Page 342 - Encyclopedia of Philosophy of Language

Formal Semantics
(c) If t has the form./(/, •••/„), for some/ef ", then
Note that the clause for function terms is recursive, and moreover, it precisely follows the recursion in the syntactic definition of such terms.
The second stage of the semantic definition process consists of explaining what it means for an arbitrary formula <p of L to be true in the model relative to an assignment s. One may recursively define a function [ • ], mapping the formulae of L to the set of truth values {0, 1} (0 for falsity, 1 for truth). The recursive definition follows the syntactic definition of the for- mulae of the language. First, the basic case is handled where <pis an atomic formula.
(a) If (p has the form ti=t2, then [<p],=l if and
(b) If<phastheformPt} •••/„,then[<p]5=7ifand oiuy if < !;,(',),..., t>,(O>e/(/>).
The logical connectives are treated as follows:
(c) If(phastheform—^,then[<?],=1ifandonly
(d) If <p has the form (i^&x). then [<?],= 1 if and
If <JPhas the form (if/ v x), then [<?],= 1 if and onlyif[*],= ! or[X],=l.
If q>has the form (\j/ -* x)» then [</>],= 0 if and onlyif[*],= ! and[xl=0.
mula in a model under an assignment s only depends on the finite part of s that assigns values to the free variables of the formula. Sentences do not contain free variables, so the truth or falsity of a sentence in a model does not depend on the assignment at all. We say that a sentence <p of L is true in a model if <p is true in the model under every assignment s. Equi- valently, we could have said that (p is true in the model if q>is true under some assignment. The notion of an assignment was a tool that can now be discarded.
The main feature of the Tarski semantics for predi- cate logic is its recursive nature: the meaning of a complex formula is recursively defined in terms of the meanings of its components. This is what com- positionality is all about.
3. Abstraction and Quantification
In the above presentation of the semantics of first- order predicate logic, quantifiers were introduced syn- categorematically, which is to say that they are not regarded as building blocks of the language in their own right. It follows that the quantifiersdo not have meanings of their own. The compositional semantics of first-order predicate logic would look more elegant if the quantifiers were to be considered building blocks. This can be done by means of the concept of abstraction.
Abstraction as a conceptual tool is already used by Frege, but his notation is rather awkward. Rather than stick with Frege's presentation a version using lambda operators or A-operators is presented. Lambda operators were introduced by Alonzo Church (1940). This device can be used to construct meanings for separate building blocks of languages. In doing so a version of typed logic is sketched. Typed logics are currently the most widely used tools for representing the semantics of natural language expressions.
The fact that in example (S) John can be replaced by Fred to form a new sentence shows abstraction focused on John.
I f <p h a s t h e f o r m OA = x X
tn e n
[ < ? ] * = 1 " °
anc
*
onlyif [^],=[x],.
Finally, the quantifiers V and 3 are considered. To start with a simple example, suppose the object is to describe the circumstances under which VxPx is true. In the description we want to refer to information about the truth or falsity of Px, for we want the account to be compositional. Saying that Px must be trueinthemodel,givens,isnotenough,because \Px\, depends on the value that s assigns to x: \Px\,= 1if and only ifs(x)eI(P). What we want to say is some- thing different: Px is true no matter which individual is assigned to x. This means that we are interested in assignments that are like s except for the fact that they may assign a different value to some variable v. Here is a precise definition (4):
John respects Bill.
(5)
s(v\d)(w) =
ifw=v.
(4)
This process of abstraction starts with a sentence, removes a proper name, and yields a function from proper names to sentences, or semantically, a function from individuals to truth values, that is, a charac- teristic function. Lambda operators allow this func- tion to be referred to explicitly (6):
Armed with this new piece of notation the quantifier case can be disposed of:
(e) If cphas the form Vr^, then [<?],= 1if and only
if for everv fyr]x*o= 1
deD-
If (p has the form 3v\l/, then [<?],= 1 if and only
Ax.x respects Bill.
(6)
if |V]j(,*/)= 1 for at least one deD.
This completes the definition of the function [ • ],. If [<p],= l we say that assignment s satisfies cp in the model, or that tp is true in the model under assignment s.
As was remarked above, truth or falsity of a for- 320
The function denoted by (6) corresponds to the prop- erty of respecting Bill. For convenience functions of this type are called 'properties.' Next, an abstraction can be made from the second proper name, and a functional expression denoting the relation of respect- ing is produced (7):