ANALYTICITY WITHOUT SYNONYMY IN SIMPLE COMPARATIVE LOGIC*

Theodore J. Everett

SUNY-Geneseo

 

 

            In a recent article[1] I gave an intuitive account of a new, simple logic of comparisons, CL.  In this paper I provide some formal schemas for the analysis of vague predicates in terms of a set of semantic relations other than classical synonymy, all of which are best represented in CL.  These relations include weak synonymy (as between "large" and "huge"), antonymy (as between "large" and "small"), relativity (as between "large" and "large for a dog"), and a kind of supervenience (as between "large" and "wide" or "long").  I use Carnapian meaning postulates to define these relations as constraints on interpretations of the formal language of CL, in accordance with the general formula: "the more something is F, the more (or less) it is G". 

 

The comparative logic CL.

Consider the following valid inference:

Frank is taller than Larry.

Larry is tall.

Therefore, Frank is tall.

I have argued that a correct interpretation of such inferences requires that we assign not just a traditional truth value, but also a "how much" value for each object with respect to each (unary) predicate, plus a separate "cut-off" value attached to the predicate itself.  So, in the above example, if the cut-off for tallness is set at six feet (or any particular height), and Larry is tall (i.e. taller than six feet), and Frank is taller than Larry, then it follows arithmetically that Frank is also taller than six feet, hence that Frank is tall.

CL is a minimal comparative logic based on these ideas, and different from standard treatments of vagueness such as fuzzy logic, supervaluation theory, or Cresswell's "semantics of degree".  It has the same syntax as a classical logical language L, except that the symbol > ("more than") serves as a two-place logical connective for atomic sentences (the other comparisons: <, £, ³, and =, are defined in the obvious way).  In the semantics of CL, the interval from 0 to 1 is used, not as an infinite set of truth values as in fuzzy logic, but as an artificial scale of sub-values or extensions for atomic sentences.  Every interpreted predicate letter in the language of CL is also assigned a minimum standard in the same range (both extensions and standards can be represented explicitly in CL using metric constants).  The truth-value of each atomic sentence is then determined by whether its extension is at least as great as the standard for its predicate.  The truth values of comparisons depend only on the sameness or difference of the extensions of their component sentences.[2]  Everything else is computed classically, on the basis of ordinary truth values alone.  CL is thus much more conservative than Casari's (1987) smallest system ("restricted" comparative logic), which allows comparisons to be formed between non-quantified molecular statements as well as atomics.[3]

            The above inference might then be translated into CL as follows:

                        Ta

                        Tb > Ta

                        \ Tb

If the standard function for some interpretation assigned the value 0.60 to T, and the extension function assigned the values 0.65 and 0.69 to the pairs <T, a> and <T, b>, respectively, then all three sentences would be satisfied by that interpretation.  It should be clear that there are no allowable interpretations in which the premises would be true and the conclusion false.

            Here is a list of the rules of CL: 

I.   Syntax

            A.   Vocabulary

                        1.         A denumerable set of variables {x, y, z, x1,...}

                        2.         A denumerable set of constants {a, b, c, a1,...}

                        3.         A denumerable set of metric constants {i, j, k, i1,...}

                        4.         For each n 1, a denumerable set of n-place predicates {F, G, H, F1,...}

                        5.         Logical symbols {Ø, &, " , >}

 

                        6.         Parentheses {(, )}

 

            B.         Formation rules

 

                        1.         If f is an n-place predicate and t1,...tn are terms (variables or constants), ft1...tn is an atomic formula.

                        2.         If f and y are atomic formulas or metric           constants, (f > y) is a formula.

                        3.         If f is a formula, ¬f is a formula.

 

                        4.         If f and y are formulas, (f & y) is a formula.

 

                        5.         If f is a formula and a is a variable, "af is a formula.

           

                        6.         If f is a formula in which no variable occurs free, f is a sentence.

           

            C.        Definitions

 

                        1.         (f v y) =df Ø(Øf & Øy)

 

                        2.         (f ® y) =df (Øf v Øy)

 

                        3.         (f « y) =df ((f ® y) & (y ® f))

 

                        4.         $af =df Ø"aØf

 

                        5.         (f < y) =df (y > f)

 

                        6.         (f ³ y) =df Ø(y > f)

 

                        7.         (f £ y) =df (y ³ f)

 

                        8.         (f = y) =df ((f ³ y) & (f £ y))

 

II.         Semantics

 

            A.        Interpretations

                        An interpretation is an ordered triple <D, s*, e*>.  D (the domain) is a non-empty set.  s* (the standard function) is a function from predicates to members of E (the interval [0, 1]).  e* (the extension function) is a function (1) from constants to members of D, (2) from n-place predicates to functions from n-tuples of members of D to members of E, (3) from metric constants to members of E.

            B.         Extensions

                        The extension rules define, for each interpreted atomic sentence or metric constant f in the language, its extension e(f) as a function of the interpretation.

                        1.         If f is a sentence of the form yt1...tn,

                                    e(f) = e*(y)(e*(t1),...e*(tn)).

                        2.         If f is a metric constant, e(f) = e*(f).

            C.        Valuations

                        The valuation rules define, for each interpreted sentence f in the language, its truth-value I(f) as a function of the interpretation.

                        1.         If f is a sentence of the form yt1...tn,

                                    I(f) =  1, if e(f) ³ s*(y)

                                                   0, if e(f) < s*(y).

                        2.         If f is a sentence of the form (y1 > y2),

                                    I(f) =  1, if e(y1) > e(y2)

                                                   0, if e(y1) £ e(y2).

                        3.         If f is a sentence of the form Øy,

                                    I(f) =  1, if I(y) = 0

                                                   0, if I(y) = 1.

                        4.         If f is a sentence of the form (y1 & y2),

                                    I(f) =  1, if I(y1) = I(y2) = 1

                                                   0, if I(y1) or I(y2) = 0.

                        5.         If f is a sentence of the form "ay,