Page 303 - Encyclopedia of Philosophy of Language

or (3):
For any thing, x, if x is a fish, then x swims. (3)
The logical form of (4):
Some fish swim (4)
deals with polyadic matters using sentences containing names, and these have their symbolic counterparts in constants (sometimes also called 'parameters'). Hence, Adelaide is between Perth and Sydney might be represented as Gabc,where a, b, and c are constants, rather than variables.
2. Logical Truth and Logical Consequence
Once the forms are understood, it becomes easy to settle the key issues of logical truth and logical conse- quence. A logical truth is a sentence whose symbolic paraphrase is true under all interpretations (in every nonempty universe). One sentence is a logical conse- quence of another (or of a set of sentences) if every interpretation (in every nonempty universe) which makes the latter true also makes the former true. Alternatively, if there is an interpretation of the set of sentences {A, B, C, D}, which makes them all true, while—under the same interpretation—a further sen- tence, E, is false, then E is not a logical consequence of the set {,4, B, C, D}.
Thanks to the completeness and consistency of both truth-functional logic and the general logic of quan- tifiers, various methods of proof (axiomatic, natural deduction, and tableau procedures) are able to give straightforward verdicts on logical truth and logical consequence. This fact is sometimes described by say- ing that semantic validity (logical truth) matches syn- tactic validity (logical provability). However, although truth tables provide a mechanical procedure for determining logical truth and consequence for truth-functions, there is no similar method available for predicate logic in general.
See also: Logic: Historical Survey; Natural Deduc- tion.
Bibliography
Frege G 1964 The Basic Laws of Arithmetic. University of California Press, Berkeley, CA
Guttenplan S 1986 The Languagesof Logic: An Introduction. Basil Blackwell, Oxford
Jeffrey R 1990 Formal Logic: Its Scope and Its Limits, 3rd edn. McGraw Hill, Maidenhead
Wittgenstein L 1961 Tractatus Logico-Philosophicus. Rout- ledge and Kegan Paul, London
is (5):
There is (that is, exists) at least one thing, x; such that x is a fish and x swims.
(5)
Using standard signs for the quantifiers, and using '=>' for //..., then..., these sentences receive the final symbolism in (6) and (7):
The x in these formulae is known as the variable of quantification, and—as Frege first observed—has a role akin to an anaphoric pronoun.
These logical forms suggest that sentences that seem simply to predicate swim of the subject all fish or some fish in fact involve something much more complex. Each involves two contained sentences, within which something is predicated of the variable of quanti- fication taken as subject. The discovery of these forms was of supreme importance in dislodging various tra-
ditional doctrines. For example, the symbolic versions above make clear that all fish swim does not imply that some fish swim (contrary to the medieval under- standing of Aristotle's logic). For, in a world con- taining no fish at all, it would be true that all fish swim, but false that there exists at least one thing which is both a fish and also swims. Frege made much of the fact that (in his version of the above symbolic forms) it is clear that the semantics of names is quite distinct from the semantics of quantifiers.
In the theory of quantification, the open sentence Fx is interpreted by giving some predicate of natural language to put in place of the predicate-letter, F (hence, x is red, x is a fish, and so on). Likewise, polyadic predicates provide interpretations for open sentences like Fxy, Gxyz, and so on (for example, x is east ofy, x is between y andz). Natural language often
All fish swim Vx(FxrsGx)
(6) Some fish swim 3x(Fx AGx). (7)
There is nothing very 'natural' about 'natural deduc- tion' systems. They are systems of rules for checking whether one 'logical formula' is provable from other
logical formulas in given systems of logic. As such they belong to the proof-theoretic approach, to logic, rather than to the model-theoretic approach: they are
Natural Deduction A. Fisher
Natural Deduction
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