Published online by Cambridge University Press: 12 March 2014
We consider here the pure functional calculus of first order as formulated by Church.
Church, l.c., p. 79, gives the definition of the validity of a formula in a given set I of individuals and shows that a formula is provable in if and only if it is valid in every non-empty set I. The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic “semantical” notion in terms of which all other semantical notions are definable.
The notion of a value of a formula retains its meaning also in the case when the set I is empty. We have only to remember that if I is empty, then an m-ary propositional function (i.e. a function from the m-th cartesian power Im to the set {f, t}) is the empty set. It then follows easily that the value of each well-formed formula with free individual variables is the empty set. The values of wffs without free variables are on the contrary either f or t. Indeed, if B has the unique free variable c and ϕ is the value of B, then the value of (c)B according to the definition given by Church is the propositional constant f or t according as ϕ(j) is f for at least one j in I or not. Since, however, there is no j in I, the condition ϕ(j) = t for all j in I is vacuously satisfied and hence the value of (c)B is t.
1 Church, Alonzo, Introduction to mathematical logic, Part I. Annals of mathematics studies Number 13, Princeton, 1944Google Scholar.
2 A quantifier is vacuous if it is followed by an expression in which the variable bound by the quantifier is not free. Cf. Quine, Willard Van Orman, Mathematical logic, Cambridge 1947, p. 74Google Scholar.
3 A different set of rules for the functional calculus, such that theorems obtained by these rules are valid in each set I whether empty or not, has been given by Jaśkowski. Cf. Jaśkowski, Stanisław, On the rules of suppositions informal logic, Studia logica. Number 1, Warszawa 1934, §5Google Scholar.
4 In the sense defined by Quine, l.c. p. 79.
5 Quine, l.c., p. 88.
6 I am indebted for this remark to Mr. Grzegorczyk, Dr. H. Hiz, and Mr. Janiczak.
To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.
To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.