Formal Logic/Sentential Logic/Translations

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Translations

The page The Sentential Language gave a very brief look at translation between English and ${\mathcal {L_{S}}}\,\!$. We look at this in more detail here.

English sentential connectives

In the following discussion, we will assume the following assignment of English sentences to ${\mathcal {L_{S}}}\,\!$ sentence letters:

$\mathrm {P} \,:\,\!$  2 is a prime number.

$\mathrm {Q} \,:\,\!$  2 is an even number.

$\mathrm {R} \,:\,\!$  3 is an even number.

Not

The canonical translation of $\lnot \,\!$ into English is 'it is not the case that'. Given the assignment above,

(1)    $\lnot \mathrm {P} \,\!$

translates as

It is not the case that 2 is a prime number.

But we usually express negation in English simply by 'not' or by adding the contraction 'n't' to the end of a word. Thus (1) can also translate either of:

2 is not a prime number.

2 isn't a prime number.

If

The canonical translation of $\rightarrow \,\!$ into English is 'if ... then ...'. Thus

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

translates into English as

(3)    If 2 is a prime number, then 2 is an even number.

Objections have been raised to the canonical translation, and our example may illustrate the problem. It may seem odd to count (3) as true; however, our semantic rules does indeed count (2) as true (because both $\mathrm {P} \,\!$ and $\mathrm {Q} \,\!$ are true). We might expect that, if a conditional and its antecedent are true, the consequent is true because the antecedent is. Perhaps we expect a general rule

(4)    if x is a prime number, then x is an even number

to be true—but this rule is clearly false. In any case, we often expect the truth of the antecedent (if it is indeed true) to be somehow relevant to the truth of the conclusion (if that is indeed true). (2) is an exception to the usual relevance of a number being prime to a number being even.

The $\rightarrow \,\!$ conditional of ${\mathcal {L_{S}}}\,\!$ is called the material conditional in contrast to strict conditional or counterfactual conditional. Relevance logic attempts to define a conditional which meets these objections. See also the Stanford Encyclopedia of Philosophy entry on relevance logic.

It is generally accepted today that not all aspects of an expression's linguistic use are part of its linguistic meaning. Some have suggested that the objections to reading 'if' as a material conditional are based on conversational implicature and so not based on the meaning of 'if'. See the Stanford Encyclopedia of Philosophy entry on implicature for more information. As much as a simplifying assumption than anything else, we will adopt this point of view. We can also point out in our defense that translations need not be exact to be useful. Even if our simplifying assumption is incorrect, $\rightarrow \,\!$ is still the closest expression we have in ${\mathcal {L_{S}}}\,\!$ to 'if'. It should also be noted that, in mathematical statements and proofs, mathematicians always use 'if' as a material conditional. They accept (2) and (3) as translations of each other and do not find it odd to count (3) as true.

'If' can occur at the beginning of the conditional or in the middle. The 'then' can be missing. Thus both of the following (in addition to (3)) translate as (2).

If 2 is a prime number, 2 is an even number.

2 is an even number if 2 is a prime number.

Implies

We do not translate 'implies' into ${\mathcal {L_{S}}}\,\!$. In particular, we reject

2 is a prime number implies 2 is an even number.

as grammatically ill-formed and therefore not translatable as (2). See the Implication section of Validity for more details.

Only if

The English

(5)    2 is a prime number only if 2 is an even number

is equivalent to the English

If 2 is not an even number, then 2 is not a prime number.

This, in turn, translates into ${\mathcal {L_{S}}}\,\!$ as

(6)    $\lnot \mathrm {Q} \rightarrow \lnot \mathrm {P} \,\!$

We saw at Conditionals section of Properties of Sentential Connectives that (6) is equivalent to

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

Many logic books give this as the preferred translation of (5) into ${\mathcal {L_{S}}}\,\!$. This allows the convenient rule ''if' always introduces an antecedent while 'only if' always introduces a consequent'.

Like 'if', 'only if' can appear in either the first or middle position of a conditional. (5) is equivalent to

Only if 2 is an even number, is 2 a prime number.

Provided that

'Provided that'—and similar expressions such as 'given that' and 'assuming that'—can be use equivalently with 'if'. Thus each of the following translate into ${\mathcal {L_{S}}}\,\!$ as (2).

2 is an even number provided that 2 is a prime number.

2 is an even number assuming that 2 is a prime number.

Provided that 2 is a prime number, 2 is an even number.

Prefixing 'provided that' with 'only' works the same as prefixing 'if' with only. Thus each of the following translate into ${\mathcal {L_{S}}}\,\!$ as (6) or (7).

2 is a prime number only provided that 2 is an even number.

2 is a prime number only assuming that 2 is an even number.

Only provided that 2 is an even number, is 2 a prime number.

Or

The canonical translation of $\lor \,\!$ into English is '[either] ... or ...' (where the 'either' is optional). Thus

(8)    $\mathrm {P} \lor \mathrm {Q} \,\!$

translates into English as

(9)    2 is a prime number or 2 is an even number

or

Either 2 is a prime number or 2 is an even number.

We saw at the Interdefinability of connectives section of Expressibility that (8) is equivalent to

$\lnot \mathrm {P} \rightarrow \mathrm {Q} \,\!$

Just as there were objections to understanding 'if' as $\rightarrow \,\!$, there are similar objections to understanding 'or' as $\lor \,\!$. We will again make the simplifying assumption that we can ignore these objections.

The English 'or' has both an inclusive and—somewhat controversially—an exclusive use. The inclusive or is true when at least one disjunct is true; the exclusive or is true when exactly one disjunct is true. The $\lor \,\!$ operator matches the inclusive use. The inclusive use becomes especially apparent in negations. If President Bush promises not to invade Iran or North Korea, not even the best Republican spin doctors will claim he can keep his promise by invading both. The exclusive reading of (9) translates into ${\mathcal {L_{S}}}\,\!$ as

$(\mathrm {P} \lor \mathrm {Q} )\land \lnot (\mathrm {P} \land \mathrm {Q} )\,\!$

or more simply (and less intuitively) as

$\mathrm {P} \leftrightarrow \lnot \mathrm {Q} \,\!$

In English, telescoping is possible with 'or'. Thus, (8) translates

2 is either a prime number or an even number.

Similarly,

$\mathrm {Q} \lor \mathrm {R} \,\!$

translates

2 or 3 is an even number.

Unless

'Unless' has the same meaning as 'if not'. Thus

(10)    $\lnot \mathrm {Q} \rightarrow \mathrm {P} \,\!$

translates

(11)    2 is a prime number unless 2 is an even number

and

(12)    Unless 2 is an even number, 2 is a prime number.

We saw at the Interdefinability of connectives section of Expressibility that (10) is equivalent to (8). Many logic books give (8) as the preferred translation of (11) or (12) into ${\mathcal {L_{S}}}\,\!$.

Nor

At the Joint denial section of Expressibility, we temporarily added $\downarrow \,\!$ to ${\mathcal {L_{S}}}\,\!$ as the connective for joint denial. If we had that connective still available to us, we could translate

Neither 2 is a prime number nor 2 is an even number

as

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

However, since $\downarrow \,\!$ is not really in the vocabulary of ${\mathcal {L_{S}}}\,\!$, we need to paraphrase. Either of the following will do:

(13)    $\lnot (\mathrm {P} \lor \mathrm {Q} )\,\!$.

(14)    $(\lnot \mathrm {P} \land \lnot \mathrm {Q} )\,\!$.

The same telescoping applies as with 'or'.

2 is neither a prime number nor an even number

translates into ${\mathcal {L_{S}}}\,\!$ as either (13) or (14). Similarly,

Neither 2 nor 3 is an even number

translates as either of

$\lnot (\mathrm {Q} \lor \mathrm {R} )\,\!$.

$(\lnot \mathrm {Q} \land \lnot \mathrm {R} )\,\!$.

And

The canonical translation of $\land \,\!$ into English is '[both] ... and ...' (where the 'both' is optional'). Thus

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

translates into English as

2 is a prime number and 2 is an even number

or

Both 2 is a prime number and 2 is an even number.

Our translation of 'and' as $\land \,\!$ is not particularly controversial. However, 'and' is sometimes used to convey temporal order. The two sentences

She got married and got pregnant.

She got pregnant and got married.

are generally heard rather differently.

'And' has the same telescoping as 'or'.

2 is both a prime number and an even number

translates into ${\mathcal {L_{S}}}\,\!$ as (15)

Both 2 and 3 are even numbers

translates as

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

If and only if

The canonical translation of $\leftrightarrow \,\!$ into English is '... if and only if ...'. Thus

(16)    $\mathrm {P} \leftrightarrow \mathrm {Q} \,\!$

translates into English as

2 is a prime number if and only if 2 is an even number.

The English sentence

(17)    2 is a prime number if and only if 2 is an even number

is a shortened form of

2 is a prime number if 2 is an even number, and 2 is a prime number only if 2 is an even number

which translates as

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

or more concisely as the equivalent formula

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

We saw at the Interdefinability of connectives section of Expressibility that (18) is equivalent to (16). Issues concerning the material versus non-material interpretations of 'if' apply to 'if and only if' as well.

Iff

Mathematicians and sometimes others use 'iff' as an abbreviated form of 'if and only if'. So

2 is a prime number iff 2 is an even number

abbreviates (17) and translates as (16).