!$ is true if at least one of $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true—is false otherwise.
• Some authors may use a vertical stroke: |. However, this comes from computer languages rather than logicians' usage. Logicians normally reserve the vertical stroke for nand (alternative denial). When used as nand, it is called the Sheffer stroke.

${\mbox{Negation:}}\ \lnot \,\!$ • Translates to English as 'it is not the case that' but is normally read 'not'.
• $\lnot \mathrm {P} \,\!$ is called a negation.
• $\lnot \mathrm {P} \,\!$ is true if $\mathrm {P} \,\!$ is false—and is false otherwise.
• Some authors use ~ (tilde) or . Some authors use an overline, for example writing
${\bar {\mathrm {P} }}\ \ {\mbox{and}}\ \ ({\overline {(\mathrm {P} \land \mathrm {Q} )}}\lor \mathrm {R} )\,\!$ $\lnot \mathrm {P} \ \ {\mbox{and}}\ \ (\lnot (\mathrm {P} \land \mathrm {Q} )\lor \mathrm {R} )\ .\,\!$ ${\mbox{Conditional:}}\ \rightarrow \,\!$ • Translates to English as 'if...then' but is often read 'arrow'.
• $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is called a conditional. Its antecedent is $\mathrm {P} \,\!$ and its consequent is $\mathrm {Q} \,\!$ .
• $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is false if $\mathrm {P} \,\!$ is true and $\mathrm {Q} \,\!$ is false—and true otherwise.
• By that definition, $\mathrm {P} \rightarrow \mathrm {Q} \,\!$ is equivalent to $(\lnot \mathrm {P} )\lor \mathrm {Q} \,\!$ • Some authors use (hook).

${\mbox{Biconditional:}}\ \leftrightarrow \,\!$ • Translates to English as 'if and only if'
• $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is called a biconditional.
• $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is true if $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ both are true or both are false—and false otherwise.
• By that definition, $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$ is equivalent to the more verbose $(\mathrm {P} \land \mathrm {Q} )\lor ((\lnot \mathrm {P} )\land (\lnot \mathrm {Q} ))\,\!$ . It is also equivalent to $(\mathrm {P} \rightarrow \mathrm {Q} )\land (\mathrm {Q} \rightarrow \mathrm {P} )\,\!$ , the conjunction of two conditionals where in the second conditional the antecedent and consequent are reversed from the first.
• Some authors use .

### Grouping

Parentheses $(\,\!$ and $)\,\!$ are used for grouping. Thus

$((\mathrm {P} \land \mathrm {Q} )\rightarrow \mathrm {R} )\,\!$ $(\mathrm {P} \land (\mathrm {Q} \rightarrow \mathrm {R} ))\,\!$ are two different and distinct sentences. Each negation, conjunction, disjunction, conditional, and biconditionals gets a single pair or parentheses.