Theorem 2-89. Preparatory theorem (a) for Theorem 2-91, Theorem 2-92 and Theorem 2-
3.1 The Calculus
With the Speech Act Calculus, the rules for assuming and inferring are established, which ultimately serve to govern the derivation of propositions from sets of propositions. In preparation, we note: An author assumes a proposition Γ by uttering the sentence Sup- pose Γ , and an author infers a proposition Γ by uttering the sentence Therefore Γ . An author utters the empty sentence sequence by not uttering anything. An author utters a non-empty sentence sequence by successively uttering i for every i ∈ Dom( ). An author extends a sentence sequence to a sentence sequence * if he has uttered and now utters a sentence sequence ' such that * = '. An author thus extends an ut- tered sentence sequence to the sentence sequence ∪ {(Dom( ), Suppose Γ )}, by assuming Γ, i.e. by uttering Suppose Γ , and an author extends an uttered sentence se- quence to the sentence sequence ∪ {(Dom( ), Therefore Γ )} by inferring Γ, i.e. by uttering Therefore Γ .12
The rules of the calculus – and only these – are to allow one to extend an already uttered sentence sequence to a sentence sequence ' with Dom( ') = Dom( )+1. After the establishment of the rules, a derivation and a consequence concept can be established, according to which derivations will be exactly those non-empty sentence sequences that can in principle be uttered in accordance with the rules of the calculus (↑ 3.2).
As is usual for pragmatised natural deduction calculi, there is a rule of assumption (Speech-act rule 3-1) and 16 inference rules (Speech-act rule 3-2 to Speech-act rule 3-17). Additionally, the calculus contains an interdiction clause (IDC, Speech-act rule 3-18),
12
For the relation between the performance of speech acts and sequences of speech acts and the uttering of sentences and sequences of sentences, see HINST, P.: Logischer Grundkurs, p. 58–71, SIEGWART, G.:
Vorfragen, p. 25–32, Denkwerkzeuge, p. 39–52, and, most recent and in English, Alethic Acts. Here, we obviously assume that the expressions and concatenations thereof stipulated by Postulate 1-1 to Postulate 1-3 are utterable entities.
which forbids all extensions that are not permitted by one of the rules from Speech-act rule 3-1 to Speech-act rule 3-17. Among the rules of inference, there are two for each of the connectives, quantificators (resp. quantifiers) and for the identity predicate. One of the rules regulates the introduction of the respective operator and the other rule regulates its elimination.
A shorthand version of the availability conception may facilitate an easier understand- ing of the presentation of the calculus: If is a sentence sequence, then (i, i) is in AVS( ) if and only if the proposition of i is available in at i. Furthermore, (i, i) is in AVAS( ) if and only if the proposition of i is available in at i and i is an assump- tion-sentence. Γ is an element of AVP( ) if and only if there is (i, i) ∈ AVS( ) such that Γ is the proposition of i, and Γ is an element of AVAP( ) if and only if there is (i,
i) ∈ AVAS( ) such that Γ is the proposition of i.
In order to give an intuitively accessible short version of the rules, we stipulate: If one has uttered a sentence sequence and Γ is available in at i, then one has gained Γ in at i. If Δ is the last assumption made in uttering that is still available, and if one has gained Γ in after or with the assumption of Δ, then one has gained Γ in departing from the assumption of Δ. If one extends to ∪ {(Dom( ), Σ)} and Δ = P( i) is an assumption that is available in at i but that is not any more available in ∪ {(Dom( ),
Σ)} at i, then one has discharged the assumption of Δ at i.
Now the short version of the rules, in which all reference to sentence sequences, posi- tions and all grammatical specifications are neglected: One may assume any proposition
Γ (AR); if one has last gained Γ departing from the assumption of Δ, then one may infer
Δ → Γ and thus discharge the assumption of Δ (CdI); if one has gained Δ and Δ →
Γ , then one may infer Γ (CdE); if one has gained Δ and Γ, then one may infer Δ∧ Γ
(CI); if one has gained Δ∧Γ or gained Γ ∧Δ , then one may infer Γ (CE); if one has gained Δ→Γ and Γ→Δ , then one may infer Δ↔Γ (BI); if one has gained Δ and
Δ ↔ Γ or gained Δ and Γ ↔Δ , then one may infer Γ (BE); if one has gained Γ or gained Δ, then one may infer Δ∨ Γ (DI); if one has gained B ∨Δ , B →Γ and Δ →Γ , then one may infer Γ (DE); if one has gained either Γ and last ¬Γ or ¬Γ and last Γ departing from the assumption of Δ, then one may infer ¬Δ and thus discharge the assumption of Δ (NI); if one has gained ¬¬Γ , then one may infer Γ (NE); if one has
gained [β, ξ, Δ], where β is not a subterm of Δ or of any available assumption, then one may infer ξΔ (UI), if one has gained ξΔ , then one may infer [θ, ξ, Δ] (UE); if one has gained [θ, ξ, Δ], then one may infer ξΔ (PI); if one has gained ξΔ , next as- sumed [β, ξ, Δ], where β is a new parameter and not a subterm of Δ, and then, departing from the assumption of [β, ξ, Δ], last gained Γ, where β is not a subterm of Γ, then one may infer Γ and thus discharge the assumption of [β, ξ, Δ] (PE); one may infer θ = θ (II); if one has gained θ0 = θ1 and [θ0, ξ, Δ], then one may infer [θ1, ξ, Δ] (IE); that is all
one is allowed to do (IDC).
Now follow the rules of the Speech Act Calculus in their authoritative formulation:
Speech-act rule 3-1.Rule of Assumption (AR)
If one has uttered ∈ SEQ and if Γ∈ CFORM, then one may extend to ∪ {(Dom( ),
Suppose Γ )}.
Speech-act rule 3-2.Rule of Conditional Introduction (CdI)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and i∈ Dom( ), and
(i) P( i) = Δ and (i, i) ∈ AVAS( ),
(ii) P( Dom( )-1) = Γ, and
(iii) There is no l such that i < l≤ Dom( )-1 and (l, l) ∈ AVAS( ),
then one may extend to ∪ {(Dom( ), Therefore Δ→Γ )}.
Note that applying the rule of conditional introduction generates CdI-closed segments according to Definition 2-23 (cf. Theorem 2-91). If one extends to ∪ {(Dom( ),
Therefore Δ→ Γ )} by CdI, then none of the propositions that one inferred or assumed by uttering after (and including) the ith
member is available in ∪ {(Dom( ), There- fore Δ→Γ )}, except for propositions that were available in before the ith
member (cf. Definition 2-26). Of course, this does not apply to the newly available conditional Δ→
Γ , as it is the proposition of the new last member and thus available in the resulting sen- tence sequence in any case (cf. Theorem 2-82). Since the proposition of the last member of a sentence sequence is always available in at Dom( )-1, it also suffices in clause (ii) of the rule to demand solely that the consequent of the conditional one wants to infer is the proposition of the last member of , without additionally demanding that that proposition is also available there. Similar remarks apply to Speech-act rule 3-10 (NI) and Speech-act rule 3-15 (PE).
Speech-act rule 3-3.Rule of Conditional Elimination (CdE)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and {Δ, Δ→Γ } ⊆ AVP( ), then one
may extend to ∪ {(Dom( ), Therefore Γ )}.
Speech-act rule 3-4.Rule of Conjunction Introduction (CI)
If one has uttered ∈ SEQ and if Δ, Γ∈ AVP( ), then one may extend to ∪ {(Dom( ),
Therefore Δ∧Γ )}.
Speech-act rule 3-5.Rule of Conjunction Elimination (CE)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and { Δ∧Γ , Γ∧Δ } ∩ AVP( ) ≠∅,
then one may extend to ∪ {(Dom( ), Therefore Γ )}.
Speech-act rule 3-6.Rule of Biconditional Introduction (BI)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and { Δ→Γ , Γ→Δ } ⊆ AVP( ), then
one may extend to ∪ {(Dom( ), Therefore Δ↔Γ )}.
Here, the meta-logical requirement of separability, according to which each rule is to regulate only one operator, is violated, because the rule-antecedent demands that certain conditionals are available. The rule of biconditional introduction is thus at the same time a rule for the elimination of conditionals in certain contexts.
Speech-act rule 3-7.Rule of Biconditional Elimination (BE)
If one has uttered ∈ SEQ and if Δ∈ AVP( ), Γ∈ CFORM, und { Δ↔Γ , Γ↔Δ } ∩
AVP( ) ≠∅, then one may extend to ∪ {(Dom( ), Therefore Γ )}.
Speech-act rule 3-8.Rule of Disjunction Introduction (DI)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and {Δ, Γ} ∩ AVP( ) ≠∅, then one may
extend to ∪ {(Dom( ), Therefore Δ∨Γ )}.
Speech-act rule 3-9.Rule of Disjunction Elimination (DE)
If one has uttered ∈ SEQ and if Β, Δ, Γ∈ CFORM and { B ∨Δ , B →Γ , Δ→ Γ } ⊆
AVP( ), then one may extend to ∪ {(Dom( ), Therefore Γ )}.
Here, the meta-logical requirement of separability is violated a second time, as the rule- antecedent demands that certain conditionals are available. The rule of disjunction elimi-
nation is thus at the same time a rule for the elimination of conditionals in certain con- texts.
Speech-act rule 3-10.Rule of Negation Introduction (NI)
If one has uttered ∈ SEQ and if Δ, Γ∈ CFORM and i, j∈ Dom( ) and
(i) i≤j,
(ii) P( i) = Δ and (i, i) ∈ AVAS( ),
(iii) P( j) = Γ and P( Dom( )-1) = ¬Γ
or
P( j) = ¬Γ and P( Dom( )-1) = Γ,
(iv) (j, j) ∈ AVS( ), and
(v) There is no l, such that i < l≤ Dom( )-1 and (l, l) ∈ AVAS( ),
then one may extend to ∪ {(Dom( ), Therefore ¬Δ )}.
Applying the rule of negation introduction generates NI-closed segments according to Definition 2-24 (cf. Theorem 2-92). Thus, if one extends to ∪ {(Dom( ), Therefore ¬Δ )} by NI, then none of the propositions that one inferred or assumed by uttering after (and including) the ith member is available in ∪ {(Dom( ), Therefore ¬Δ )}, except for propositions that were available in before the ith
member (cf. Definition 2-26). Of course, this does not apply to the newly available negation ¬Δ . Since the proposition of the last member of a sentence sequence is always available in at Dom( )-1 (cf. Theorem 2-82), it also suffices in clause (iii) of the rule to demand that one of he two contradictory statements is available at j and that the second part of the contradiction is the proposition of the last sentence of .
Speech-act rule 3-11.Rule of Negation Elimination (NE)
If one has uttered ∈ SEQ and if Γ∈ CFORM and ¬¬Γ ∈ AVP( ), then one may extend
to ∪ {(Dom( ), Therefore Γ )}.
Speech-act rule 3-12.Rule of Universal-quantifier Introduction (UI)
If one has uttered ∈ SEQ and if β∈ PAR, ξ∈ VAR, Δ∈ FORM, where FV(Δ) ⊆ {ξ}, [β,
ξ, Δ] ∈ AVP( ) and β∉ STSF({Δ} ∪ AVAP( )), then one may extend to ∪ {(Dom( ),
Speech-act rule 3-13.Rule of Universal-quantifier Elimination (UE)
If one has uttered ∈ SEQ and if θ∈ CTERM, ξ ∈ VAR, Δ∈ FORM, where FV(Δ) ⊆ {ξ},
and ξΔ ∈ AVP( ), then one may extend to ∪ {(Dom( ), Therefore [θ, ξ, Δ] )}.
Speech-act rule 3-14.Rule of Particular-quantifier Introduction (PI)
If one has uttered ∈ SEQ and if θ∈ CTERM, ξ ∈ VAR, Δ∈ FORM, where FV(Δ) ⊆ {ξ},
and [θ, ξ, Δ] ∈ AVP( ), then one may extend to ∪ {(Dom( ), Therefore ξΔ )}.
Speech-act rule 3-15.Rule of Particular-quantifier Elimination (PE)
If one has uttered ∈ SEQ and if β∈ PAR, ξ∈ VAR, Δ∈ FORM, where FV(Δ) ⊆ {ξ}, Γ∈
CFORM and i∈ Dom( ), and
(i) P( i) = ξΔ and (i, i) ∈ AVS( ),
(ii) P( i+1) = [β, ξ, Δ] and (i+1, i+1) ∈ AVAS( ),
(iii) P( Dom( )-1) = Γ,
(iv) β∉ STSF({Δ, Γ}),
(v) There is no j≤i such that β∈ ST( j),
(vi) There is no m such that i+1 < m≤ Dom( )-1 and (m, m) ∈ AVAS( ),
then one may extend to ∪ {(Dom( ), Therefore Γ )}.
Applying the rule of particular-quantifier elimination generates PE-closed segments ac- cording to Definition 2-25 (cf. Theorem 2-93). Thus, if one extends to ∪ {(Dom( ),
Therefore Γ )} by PE, then none of the propositions that one inferred or assumed by uttering after the ith
member is available in ∪ {(Dom( ), Therefore Γ )}, except for propositions that were available in before the i+1th member (cf. Definition 2-26). Of course, this does not apply to the last inferred proposition, i.e. Γ, which is in any case available in the resulting sentence sequence. Since the proposition of the last member of a sentence sequence is always available in at Dom( )-1 (cf. Theorem 2-82), it also sufficises in clause (iii) of the rule, to demand solely that Γ is the proposition of the last member of .
Speech-act rule 3-16.Rule of Identity Introduction (II)
If one has uttered ∈ SEQ and if θ ∈ CTERM, then one may extend to ∪ {(Dom( ),
Speech-act rule 3-17.Rule of Identity Elimination (IE)
If one has uttered ∈ SEQ and if ξ ∈ VAR, Δ ∈ FORM, where FV(Δ) ⊆ {ξ}, θ0, θ1 ∈
CTERM and { θ0 = θ1 , [θ0, ξ, Δ]} ⊆ AVP( ), then one may extend to ∪ {(Dom( ),
Therefore [θ1, ξ, Δ] )}.
Last, we formulate a prohibition that makes the interdictory status of the rules explicit. For this, all 17 rule-antecedents for the extension of to ' are required to be unsatisfied. This condition is then sufficient for one not being allowed to extend to '.
Speech-act rule 3-18.Interdiction Clause (IDC)
If ∉ SEQ or if one has not uttered or if there are no B, Γ, Δ ∈ CFORM and θ0, θ1 ∈
CTERM and β ∈ PAR and ξ ∈ VAR and Δ' ∈ FORM, where FV(Δ') ⊆ {ξ}, and i, j ∈
Dom( ) such that
(i) ' = ∪ {(Dom( ), Suppose Γ )} or
(ii) P( i) = Δ, (i, i) ∈ AVAS( ), P( Dom( )-1) = Γ, there is no l such that i < l ≤
Dom( )-1 and (l, l) ∈ AVAS( ), and ' = ∪ {(Dom( ), Therefore Δ→Γ )} or
(iii) {Δ, Δ→Γ } ⊆ AVP( ) and ' = ∪ {(Dom( ), Therefore Γ )} or
(iv) {Δ, Γ} ⊆ AVP( ) and ' = ∪ {(Dom( ), Therefore Δ∧Γ )} or
(v) { Δ∧Γ , Γ∧Δ } ∩ AVP( ) ≠∅ and ' = ∪ {(Dom( ), Therefore Γ )} or
(vi) { Δ→Γ , Γ→Δ } ⊆ AVP( ) and ' = ∪ {(Dom( ), Therefore Δ↔Γ )} or
(vii) Δ∈ AVP( ), { Δ↔Γ , Γ↔Δ } ∩ AVP( ) ≠∅, and ' = ∪ {(Dom( ), There-
fore Γ )} or
(viii) {Δ, Γ} ∩ AVP( ) ≠∅ and ' = ∪ {(Dom( ), Therefore Δ∨Γ )} or
(ix) { B ∨Δ , B →Γ , Δ→Γ } ⊆ AVP( ) and ' = ∪ {(Dom( ), Therefore Γ )}
or
(x) i≤ j, P( i) = Δ, (i, i) ∈ AVAS( ), P( j) = Γ and P( Dom( )-1) = ¬Γ or P( j) =
¬Γ and P( Dom( )-1) = Γ, (j, j) ∈ AVS( ), there is no l such that i < l ≤ Dom( )-1
and (l, l) ∈ AVAS( ), and ' = ∪ {(Dom( ), Therefore ¬Δ )} or
(xi) ¬¬Γ ∈ AVP( ) and ' = ∪ {(Dom( ), Therefore Γ )} or
(xii) [β, ξ, Δ'] ∈ AVP( ), β ∉ STSF({Δ'} ∪ AVAP( )) and ' = ∪ {(Dom( ), There-
fore ξΔ' )} or
(xiii) ξΔ' ∈ AVP( ) and ' = ∪ {(Dom( ), Therefore [θ0, ξ, Δ'] )} or
(xiv) [θ0, ξ, Δ'] ∈ AVP( ) and ' = ∪ {(Dom( ), Therefore ξΔ' )} or
(xv) P( i) = ξΔ' , (i, i) ∈ AVS( ), P( i+1) = [β, ξ, Δ'], (i+1, i+1) ∈ AVAS( ),
P( Dom( )-1) = Γ, β∉ STSF({Δ', Γ}), there is no l≤i such that β∈ ST( l), there is no
m such that i+1 < m≤ Dom( )-1 and (m, m) ∈ AVAS( ), and ' = ∪ {(Dom( ),
Therefore Γ )} or
(xvii) { θ0 = θ1 , [θ0, ξ, Δ]} ⊆ AVP( ) and ' = ∪ {(Dom( ), Therefore [θ1, ξ, Δ] )},
then one may not extend to '.
Informally, Speech-act rule 3-18 says: If none of the rules from Speech-act rule 3-1 to Speech-act rule 3-17 allows the extension of to ', then one may not extend to '.
By setting the 18 rules, the calculus has now been established and can already be used. If one wants to add further rules later, e.g. rules for adducing-as-reason, stating, the posit- ing-as-axiom or defining, one has to adapt Speech-act rule 3-18 accordingly. In the next section, we will now establish a derivation concept and a consequence concept for the calculus (3.2). Then, we will prove some theorems that shed some light on the way in which the calculus works (3.3).