Consider the argument:
Bill is a brother
-----
Bill is a sibling.
Is this little argument valid or invalid? It depends on what we mean by 'valid.' Intuitively, the argument is valid in the following sense:
D1. An argument is valid if and only if it is impossible that its premise(s) be true and its conclusion false.
(D1) may be glossed by saying that there are no possible circumstances in which the premises are true and the conclusion false. Equivalently, in every possible circumstance in which the premises are true, the conclusion is true. Since it is impossible that Bill be a brother without his being a sibling, the opening argument is valid by (D1).
(D1), though correct as far as it goes, leaves unspecified the source or ground of a valid argument's validity. This is the philosophically interesting question. What makes a valid argument valid? What is the ground of the impossibility of the premises' being true and the conclusion's being false? One answer is that the source of validity is narrowly logical or purely syntactic: the validity of a valid argument derives from its instantiation of valid argument-forms.
Now it is obvious that the validity of the above argument does not derive from its logical form. The logical form is
Fa
-----
Ga
where 'a' is an arbitrary individual constant and 'F' an arbitrary predicate constant. The above argument-form is invalid since it is easy to interpret the place-holders so as to make the premise true and the conclusion false: let 'a' stand for Al, 'F' for fat and 'G' for gay.
Valid arguments are either syntactically valid or semantically valid. The opening argument is not syntactically valid but it is semantically valid.
D2. An argument is syntactically valid iff it is narrowly-logically impossible that there be an argument of that form having true premises and a false conclusion.
According to (D2), a valid argument inherits its validity from the validity of its form, or logical syntax. So on (D2) it is primarily argument-forms that are valid or invalid; arguments are valid or invalid only in virtue of their instantiation of valid or invalid argument-forms. (D2) is thus a specification of the generic (D1).
But there is a second specification of (D1) according to which validity/invalidity has its source in the constituent propositions of the arguments themselves and so depends on their extra-syntactic content:
D3. An argument is extra-syntactically valid iff (i) it is impossible that its premises be true and its conclusion false; (ii) this impossibility is grounded neither in any contingent matter of fact nor
in formal logic proper, but in some necessary connection between the senses or the referents of the extra-logical terms of the argument.
A specification of (D3) is
D4. An argument is semantically valid iff (i) if it is impossible that its premises be true and its conclusion false; and (ii) this impossibility is grounded in the senses of the extra-logical terms of
the argument.
Thus to explain the semantic validity of the opening argument we can say that the sense of 'brother' includes the sense of 'sibling.' There is a necessary connection between the two senses, one that does not rest on any contingent matter of fact and is also not mediated by any purely formal law of logic. Note that logic allows (does not rule out) a brother who is not a sibling. Logic would rule out a non-sibling brother only if 'x is F & x is not G' had only false substitution-instances -- which is not the case. To put it another way, a brother that is not a sibling is a narrowly-logical possibility. But it is not a broadly-logical possibility due to the necesssary connection of the two senses.
So it looks as if analytic entailments like Bill is a brother, ergo, Bill is a sibling show that subsumability under purely formal logical laws is not necessary for (generically) valid inference. Sufficient, but not necessary. Analytic entailments appear to be counterexamples to the thesis that inferences in natural language can be validated only by subsumption under logical laws.
One might wonder what philosophers typically have in mind when they speak of validity. I would say that most philosophers today have in mind (D1) as specified by (D2). Only a minority have in mind (D3) and its specification (D4). I could easily be wrong about that. Is there a sociologist of philosophers in the house?
Consider the Quineans and all who reject the analytic/synthetic distinction. They of course will have no truck with analytic entailments and talk of semantic validity. Carnapians, on the other hand, will uphold the analytic/synthetic distinction but validate all entailments in the standard (derivational) way by importing all analytic truths as meaning postulates into the widened category of L-truths.
Along broadly Carnapian lines one could argue that the above argument is an enthymeme which when spelled out is
Every brother is a sibling
Bill is a brother
-----
Bill is a sibling.
Since this expanded argument is syntactically valid, the original argument -- construed as an enthymeme -- is also syntactically valid. When I say that it is syntactically valid I just mean that the
conclusion can be derived from the premises using the resources of standard logic, i.e. the Frege-inspired predicate calculus one finds in logic textbooks such as I. Copi's Symbolic Logic. In the above example, one uses two inference rules, Universal Instantiation and Modus Ponens, to derive the conclusion.
If this is right, then the source of the expanded argument's validity is not in a necessary connection between the senses of the 'brother' and 'sibling' but in logical laws. The question, however, was whether the opening argument as stated is valid or invaid. I say it is semantically, but not syntactically valid.
A juicier example is the Cartesian cogito:
I think
-----
I am.
This looks to be semantically valid and thus valid without the need of an auxiliary premise to mediate the inferential transition from premise to conclusion. It is valid absent an auxiliary major premise such as 'Whatever thinks, is.'
Recent Comments