**'and' and Analytic Validity**

In
order to understand A.N Prior's use of the neologism 'tonk', we
firstly need to understand the way in which he takes the connective
of conjunction – namely, 'and'.

Prior
makes the counterintuitive claim that “any statement whatever may
be inferred, in an analytically valid way, from any other” (130).

*Prima facie*, that raises a question:*Does that mean that any statement with any content can be inferred from any other with any content?*
The
word 'and' (in this logical sense at least) is understood by seeing
its use in statements or propositions.

We
start off with two propositions: P and Q; which begin as separate
entities in this context. Prior argues that we can “infer”
P-and-Q from statements P and Q. The former symbolism, “P-and-Q”
(i.e., with hyphens) signifies the conjunction; whereas “P and Q”
(i.e., without hyphens) signifies two statements taken separately.
However, we can infer

*P-and-Q*from any P and Q. That is, from P on its own, and Q on its own, we can infer*P-and-Q. In other words,*statements P and Q can be joined together to form a compound statement.
Two
questions which we can raise now are. One, do the truth-values of
both P and Q matter at this juncture? Two, do the contents of both P
and Q matter at this point?

In
basic logic, the answer to both questions is 'no'. It's primarily
because of this that some of the counterintuitive elements of this
account become apparent.

For
example, Prior says that “for any pair of statements P and Q, there
is always a statement R such that given P and given Q we can infer R”
(129). The important word to note here is “any” (as in “for any
pair of statements P and Q”). This leads to the conclusion (just
mentioned) that the truth-values and/or contents of both P and Q
don't matter within the logical context of defining the connective
'and'. It's partly because of this that Prior tells us that “we can
infer R” from P and Q. Thus:

(P)
The sun is in the solar system.

__(P & Q) Therefore the sun is in the solar system and peas are green.__

(R/Q)
Therefore peas are green.

All
those statements are true; yet they have unconnected contents and a
conclusion which doesn't follow from (the content) of the premises.
Similarly with two false premises. Thus:

(P)
The sun is in the bathroom.

__(P & Q) Therefore the sun is in the bathroom and peas are blue.__

R/Q)
Therefore peas are blue.

It's
because of this irrelevance of contents and truth-values that R will
follow from any P and any Q.

Thus
it's no surprise that Prior also says that “given R we can infer P
and can also infer Q”. As an this example:

__(R/P) Peas are green__

(P
and Q) Therefore peas are green and the sun is in the solar system.

The
difference here is that a compound statement (i.e.,

*P-and-Q*) is derived from an atomic statement (i.e., R/P). (Except, in this instance, R should be P and*P and Q*should be R.) Nonetheless, contents and truth-values still don't matter. Another of putting this is (as in the argument-form above) that premises and conclusions can change places without making a difference.**'tonk'**

If
we still have problems with Prior's 'tonk', that situation arises
because we fail to see that the “meaning” of any connective “is
completely given by the rules” (130).

Prior
gives the following example of this logical phenomenon:

(P)
2 and 2 are 4.

__(Q)Therefore, 2 and 2 are 4__

__tonk____2 and 2 are 5.__

(R/Q)Therefore
2 and 2 are 5.

Clearly
the connective 'tonk' is doing something to the proceeding '2 and 2
are 4' - but what? Could it be that 'tonk' seems to mean

*add 1 -*at least in this instance? That would mean, however, that 'tonk' is the operation of*adding 1,*which isn't (really?) a connective of any kind.
The
new connective 'tonk' works like the connective 'and'. Or as Prior
puts it:

“

**Its meaning is completely given by the rules that (i) from any statement P we can infer any statements formed by joining P to any statement Q by 'tonk'... and that (ii) from any 'tonktive' statement P-****tonk****-Q we can infer the contained statement Q.”****(130)**
Thus,
at a symbolic level, 'tonk

*'*works like 'and'. And just as Prior symbolised P and Q taken together as*P-and-Q;*so he takes P and Q taken together with tonk as*P-tonk-Q*.
In
this case, '2 and 2 are 4' (P) is being conjoined with '2 and 2 are
5' (Q). Thus the conclusion, 'therefore, 2 and 2 are 5' (R) follows
from '2 and 2 are 5' (Q), though not from '2 and 2 are 4'. In other
words, R only needs to follow from either P or Q, not from both. Thus
when P and Q are, as it were,

*tonked*, we get: '2 and 2 are 4*tonk*2 and 2 are 5'. And the conclusion is: 'Therefore 2 and 2 are 5'.
To
express all this in argument-form, take this example:

(P)
Cats have four legs.

(P
& Q) Therefore cats have four legs

*tonk*Cats have three legs.
(R/Q)
Therefore cats have three legs.

What
is 'tonk' doing in the above? It seems to be cancelling out the statement before (i.e., 'Cats have four legs'). Thus if 'tonk' comes
after any P in any compound statement, then Q will cancel out P. If
that appears odd (especially with the seeming contradiction), that's
simply because, as Prior puts it, “there is simply nothing more to
knowing the meaning of ['tonk'] than being able to perform these
inferences” (129). In this case, we firstly state P, and then

*P-tonk-Q*(in which Q cancels out P), from which we conclude R.
Nuel
D. Belnap helps us understand what's happening here by offering
different symbols and a different scheme. Instead of the
argument-form above (which includes P, Q and R), we have the
following:

i)
A ⊢ A-

*tonk*-B
ii)
A

*-tonk*-B ⊢ B
iii) A ⊢ B

Quite
simply, one can deduce A-

*tonk*-B from A. Then one can deduce B from A-*tonk*-B. Finally, this means that one can derive B from A.
In
our example, by a simple rule of inference, one can derive 'Cats have
four legs

*tonk*cats have three legs' (A-*tonk*-B) from 'Cats have four legs' (A). And then one can derive 'Cat have three legs' (B) from 'Cats have four legs tonk cats have three legs' (A-*tonk*-B). Finally, one can derive 'Cats have three legs' (B) from 'Cats have four legs' (A).
Belnap
claims that an arbitrary creation of a connective (through

*implicit**definition*) could or can result in a contradiction. Thus, the '?' in the following
(

__a___{?}_{ }__c__)_{= df }_{ }_{ }__a + c__
(b d) b + d.

could
result in:

__2___{=}__3__
3 5

However, doesn't Prior's 'Therefore 2 and 2 are 4 tonk 2 and 2 are 5' also contain a
contradiction? Prior seems to be stressing the point
that in the definitions of connectives, even counterintuitive ones,
such contradictions are to be expected. Isn't that the point?

**References**

Belnap,
Nuel. (1962) 'Tonk,
Plonk and Plink'.

Prior,
A.N. (1960) 'The
Runabout Inference Ticket'.

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