Formal Logic/Sentential Logic/The Sentential Language

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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

$\mathrm {PQ} \,\!$

instead of our

$\mathrm {P} \land \mathrm {Q} \ .\,\!$

${\mbox{Disjunction:}}\ \lor \,\!$

• 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

${\bar {\mathrm {P} }}\ \ {\mbox{and}}\ \ ({\overline {(\mathrm {P} \land \mathrm {Q} )}}\lor \mathrm {R} )\,\!$

instead of

$\lnot \mathrm {P} \ \ {\mbox{and}}\ \ (\lnot (\mathrm {P} \land \mathrm {Q} )\lor \mathrm {R} )\ .\,\!$

${\mbox{Conditional:}}\ \rightarrow \,\!$

• 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.

Notes

(1) An atomic sentence is a sentence consisting of just a single sentence letter. A molecular sentence is a sentence with at least one sentential connective. The main connective of a molecular formula is the connective that governs the entire sentence. Atomic sentences, of course, do not have a main connective.

(2) The ⊃ and ≡ signs for conditional and biconditional are historically older, perhaps a bit more traditional, and definitely occur more commonly in WikiBooks and Wikipedia than our arrow and double arrow. They originate with Alfred North Whitehead and Bertrand Russell in Principia Mathematica. Our arrow and double arrow appear to originate with Alfred Tarski, and may be a bit more popular today than the Whitehead and Russell's ⊃ and ≡.

(3) Sometimes you will see people reading our arrow as implies. This is fairly common in WikiBooks and Wikipedia. However, most logicians prefer to reserve 'implies' for metalinguistic use. They will say:

If P then Q

or even

P arrow Q

They approve of:

'P' implies 'Q'

but will frown on:

P implies Q

Translation

Consider the following English sentences:

If it is raining and Jones is out walking, then Jones has an umbrella.

If it is Tuesday or it is Wednesday, then Jones is out walking.

To render these in ${\mathcal {L_{S}}}\,\!$, we first specify an appropriate English translation for some sentence letters.

$\mathrm {P} \ :\,\!$ It is raining.

$\mathrm {Q} \ :\,\!$ Jones is out walking.

$\mathrm {R} \ :\,\!$ Jones has an umbrella.

$\mathrm {S} \ :\,\!$ It is Tuesday.

$\mathrm {T} \ :\,\!$ It is Wednesday.

We can now partially translate our examples as:

${\mbox{If}}\ \mathrm {P} \ {\mbox{and}}\ \mathrm {Q} ,\ {\mbox{then}}\ \mathrm {R} \,\!$

${\mbox{If}}\ \mathrm {S} \ {\mbox{or}}\ \mathrm {T} ,\ {\mbox{then}}\ \mathrm {Q} \,\!$

Then finish the translation by adding the sentential connectives and parentheses:

$((\mathrm {P} \land \mathrm {Q} )\rightarrow \mathrm {R} )\,\!$

$((\mathrm {S} \lor \mathrm {T} )\rightarrow \mathrm {Q} )\,\!$

Quoting convention

For English expressions, we follow the logical tradition of using single quotes. This allows us to use ' 'It is raining' ' as a quotation of 'It is raining'.

For expressions in ${\mathcal {L_{S}}}\,\!$, it is easier to treat them as self-quoting so that the quotation marks are implicit. Thus we say that the above example translates $\mathrm {S} \rightarrow \mathrm {P} \,\!$ (note the lack of quotes) as 'If it is Tuesday, then It is raining'.

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