(p.281) Appendix
(p.281) Appendix
A Ttheory for a Fragment of English^{1}
We shall now consider a very simple fragment of English that we shall call ‘QE’, and show how to construe a Ttheory for it. The vocabulary of QE contains the following:
1. Proper names: 
‘Elsa’, ‘John’, ‘Mary’, ‘Paul’ 
2. Predicates: 
‘sleeps’, ‘loves’, ‘envies’, ‘helps’ 
3. Logical particles: 
‘and’, ‘or’, ‘it is not the case that’, and ‘if, then’ 
4. Quantifiers: 
‘everyone’, ‘someone’ 
The grammar is very simple, too. Proper names and quantifiers are both noun phrases (NPs). The category S is that of sentences. The rules are then:
1. NP + ‘sleeps’ is an S
2. NP_{1} + ‘loves’—‘envies’—‘helps’ + NP_{2} is an S
3. ‘it is not the case that’ + S is an S
4. S_{1} + ‘and’—‘or’—‘if, then’ + S_{2} is an S
(p.282) By these rules we can e.g. form the following sentences:
(5) Mary sleeps or John loves Mary.
(6) Everyone loves someone.
(7) If Elsa helps Paul, then Paul loves Elsa and John envies Paul.
Even though this fragment is very simple, the examples show that it has ambiguous sentences. (6) can mean that there is some person such that everyone loves that person, or that for each person there is some person, not necessarily the same, that the first person loves. And (7) can mean either that John envies Paul (unconditionally) and if Elsa helps Paul, then Paul loves Elsa, and it can mean that If Elsa helps Paul, then Paul loves Elsa and (under this condition) John envies Paul. These ambiguities matter semantically So we need to disambiguate the sentences before they can be interpreted.
We disambiguate by means of bracketing sentences and indexing the quantifiers for scope relations. Using these devices we get the following readings corresponding to the readings just specified:
(6′) Everyone_{2} loves someone_{1}.
(6″) Everyone_{1} loves someone_{2}.
(7′) (If Elsa helps Paul, then Paul loves Elsa) and John envies Paul.
(7″) If Elsa helps Paul, then (Paul loves Elsa and John envies Paul).
In order to apply the Tarskian semantics, the fragment needs to be regimented into the apparatus of quantifiers and variables. Thus we will have
1. Proper names: 
‘Elsa’, ‘John’, ‘Mary’, ‘Paul’ 
2. Predicates: 
‘sleeps’, ‘loves’, ‘envies’, ‘helps’ 
3. Logical connectives: 
‘&’, ‘∨’, ‘¬’, ‘→’ 
4. Quantifiers: 
‘∀’, ‘∃’ 
5. Individual variables: 
x_{1}, x_{2}, … 
(p.283) The grammar is the usual formation rules for a firstorder language. The translation rules are again very simple (we will not spell them out). What we need to take account of is the relative scope of simple sentences containing (different) quantifiers. Corresponding to (6′) and (6″) we have
(6*) ∃x_{1}∀x_{2}(x_{2} loves x_{1})
(6*) ∀x_{1}∃x_{2}(x_{1} loves x_{2})
One must also take care that when a new variable is introduced in the translation, it does not already occur in the scope of the new quantifier.
It should be noted at this point that the firstorder language is expressively richer than the QE fragment. For instance, a firstorder sentence such as
(8) ∀x_{1}(∃x_{2}(x_{1} loves x_{2}) → ∃x_{3}(x_{3} loves x_{1}))
does not have a counterpart in QE. It would need forms such as
(8′) Everyone who loves someone is loved by someone.
(8″) Everyone is such that if he/she loves someone, then he/she is loved by someone.
In (8′) we use a subordinate relative clause, and in (8″) bound pronouns, to express the same proposition, and QE does not contain either of these devices. Adding them to QE would complicate the grammar considerably. It is not needed in order to exemplify the basic features of Ttheories. Still, the Ttheory that we give below, applied to just the regimented language, does provide an interpretation of (8) (after going through the example derivation below, you could try it as an exercise).
We can now set out the basic clauses for the Ttheory for QE. We shall use‘s’ as a metalanguage variable for infinite sequences (s = 〈s_{1}, s_{2}, …〉), and ‘a_{1}, ‘a_{2}’, … as metalanguage individual variables. The axioms and axiom schemata are the following: (p.284)
1. Singular terms: 
⊦ Ref(‘Elsa’) = Elsa 
⊦ Ref(‘Mary’) = Mary 

⊦ Ref(‘John’) = John 

⊦ Ref(‘Paul’) = Paul 

2. Predicates: 
⊦ ∀s (sat(s,⌜x_{i} sleeps⌝ iff s_{i} sleeps) 
⊦ ∀s (sat(s, ⌜x_{i} loves x_{j}⌝) iff s_{i} loves s_{j}) 

⊦ ∀s (sat(s, ⌜x_{i} helps x_{j}⌝) iff s_{i} helps s_{j}) 

⊦ ∀s (sat(s, ⌜x_{i} envies x_{j}⌝) iff s_{i} envies s_{j}) 

3. Connectives: 
⊦∀s (sat(s, ⌜S_{1} & S_{2}⌝)iff sat(s,S_{1})and sat(s,S_{2})) 
⊦ ∀s (sat(s, ⌜¬S⌝) iff it is not the case that sat(s,S 

⊦∀s(sat(s, ⌜S_{1} ∨ S_{2}⌝iff sat(s,S_{1}) orsat(s,S_{2}))) 

⊦ ∀s (sat(s, ⌜S_{1} → S_{2}⌝) iff: if sat(s, S_{1} ), then sat(s, S_{2})) 

4. Quantifiers: 
⊦ ∀s (sat(s, ⌜∃x_{i}Px_{i}⌝) iff for some a_{i} (sat(s[a_{i}/s_{i}], ⌜Px_{i}⌝ ))) 
⊦ ∀s (sat(s, ⌜∀x_{i}Px_{i}⌝) iff for every a_{i} (sat(s[a_{i}/s_{i}], ⌜Px_{i} ⌝))) 

where ⌜x_{i}⌝ etc. is schematic, with instances like ‘x_{13}’, 

and where s[b/s_{i}] is the sequence differing from s at most by having b in its i:th position.^{2} 

5. Terms in formulas: 
⊦ sat(s,⌜ Pt⌝) iff sat(s[Ref (t)/s_{i}], Px_{i}), provided ⌜ x_{i}⌝ does not already occur in ⌜P⌝. 
We will also need rules of inference in order to derive the right theorems. But we don’t want the full power of frstorder logic, since that would have the consequence, for instance, that if ⊦ A iff B is a theorem, so is ⊦ A iff (B or (p and not p)). But the latter would not be interpretive if the former is. Therefore, we limit ourselves to a few more specialized rules of inference. They provide one alternative for capturing what Davidson had in mind when speaking of “canonical derivations” (see above, p. 65). (p.285)
Rule 1: 
If ⊦ a = b and ⊦ X(a), then ⊦ X(b) 
Rule 2: 
If ⊦ ∀s(A iff B) and ⊦ ∀s(X(A)), then ⊦ ∀s(X(B)) 
Rule 3: 
If ⊦ ∀s(A(s) iff B) and ‘s’ does not occur in B, then ⊦ ∀s(A(s)) iff B 
Rule 4: 
If ⊦ A iff B and ⊦ B iff C, then ⊦ A iff C 
Rule 5: 
⊦ True (s) iff True (s′), if s′ is the regimentation of s 
Rule 6: 
⊦ for all s, a, i (s[a/s_{i}]i = a) 
In Rules 1 and 2 ‘X’ is schematic for the sentential context. We can now define truth:
Rule 7: 
⊦ True S iff for all s (sat(s, S)) 
To get a bit of a feeling for how this works, let’s run through a derivation. Below, for the sake of readability, we shall use italics instead of quote marks for expressions that are mentioned.
(9)
i)
⊦ True(If someone sleeps, then Paul sleeps) iff
True(∃x_{1}(x_{1} sleeps) → Paul sleeps)
(rule 5)
ii)
⊦ True(∃x_{1}(x_{1} sleeps) → Paul sleeps) iff
for all s(sat(s,∃x_{1}(x _{1} sleeps) → Paul sleeps))
(rule 7)
iii)
⊦ For all s (sat(s, ∃x_{1}(x_{1} sleeps) → Paul sleeps) iff
if sat(s, ∃x_{1} (x_{1} sleeps)), then sat(s,Paulsleeps))
(Connectives)
iv)
⊦ For all s (sat(s, ∃x_{1} (x_{1} sleeps)) iff
forsome a_{1}(sat(s[a1/s_{1}],x_{1} sleeps)))
(quantifiers)
v)
⊦ For alls (forsome a_{1}(sat(s[a_{1}/s_{1}],x_{1} sleeps)) iff
for some a_{1}(s[a_{1}/s_{1}]_{1} sleeps))
(predicates, rule 2)
vi)
⊦ For all s (forsome a_{1}(s[a_{1}/s_{1}]_{1} sleeps)) iff
for some a_{1}(a_{1} sleeps))
(rule 6, rule 1)
vii)
⊦ For all s (sat (s, Paul sleeps) iff
sat(s[Ref (Paul)/s_{2}], x_{2} sleeps))
(terms in formulas)
viii)
⊦ For all s (sat(s[Ref(Paul)/s_{2}], x_{2} sleeps) iff
s[Ref(Paul)/s_{2}]_{2} sleeps)
(predicates)
ix)
⊦ For all s (s[Ref(Paul) /s_{2}]_{2} sleeps iff
Ref(Paul) sleeps)
(rule 6, rule 1 )
x)
⊦ For all s (Ref(Paul) sleeps iff
Paul sleeps)
(singular terms, rule 1)
xi)
⊦ For all s (if sat(s, ∃x_{1} (x_{1} sleeps)), then sat(s, Paul sleeps) iff
if for some a_{1} (a_{1} sleeps), then Paul sleeps)
((9iv)(9x), rule 2)
xii)
⊦ For all s (sat(s, ∃x_{1} (x_{1} sleeps) → Paul sleeps) iff
if for some a_{1} (a_{1} sleeps), then Paul sleeps)
((9iii), (9xi), rule 2)
xiii)
⊦ For all s (sat(s, ∃x_{1} (x_{1} sleeps) → Paul sleeps)) iff
if for some a_{1} (a_{1} sleeps), then Paul sleeps
((9xii), rule 3)
xiv)
⊦ True(∃x_{1} (x_{1} sleeps) → Paul sleeps) iff
if for some a_{1} (a_{1} sleeps), then Paul sleeps
((9ii), (9xiii), rule 4)
xv)
⊦ True(If someone sleeps, then Paul sleeps) iff
if for some a_{1} (a_{1} sleeps), then Paul sleeps.
((9i), (9xiv), rule 4).
Notes:
(1.) Special thanks to Peter Pagin for helping me make this appendix.
(2.) The method of quantifying over individual metalanguage variables applied in the quantifer clauses is suggested in Wiggins 1980, 325.