The Meanings of Logical Constants
It seems to illuminate the meanings of logical constants to say how they contribute to the truth conditions of propositions containing them, as in the account of certain sentential connectives such as truth functions, the Frege-Tarski analysis of quantification, Kripke's semantics for modal operators in a quantified modal logic, and Davidson's treatment of certain sorts of adverbial modification in his paper about the logical form of action sentences. 1
In all these instances, an account of contribution to truth conditions seems to tell us significant things about the meanings of the logical constants involved.
It is not clear why this should be so. Compare the following examples:
(1) the predicate `horse' is true of something if and only if it is a horse.
(2) a construction of the form P and Q is true if and only if P is true and Q is true.
There is something trivial about these clauses since you can know they are true simply by knowing that `horse' is an extensional (one-place) predicate and that `and' is a (two-place) truth functional connective. You do not even have to know what these expressions mean. Compare:
(3) the predicate `borogrove' is true of something if and only if it is a borogrove.
(4) a construction of the form P zop Q is true if and only if P is true zop Q is true.
If you know that `borogrove' is a one-place predicate and `zop' is a two-place truth functional sentential connective, then you know that (3) and (4) hold even though you do not know what `borogrove' and `zop' mean, as long as you know what `true' and the other words in these sentences mean
Yet there seems to be an important difference between predicates and logical constants. A theory of truth of the sort Tarski describes, 2
containing clauses like (1) and (2), is often thought of as giving the meanings of sentential connectives corresponding to `and', `or' and `not' and quantifiers corresponding to `everything' and `something'. The theory is not normally thought of as giving the meaning of predicates like `horse'. People may refer to Tarski's account of the meanings of certain quantifiers, but not to his account of the meaning of the predicate `horse'.
(Well, the predicate `horse' does not appear in the example Tarski actually develops. The only non-logical predicate is the relation of set inclusion. So I should really say here that people do not refer to Tarski's account of the meaning of the predicate `is included in'.) 3
At one time, I thought that the clauses for logical constants in the relevant sort of theory of truth say more about their meanings than do the clauses for non-logical predicates for the following reason. The meanings of logical constants are determined by the role these constants play in reasoning, whereas the meanings of non-logical predicates are not to the same extent determined by the role such predicates play in reasoning. Furthermore, the roles logical constants play in reasoning are determined by the logical implications that depend on those logical constants, where these implications are determined by the contribution logical constants make to truth conditions. In other words, I saw the following dependencies, each determining the next:
contribution to truth conditions
relevant logical implications
role in reasoning
I no longer believe this. I still think the meanings of logical constants arise from their roles in reasoning, but these roles are not determined by relevant logical implications. The trouble is that logically equivalent connectives can have different meanings. For example, consider the two connectives that combine propositions P and Q to get the following results, respectively:
P and Q
not ((not P) or (not Q))
Since these results are logically equivalent, the two connectives make exactly the same contributions to truth conditions and are subject to exactly the same rules of logical implication, even though they do not have the same meaning
Still, I feel that something might be salvageable from the attempt to see a particular sort of connection between principles of conceptual role and truth clauses. One might attempt to characterize the meanings of logical constants in terms of principles of natural deduction that resemble or parallel the truth clauses for such logical constants. I will discuss this idea in the next section of the paper. To be sure, this does not yet avoid the problem that logically equivalent constants are assigned the same meaning. But I will suggest that we might avoid this problem by thinking of the relevant rules as rules of immediate implication, and perhaps also rules of immediate exclusion, not just rules of implication, although this loses the parallel with the truth clauses.
The idea that the meanings of logical constants are determined by certain characteristic implications has been elaborated in theories of `natural deduction'. 4
In this view, each logical constant is associated with `introduction rules' and `elimination rules' which fix its meaning. For example, logical conjunction, i.e. `and', is defined as that sentential connective C such that, for any propositions, P and Q,
P, Q logically imply C(P,Q)
C(P,Q) logically implies P
C(P,Q) logically implies Q
The first of these clauses gives an introduction rule for logical conjunction in the sense that it permits the introduction of a conjunctive statement into a proof. The second and third give elimination rules for conjunction in the sense that they allow a consequence not containing conjunction to be derived from a conjunctive statement.
Now, to say one proposition logically implies another is to say the first cannot be true without the second being true. So, the introduction rule for conjunction is just another way of saying that, necessarily.
C(P,Q) is true if P is true and Q is true.
Similarly, the elimination rule is just another way of saying that, necessarily,
C(P,Q) is true only if P is true and Q is true.
Putting these together yields the clause for conjunction in a theory of truth, supposing that theory is supposed to be necessary in the relevant sense (saying what holds necessarily, given what the relevant terms mean). So far, then, there is a close connection between the idea that the meanings of logical constants are determined by their truth conditions and the idea that these meanings are determined by characteristic logical implications.
But things are less simple for logical disjunction (`or') and logical negation (`not'). Consider logical disjunction. The introduction rule is easy:
P implies D(P,Q)
Q implies D(P,Q)
And this is just another way of expressing the `if' half of the truth conditions for logical disjunction:
D(P,Q) is true if P is true or Q is true.
The trouble comes with the elimination rule. Notice, first of all, that the disjunction D(P,Q) by itself does not logically imply either of its disjuncts, P or Q, although it does imply one of its disjuncts (e.g. P) given also the negation of the other disjunct (N(P)). This may seem to suggest taking the following as elimination rule:
D(P,Q), N(P) logically imply Q
D(P,Q), N(Q) logically imply P
But this has the disadvantage of appealing to one logical constant (negation) while defining another (disjunction). It leaves it unclear whether disjunction is being defined, or negation, or both simultaneously. The definition also lacks generality, since it would not work for a language containing disjunction but lacking negation.
So it is customary in systems of natural deduction to adopt a more complex rule:
IF P and certain other assumptions logically imply C,
AND Q and those other assumptions also logically imply C
THEN D(P,Q) and those other assumptions logically imply C.
However, this yields an elimination rule that is not just another way of expressing part of the truth conditions for disjunction.
Multiple conclusion logic
This difference between conjunction and disjunction can be avoided by giving up appeal to rules of natural deduction in favor of rules of a `sequent calculus'. A sequent calculus is a `multiple conclusion logic', whose multiple conclusions are in a certain sense understood disjunctively. Where an ordinary argument is valid if and only if the premises cannot all be true unless its single conclusion is also true, a multiple conclusion argument is valid if and only if its premises cannot all be true without at least one of its possibly many conclusions being true. Instead of saying that the premises in a valid multiple conclusion argument logically imply its conclusions, we can say that they logically involve its conclusions.
In a multiple conclusion logic, the following might be used as an elimination rule for disjunction:
D(P,Q) logically involves P,Q 5
And this rule does express exactly what is expressed in the clause giving the `only if' part of the truth conditions of disjunction.
D(P,Q) is true only if P is true or Q is true.
However, it must be confessed that appeal to the notion of involvement seems a trifle ad hoc in this connection.
Now, consider how negation might be defined in terms of introduction and elimination rules. First, the elimination rule for negation can be taken to be this:
If N(P) and certain other assumptions logically imply P, then those other assumptions by themselves logically imply P.
Alternatively, we can use this:
P, N(P) logically imply anything.
The following introduction rule corresponds to the latter elimination rule:
If P and certain other premises logically imply everything, then those other premises logically imply N(P)
It might seem this would not be useful in a system of natural deduction, in which one is interested in using such rules in proofs, since it might seem one would first have to prove infinitely many things to order to show that P and the other premises logically imply everything before being able to conclude N(P). But, given the second elimination rule, it is enough to show that P and the other premises logically imply N(P), for in that case they imply anything. So, the introduction rule could also be stated like this:
If P and certain other assumptions logically imply N(P), then those other assumptions by themselves logically imply N(P).
So, we can state introduction and elimination rules of natural deduction for negation, but clearly these rules are not simply restatements of the truth conditions of negated propositions. Nor does it help to state introduction and elimination rules in a sequent calculus or multiple conclusion logic for negation. It turns out that we still need basically the same rules.
Negative conclusion logic
Now, in the case of disjunction, we can get introduction and elimination rules that express truth conditions by appealing to a multiple conclusion or disjunctive conclusion logic. We could do the analogous thing for negation by introducing a negative conclusion logic. 6
Let us say that a negative conclusion argument is valid if and only if the premises cannot all be true unless the conclusion is not true. We can say that the premises of a valid negative conclusion argument logically exclude its conclusion. Then it is easy to give simple, straightforward introduction and elimination rules for negation:
P logically excludes N(P)
N(P) logically excludes P.
These rules express the truth conditions for negative propositions in exactly the way in which the rules for disjunction in a multiple conclusion logic express the truth conditions of disjunctive propositions.
In short, we can equate the usual sorts of clauses in a truth theory for logical connectives with rules of natural deduction only if we extend the notion of implication in various seemingly ad hoc ways. More straightforward rules of natural deduction in terms of logical implication do not coincide with clauses giving truth conditions except in the case of conjunction.
Improved rules: immediate implication
We have, then, two rather different ideas about the meaning of logical constants. One is that the meaning of a logical constant is determined by the contribution that the constant makes to truth conditions. The other is that the meaning of a logical constant is determined by certain principles of logical implication involving that constant.
Even though these ideas do not coincide, I have remarked already that they are both subject to the objection that they assign logically equivalent connectives the same meaning. To repeat my earlier example, consider the two connectives that combine propositions P and Q to get the following results, respectively:
P and Q
not ((not P) or (not Q))
Since these results are logically equivalent, the two connectives make exactly the same contributions to truth conditions and are subject to exactly the same rules of logical implication, even though they do not have the same meaning.
I believe the difference m meaning in this case has to do with how immediate certain implications are. In particular, the simple conjunction, `P and Q', immediately implies `P' in a way that the more complicated proposition, `not ((not P) or (not Q))', does not.
Immediate implication is a psychological notion. An immediate implication is one that is immediately obvious, one that can be immediately recognized. I try to say more about this elsewhere. 7
For present purposes let us simply take this notion as primitive. Then we can ask whether we can reformulate rules of natural deduction as rules of immediate implication and use these reformulated rules to define logical constants.
So, we might define conjunction as that sentential connective C such that
C(P,Q) immediately implies P
C(P,Q) immediately implies Q
P, Q immediately imply C(P,Q)
By this criterion, `not ((not P) or (not Q))' fails to be the conjunction C(P,Q), for example, because it does not immediately imply P, etc.
Notice that, although there is a certain similarity between this definition and the clause giving the contribution to truth conditions of `and', the present definition is by no means equivalent to that clause. Even if logical implication is solely a matter of truth conditions, immediate implication is not.
Logical disjunction might be defined as that connective D that satisfies the following conditions, letting "q" stand for any collection of sentences:
P immediately implies D(P,Q)
Q immediately implies D(P,Q)
`P, q imply C' and `Q, q imply C' immediately imply `D(P,Q), q imply C'.
These conditions parallel the usual natural deduction rules but are not at all similar to the truth conditions for disjunction. Nor does the last condition seem correct as a principle about immediate implication.
What principles should be adopted for logical negation? Certain principles, paralleling natural deduction rules, seem clearly incorrect, for example:
P, N(P) immediately imply anything.
This implication seems to be mediated by several steps. Similarly for
`P, q imply everything' immediately implies `q imply N(p)'.
The following are not so clearly inadequate:
`N(P), q imply P' immediately implies `q imply P'.
`P, q imply N(P)' immediately implies `q imply N(P)'.
They do not suffice to establish all the relevant principles involving negation, for example that a contradiction implies everything. However, these related principles of reductio ad absurdum proof will do the trick:
`N(Q), q imply P' and `N(Q), q imply N(P)' immediately imply `q imply Q'.
`Q and q imply P' and `Q and q imply N(P)' immediately imply `q imply N(Q)'.
Still, it is far from clear that these principles are correct as principles of immediate implication.
It is not clear that reductio ad absurdum proof is immediately perspicuous for everyone who grasps negation. A better idea might be to borrow the notion of exclusion from the imaginary negative conclusion logic I have mentioned. More precisely, the idea would be to appeal to a notion of immediate exclusion or immediate inconsistency. Things are immediately inconsistent for someone if he or she can immediately see that they are incompatible, if they are obviously inconsistent. Using this notion, we can then define negation as that (one-place) sentential connective N such that
N(P) is immediately inconsistent with P and is immediately implied by any set of propositions immediately inconsistent with P; furthermore, any set of propositions immediately inconsistent with N(P) immediately imply P.
Given this, it is possible to derive a principle of reductio ad absurdum proof. (To show this we appeal to the principle that, if P and A are immediately inconsistent, so are P, q and `P, q imply A'. Then, by the definition of negation, q and `P, q imply A' imply N(P).)
Using immediate inconsistency in addition to immediate implication allows a more plausible elimination rule for disjunction:
D(P,Q) plus any set of propositions immediately inconsistent with P immediately imply Q.
D(P,Q) plus any set of propositions immediately inconsistent with Q immediately imply P.
This would allow us to avoid the somewhat implausible claims about immediate implication in the previous rules based on the usual natural deduction rules.
One might instead try to account for disjunction by appeal to the principle of `immediate involvement' in a multiple conclusion logic. However, the notion of immediate involvement is not a natural notion like immediate implication and immediate inconsistency. Propositions do not immediately involve one another in the way they immediately imply or exclude one another.
Rules for quantifiers might be adapted from the introduction and elimination rules in some system of natural deduction, for example:
A universal quantification (x)P(x) immediately implies any instance P(a).
`q implies P(a)' and `a does not occur in q' immediately imply `q implies (x)P(x)'.
A singular proposition P(a) immediately implies any existential generalization (Ex)P(x).
`q, P(a) imply C and `a does not occur in any other relevant place, i.e. in q, C, or (Ex)P(x)' immediately imply `q, (Ex)P(x) imply C'.
However, these rules do not fully characterize ordinary `objectual' quantification, since they do not distinguish `objectual' from `substitutional' quantification. (A universal substitutional quantification is true if and only if all its substitution instances are true, whereas a universal objectual quantification is true if and only if all things in the range of the quantifier satisfy the predicate or open sentence to which the quantifier is attached.) Furthermore, the rules of universal quantifier introduction and existential quantifier elimination do not seem true as principles of immediate implication.
To meet the first complaint, what is needed are rules like the following:
`For every referring name or pronoun or variable a, no matter what it is taken to refer to, `q implies P(a)' immediately implies `q implies (x)P(x)'.
`For every a, no matter what it is taken to refer to, q, P(a) imply C' immediately implies `q, (Ex)P(x) imply C'.
These would yield ordinary objectual quantification and not just substitutional quantification. The first seems to me true as a claim about immediate implication. I am not sure about the second, but maybe it is true too.
A caveat about ordinary language
This kind of account of logical concepts is not intended as analysis of ordinary language. If definitions of this sort are correct, they say what it is for a concept to be the concept of classical negation, classical disjunction, or whatever. The definitions do not imply that such concepts occur in ordinary language or are actually used by anyone. If these logical concepts are used at all, it may well be in some special calculus that has been devised for some special purpose. Such a `calculus' would be used in the first instance for a certain sort of `calculation' rather than for communication. Furthermore, it may be that we never actually use it for calculation but merely reflect on certain aspects of what it would be like to use it in that way. So, to talk of immediate implication and exclusion in this connection is to talk about what would be immediately obvious to users of such a calculus.
Even if the meanings of logical constants are determined by their roles in inference, it is imprecise to say that these roles are determined by characteristic implications. It is important which implications (and perhaps also which exclusions) are immediate. And this is not just a matter of the contribution a construction makes to truth conditions. So there is no argument here for thinking that truth conditions are more relevant to the meanings of logical constants than to the meanings of nonlogical predicates. 8