=Paper=
{{Paper
|id=None
|storemode=property
|title=A tableaux method for term logic
|pdfUrl=https://ceur-ws.org/Vol-2264/paper1.pdf
|volume=Vol-2264
|authors=J.-Martín Castro-Manzano
|dblpUrl=https://dblp.org/rec/conf/lanmr/Castro-Manzano18
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==A tableaux method for term logic==
https://ceur-ws.org/Vol-2264/paper1.pdf
A Tableaux Method for Term Logic
J.-Martı́n Castro-Manzano
Faculty of Philosophy & Humanities, UPAEP
21 sur 1103, Puebla, Mexico
josemartin.castro@upaep.mx
Abstract. In this contribution we present a full tableaux method for
Term Functor Logic and we discuss its features. This goal should be of
interest because the plus-minus calculus of Term Functor Logic features
a peculiar algebra that, as of today, has not been used to produce a full
tableaux method. To reach this goal we briefly present Term Functor
Logic, then we introduce our contribution and, at the end, we discuss
some of its features.
Keywords: Term logic, semantic tree, syllogistic.
1 Introduction
In this contribution we present a full tableaux method for Term Functor
Logic [29,30,32,9,12,13] and we discuss its features. This goal should be of inter-
est because the plus-minus calculus of Term Functor Logic features a peculiar
algebra that, as of today, has not been used to produce a full tableaux method
(cf. [8,26]).1 To reach this goal we briefly present Term Functor Logic (with spe-
cial emphasis on syllogistic), then we introduce our contribution and, at the end,
we discuss the features of the method.
2 Preliminaries
Syllogistic is a term logic that has its origins in Aristotle’s Prior Analytics [1]
and deals with the consequence relation between categorical propositions. A cat-
egorical proposition is a proposition composed by two terms, a quantity, and
a quality. The subject and the predicate of a proposition are called terms: the
term-schema S denotes the subject term of the proposition and the term-schema
P denotes the predicate. The quantity may be either universal (All ) or particu-
lar (Some) and the quality may be either affirmative (is) or negative (is not).
These categorical propositions have a type denoted by a label (either a (universal
affirmative, SaP), e (universal negative, SeP), i (particular affirmative, SiP), or
o (particular negative, SoP)) that allows us to determine a mood. A categorical
1
[32, p.183ff] have already advanced a proposal, but its scope is limited to proposi-
tional logic.
1
syllogism, then, is a sequence of three categorical propositions ordered in such a
way that two propositions are premises and the last one is a conclusion. Within
the premises there is a term that appears in both premises but not in the conclu-
sion. This particular term, usually denoted with the term-schema M, works as a
link between the remaining terms and is known as the middle term. According
to the position of this last term, four figures can be set up in order to encode
the valid syllogistic moods or patterns (Table 1).2
Table 1: Valid syllogistic moods
Figure 1 Figure 2 Figure 3 Figure 4
aaa eae iai aee
eae aee aii iai
aii eio oao eio
eio aoo eio
Term Functor Logic (TFL) is a plus-minus calculus, developed by Som-
mers [29,30,32] and Englebretsen [9,12,13], that deals with syllogistic by using
terms rather than first order language elements such as individual variables or
quantifiers.3 According to this algebra, the four categorical propositions can be
represented by the following syntax:4
• SaP := −S + P = −S − (−P) = −(−P) − S = −(−P) − (+S)
• SeP := −S − P = −S − (+P) = −P − S = −P − (+S)
• SiP := +S + P = +S − (−P) = +P + S = +P − (−S)
• SoP := +S − P = +S − (+P) = +(−P) + S = +(−P) − (−S)
Given this algebraic representation, the plus-minus algebra offers a simple
method of decision for syllogistic: a conclusion follows validly from a set of
premises if and only if i) the sum of the premises is algebraically equal to the
conclusion and ii) the number of conclusions with particular quantity (viz., zero
or one) is the same as the number of premises with particular quantity [12,
p.167]. Thus, for instance, if we consider a valid syllogism from figure 1, we can
see how the application of this method produces the right conclusion (Table 2).
2
For sake of brevity, but without loss of generality, here we omit the syllogisms that
require existential import.
3
That we can represent and perform inference without first order language elements
such as individual variables or quantifiers is not news (cf. [27,23,18]), but Sommers’
logical project has a wider impact: that we can use a logic of terms instead of a first
order system has nothing to do with the mere syntactical fact, as it were, that we
can reason without quantifiers or variables, but with the general view that natural
language is a source of natural logic (cf. [30,31,20]).
4
We mainly focus on the presentation by [12].
2
Table 2: A valid syllogism: aaa-1
Proposition TFL
1. All dogs are animals. −D + A
2. All German Shepherds are dogs. −G + D
` All German Shepherds are animals. −G + A
In the previous example we can clearly see how the method works:
i) if we add up the premises we obtain the algebraic expression
(−D + A) + (−G + D) = −D + A − G + D = −G + A, so that the sum of the
premises is algebraically equal to the conclusion and the conclusion is −G + A,
rather than +A − G, because ii) the number of conclusions with particular quan-
tity (zero in this case) is the same as the number of premises with particular
quantity (zero in this case).
This algebraic approach is also capable of representing relational, singular,
and compound propositions with ease and clarity while preserving its main idea,
namely, that inference is a logical procedure between terms. For example, the
following cases illustrate how to represent and perform inferences with relational
(Table 3), singular5 (Table 4), or compound propositions6 (Table 5).
Table 3: Relational propositions
Proposition TFL
1. Some horses are faster than some dogs. +H1 + (+F12 + D2 )
2. Dogs are faster than some men. −D2 + (+F23 + M3 )
3. The relation faster than is transitive. −(+F12 + (+F23 + M3 )) + (+F13 + M3 )
` Some horses are faster than some men. +H1 + (+F13 + M3 )
Table 4: Singular propositions Table 5: Compound propositions
Proposition TFL Proposition TFL
1. All men are mortal. −M + L 1. If P then Q. −[p] + [q]
2. Socrates is a man. +s + M 2. P . [p]
` Socrates is mortal. +s + L ` Q. [q]
5
Provided singular terms, such as Socrates, are represented by lowercase letters.
6
Given that compound propositions can be represented as follows, P := +[p], Q :=
+[q], ¬P := [−p], P ⇒ Q := −[p]+[q], P ∧Q := +[p]+[q], and P ∨Q := −−[p]−−[q],
the method of decision behaves like resolution (cf. [24]).
3
These examples are designed to show that TFL is capable of dealing with
a wide range of inferences, namely, those classical first order logic is capable
to deal with. However, in certain sense, TFL is arguably more expressive than
classical first order logic in so far as it is capable of dealing with active-passive
voice transformations, associative shifts, and polyadic simplifications [9, p.172ff]:
we will refer to these features later.
3 TFL tableaux
As we can see, the peculiar algebra of TFL has some interesting capacities (and
inference rules); however, as of today, this algebra has not been exploited as to
produce a full tableaux method (cf. [8,32,26]): so here we propose one in three
steps. First, we start by offering some rules; then we show how can we apply those
rules in three different inferential contexts (basic syllogistic, relational syllogistic,
and propositional logic); and finally, we offer some evidence to the effect that
the method is reliable.
As usual, and following [8,26], we say a tableau is an acyclic connected graph
determined by nodes and vertices. The node at the top is called root. The nodes
at the bottom are called tips. Any path from the root down a series of vertices
is a branch. To test an inference for validity we construct a tableau which begins
with a single branch at whose nodes occur the premises and the rejection of the
conclusion: this is the initial list. We then apply the rules that allow us to extend
the initial list:
−A ± B +A ± B
−Ai ±Bi +Ai
±Bi
Diagram 1.1: TFL tableaux rules
In Diagram 1.1, from left to right, the first rule is the rule for a (e) type
propositions, and the second rule is the rule for i (o) type propositions. Notice
that, after applying the rule, we introduce some index i ∈ {1, 2, 3, . . .}. For
propositions a and e, the index may be any number; for propositions i and o,
the index has to be a new number if they do not already have an index. Also,
following TFL tenets, we assume the followings rules of rejection: −(±A) = ∓A,
−(±A ± B) = ∓A ∓ B, and −(− − A − −A) = +(−A) + (−A).
As usual, a tableau is complete if and only if every rule that can be applied
has been applied. A branch is closed if and only if there are terms of the form ±Ai
and ∓Ai on two of its nodes; otherwise it is open. A closed branch is indicated
by writing a ⊥ at the end of it; an open branch is indicated by writing ∞. A
tableau is closed if and only if every branch is closed; otherwise it is open. So,
4
again as usual, A is a logical consequence of the set of terms Γ (i.e., Γ ` A)
if and only if there is a complete closed tableau whose initial list includes the
terms of Γ and the rejection of A (i.e., Γ ∪ {−A} ` ⊥).
Accordingly, up next we show the method works for syllogistic by proving
the four basic syllogisms of the first figure, namely, moods aaa, eae, aii, and
eio (Diag. 1.2). Also, as expected, this method can also be used with relational
propositions. Consider the tableau in Diagram 1.3 corresponding to the example
given in Table 6 (in the following tableaux we omit the subscript indexes since
there is no ambiguity). Also, in compliance with the tenets of TFL, the method
preserves the expressive power of TFL with respect to active-passive voice trans-
formations, associative shifts, and polyadic simplifications as shown in Table 7
and Diagram 1.4.
−M + P −M − P −M + P −M − P
−S + M −S + M +S + M +S + M
` −S + P ` −S − P ` +S + P ` +S − P
−(−S + P) −(−S − P) −(+S + P) −(+S − P)
+S − P +S + P −S − P −S + P
+S1 +S1 +S1 +S1
−P1 +P1 +M1 +M1
−S1 +M1 −S1 +M1 −M1 +P1 −M1 −P1
⊥ ⊥ ⊥ ⊥
−M1 +P1 −M1 −P1 −S1 −P1 −S1 +P1
⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
Diagram 1.2: TFL proofs
Table 6: Relational syllogistic example (adapted from [12, p.172])
Proposition TFL
1. Every boy loves some girl. −B1 + (+L12 + G2 )
2. Every girl adores some cat. −G1 + (+A12 + C2 )
3. All cats are mangy. −C + M
4. Whoever adores something mangy is a fool. −(+A12 + M1 ) + F2
` Every boy loves something fool. −B1 + (+L12 + F2 )
5
Table 7: Passive-active voice transformation, associative shift, and polyadic sim-
plification examples (adapted from [12, p.174])
Proposition TFL
1. Some man loves some woman. +M1 + (+L12 + W2 )
2. What a man loves is a woman. +(+M1 + L12 ) + W2
3. A woman is something a man loves. +W2 + (+M1 + L12 )
4. A woman is loved by a man. +W2 + (+L12 + M1 )
5. Some lover is a man. +L12 + M1
−B + (+L + G)
−G + (+A + C)
−C + M
−(+A + M) + F
` −B + (+L + F)
−(−B + (+L + F))
+B − (+L + F)
+B1
−(+L + F)1
−L − F1
−B1 +(+L + G)1
⊥
+L1
+G1
−G1 +(+A + C)1
⊥
+A1
+C1
−C1 +M1
⊥
−L1 −F1
⊥
+F1 −(+A + M)1
⊥ −A − M1
−A1 −M1
⊥ ⊥
Diagram 1.3: Relational syllogistic tableau
6
+M + (+L + M) +M + (+L + W) +M + (+L + W)
` +W + (+L + M) ` +(+M + L) + W ` +L + M
−(+W + (+L + M)) −(+(+M + L) + W) −(+L + M)
−W − (+L + M) −(+M + L) − W −L − M
+M1 +M1 +M1
+L1 +L1 +L1
+W1 +W1 +W1
−W1 −(+L + M)1 −W1 −(+M + L)1 −L1 −M1
⊥ −L − M1 ⊥ −M − L1 ⊥ ⊥
−L1 −M1 −M1 −L1
⊥ ⊥ ⊥ ⊥
Diagram 1.4: Active-passive voice transformation, associative shift, and polyadic
simplification
Finally, we produce some evidence to the fact that this method is reliable in
the sense that what can be proven using the inference rules (say, TFLrules
valid ) can
be proven using the tableaux method (say, TFLtableaux
valid ), and vice versa.
Proposition 1. If an inference is TFLrules tableaux
valid , it is TFLvalid .
Proof. We proceed by cases. We check each mediate inference rule of TFL (vide
Appendix) is TFLtableaux
valid . For the rule DON we only need to retort to Section 2:
in there we can observe the four possible ocurrences of DON and how they are
TFLtableaux
valid . For the rule Simp we can build the next tableaux and observe they
are all valid:
+(+X + Y) + Z +(+X + Y) − Z +(+X + Y) + Z
` +X + Y ` +X + Y ` +X + Z
−(+X + Y) −(+X + Y) −(+X + Z)
−X − Y −X − Y −X − Z
+X1 +X1 +X1
+Y1 +Y1 +Y1
+Z1 −Z1 +Z1
−X1 −Y1 −X1 −Y1 −X1 −Z1
⊥ ⊥ ⊥ ⊥ ⊥ ⊥
7
+(+X + Y) − Z +(+X + Y) + Z +(+X + Y) − Z
` +Y − Z ` +Y + Z ` +Y − Z
−(+X − Z) −(+Y + Z) −(+Y − Z)
−X + Z −Y − Z −Y + Z
+X1 +X1 +X1
+Y1 +Y1 +Y1
−Z1 +Z1 −Z1
−X1 +Z1 −Y1 −Z1 −Y1 +Z1
⊥ ⊥ ⊥ ⊥ ⊥ ⊥
Finally, for the rule Add we can build the next tableaux:
±A −X + Y
±B −X + Z
` +(±A) + (±B) ` −X + (+Y + Z)
−(+(±A) + (±B)) −(−X + (+Y + Z))
−(±A) − (±B) +X − (+Y + Z)
+X1
−(±A) −(±B)
∓A ∓B 1
−(+Y + Z)
⊥ ⊥
−Y − Z1
−X1 +Y1
⊥
−X1 +Z1
⊥
−Y1 −Z1
⊥ ⊥
Proposition 2. If an inference is TFLtableaux
valid , it is TFLrules
valid .
Proof. We proceed by reductio. Suppose we pick an arbitrary inference that is
TFLtableaux
valid but is not TFLrules
valid . Then there is a complete closed tableau whose
initial list includes the set of terms Γ (possibly empty) and the rejection of the
conclusion (say, Γ + = Γ ∪ {−(±A)}); but from Γ alone we cannot construct
a proof of the conclusion by using any of the rules of mediate inference. There
are five general cases using the tableaux rules: a complete closed tableau whose
conclusion is −A + B, −A − B, +A + B, +A − B, or −A + A when Γ is empty.
Since in each case the tableau is complete, the corresponding rules have been
applied; and since each tableau is closed, each tableau must be of one of the
following general forms:
8
Γ Γ ` −A + A
` −A ± B ` +A ± B −(−A + A)
−(−A ± B) −(+A ± B) +A − A
+A ∓ B −A ∓ B
+Ai
i i
+A +A ∈ Γ
−Ai
i i
∓B ±B ∈ Γ ⊥
.. ..
. .
−Ai ∓Bi
i i
−A ∈ Γ ±B ∈ Γ ⊥ ⊥
⊥ ⊥
So, suppose we have an instance of the first general form but the correspond-
ing inference is not TFLrules
valid , i.e., where Γ
+
= Γ ∪ {+A ∓ B}, Γ + ` ⊥, but from
any application of DON, Simp, and Add to Γ , the conclusion −A ± B does not
obtain. Now, by following the paths of the tableau of the first general form,
we observe that, at the bottom, the tableau has a couple of closed branches.
Hence, at some previous nodes the tableau has to include something of the form
−A + X, −X ± B, that is to say, we need Γ = {. . . , −A + X, −X ± B, . . .}. But if
so, by applying DON, we obtain −A ± B from Γ , which contradicts the assump-
tion. Similarly, suppose we have an instance of the second general form but the
corresponding inference is not TFLrules valid , i.e., where Γ
+
= Γ ∪ {−A ∓ B}, Γ + ` ⊥,
but from any application of DON, Simp, and Add to Γ , the conclusion +A ± B
does not obtain. By following the paths of the tableau of the second general form,
we observe that, at the bottom, the tableau has a couple of closed branches, and
for those branches to be closed, we need something of the form +A, ±B or some-
thing of the form +(+A + X) ± B at some previous nodes of the tableau, that is
to say, we need either Γ = {. . . , +A, ±B, . . .} or Γ = {. . . , +(+A + X) ± B, . . .}.
But if so, in either case, by applying Add, we obtain +A ± B from Γ ; and by
applying Simp, we obtain +A ± B from Γ , which contradicts the assumption.
Finally, assume we have an instance of the third general form. In that case, the
path of the tableau consists only of Γ + = {+A − A}, and so, trivially, Γ + ` ⊥.
But since Γ is empty, −A + A has to be a tautology (i.e., All A is A) (cf. [12,
p.168]).
4 Concluding remarks
In this contribution we have attempted to offer a full tableaux method for Term
Functor Logic. Here are some remarks we can extract from this attempt: a) the
tableaux method we have offered avoids condition ii) of the method of decision
for syllogistic (namely, that the number of particular premises has to be equal
to the number of particular conclusions), thus allowing its general application
9
for any number of premisses. b) The method preserves the power of TFL with
respect to relational inferences, passive-active voice transformations, associative
shifts, and polyadic simplifications, which is something that gives this method a
competitive advantage over classical first order logic (tableaux). c) As a partic-
ular case, when no superscript index is used, we just obtain a tableaux method
for propositional logic.7 d) Due to the peculiar algebra of TFL, we have no use
for quantification rules nor skolemization, which could be useful in relation to
resolution and logic programming. e) The number of inference rules (cf. [12,
p.168-170]) gets drastically reduced to a shorter, simpler, and uniform set of
tableaux rules that preserves the capacity of TFL to perform inference in differ-
ent inferential contexts (basic syllogistic, relational syllogistic, and propositional
logic). f ) Also, we have to mention that for the purposes of this paper we have
focused only on the terministic features of TFL, but further comparison is re-
quired with the algebraic proof systems introduced in the 2000s by [6,4], since
these systems allow us to reconstruct Boole’s analysis of syllogistic logic by em-
ploying polynomial formatted proofs [5] and can also be extended to several other
logics, like modal logic [2,7]. g) Finally, we need to add that, due to reasons of
space, we are unable to introduce the modal [10,28,19], intermediate [25,33] or
numerical [22] extensions of the method that allow us to represent and reason
with modal propositions or non-classical quantifiers; however, we need to stress
that the inferential and representative powers of term logics go far beyond the
limits of the traditional or first order logic frontiers (cf. [20]).
For all these reasons, we believe this proof procedure is not only novel, but
also promising, not just as yet another critical thinking tool or didactic contrap-
tion, but as a research device for probabilistic and numerical reasoning (in so
far as it could be used to represent probabilistic (cf. [34]) or numerical reasoning
(cf. [22])), diagrammatic reasoning (as it finds its natural home in a project of
visual reasoning (cf. [11,32])), psychology (as it could be used to approximate
a richer psychological account of syllogistic reasoning (cf. [17,16])) , artificial
intelligence (as it could be used to develop tweaked inferential engines for Aris-
totelian databases (cf. [21])), and of course, philosophy of logic (as it promotes
the revision and revival of term logic (cf. [35,30,12,13]) as a tool that might
be more interesting and powerful than once it seemed (cf. [3,14,15])). We are
currently working on some of these issues.
Acknowledgments
We would like to thank the anonymous reviewers for their precise corrections
and useful comments. Financial support given by UPAEP Research Grant.
7
The tableaux method can be used for propositional logic. Recall the propositional
logic to TFL transcription is as follows: P := +[p], Q := +[q], ¬P := −[p], P ⇒ Q :=-
−[p] + [q], P ∧ Q := +[p] + [q], and P ∨ Q := − − [p] − −[q]; and notice that for the
propositional case we need not use the superscript indexes.
10
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12
Appendix. Rules of inference for TFL
In this Appendix we expound the rules of inference for TFL as they appear in [12,
p.168-170].
Rules of immediate inference
1. Premise (P ): Any premise or tautology can be entered as a line in proof.
(Tautologies that repeat the corresponding conditional of the inference are
excluded. The corresponding conditional of an inference is simply a con-
ditional sentence whose antecedent is the conjunction of the premises and
whose consequent is the conclusion.)
2. Double Negation (DN ): Pairs of unary minuses can be added or deleted from
a formula (i.e., − − X = X).
3. External Negation (EN ): An external unary minus can be distributed into
or out of any phrase (i.e., −(±X ± Y ) = ∓X ∓ Y ).
4. Internal Negation (IN ): A negative qualifier can be distributed into or out
of any predicate-term (i.e.,±X − (±Y ) = ±X + (±Y )).
5. Commutation (Com): The binary plus is symmetric (i.e., +X+Y = +Y +X).
6. Association (Assoc): The binary plus is associative (i.e., +X + (+Y + Z) =
+(+X + Y ) + Z).
7. Contraposition (Contrap): The subject- and predicate-terms of a universal
affirmation can be negated and can exchange places (i.e., −X +Y = −(−Y )+
(−X)).
8. Predicate Distribution (PD): A universal subject can be distributed into or
out of a conjunctive predicate (i.e., −X + (+Y + Z) = +(−X + Y ) + (−X +
Z)) and a particular subject can be distributed into or out of a disjunctive
predicate (i.e., +X + (−(−Y ) − (−Z)) = − − (+X + Y ) − −(+X + Z)).
9. Iteration (It): The conjunction of any term with itself is equivalent to that
term (i.e., +X + X = X).
Rules of mediate inference
1. (DON ): If a term, M , occurs universally quantified in a formula and either
M occurs not universally quantified or its logical contrary occurs universally
quantified in another formula, deduce a new formula that is exactly like the
second except that M has been replaced at least once by the first formula
minus its universally quantified M .
2. Simplification (Simp): Either conjunct can be deduced from a conjunctive
formula; from a particularly quantified formula with a conjunctive subject-
term, deduce either the statement form of the subject-term or a new state-
ment just like the original but without one of the conjuncts of the subject-
term (i.e., from +(+X +Y )±Z deduce any of the following: +X +Y , +X ±Z,
or +Y ± Z), and from a universally quantified formula with a conjunctive
predicate- term deduce a new statement just like the original but without
one of the conjuncts of the predicate-term (i.e., from −X ± (+Y + Z) deduce
either −X ± Y or −X ± Z).
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3. Addition (Add ): Any two previous formulae in a sequence can be conjoined
to yield a new formula, and from any pair of previous formulae that are
both universal affirmations and share a common subject-term a new formula
can be derived that is a universal affirmation, has the subject-term of the
previous formulae, and has the conjunction of the predicate- terms of the
previous formulae as its predicate-term (i.e., from −X + Y and −X + Z
deduce −X + (+Y + Z)).
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