Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related. They may be viewed as a family of substructural or modal logics. (It is generally, but not universally, called relevant logic by Australian logicians, and relevance logic by other English-speaking logicians.)

Relevance logic aims to capture aspects of implication that are ignored by the "material implication" operator in classical truth-functional logic, namely the notion of relevance between antecedent and conditional of a true implication. This idea is not new: C. I. Lewis was led to invent modal logic, and specifically strict implication, on the grounds that classical logic grants paradoxes of material implication such as the principle that a falsehood implies any proposition. Hence "if I'm a donkey, then two and two is four" is true when translated as a material implication, yet it seems intuitively false since a true implication must tie the antecedent and consequent together by some notion of relevance. And whether or not I'm a donkey seems in no way relevant to whether two and two is four.

How does relevance logic formally capture a notion of relevance? In terms of a syntactical constraint for a propositional calculus, it is necessary, but not sufficient, that premises and conclusion share atomic formulae (formulae that do not contain any logical connectives). In a predicate calculus, relevance requires sharing of variables and constants between premises and conclusion. This can be ensured (along with stronger conditions) by, e.g., placing certain restrictions on the rules of a natural deduction system. In particular, a Fitch-style natural deduction can be adapted to accommodate relevance by introducing tags at the end of each line of an application of an inference indicating the premises relevant to the conclusion of the inference. Gentzen-style sequent calculi can be modified by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequents.

A notable feature of relevance logics is that they are paraconsistent logics: the existence of a contradiction will not cause "explosion". This follows from the fact that a conditional with a contradictory antecedent that does not share any propositional or predicate letters with the consequent cannot be true (or derivable).

History

Relevance logic was proposed in 1928 by Russian Soviet philosopher Ivan E. Orlov (1886–circa 1936) in his strictly mathematical paper "The Logic of Compatibility of Propositions" published in Matematicheskii Sbornik. The basic idea of relevant implication appears in medieval logic, and some pioneering work was done by Ackermann,^[1] Moh,^[2] and Church^[3] in the 1950s. Drawing on them, Nuel Belnap and Alan Ross Anderson (with others) wrote the magnum opus of the subject, Entailment: The Logic of Relevance and Necessity in the 1970s (the second volume being published in the nineties). They focused on both systems of entailment and systems of relevance, where implications of the former kinds are supposed to be both relevant and necessary.

Axioms

The early developments in relevance logic focused on the stronger systems. The development of the Routley-Meyer semantics brought out a range of weaker systems. The weakest of these systems is the relevance logic B. It is axiomatized with the following axioms and rules.

$A\to A$
$A\land B\to A$
$A\land B\to B$
$(A\to B)\land (A\to C)\to (A\to B\land C)$
$A\to A\lor B$
$B\to A\lor B$
$(A\to C)\land (B\to C)\to (A\lor B\to C)$
$A\land (B\lor C)\to (A\land B)\lor (A\land C)$
$\lnot \lnot A\to A$

The rules are the following.

$A,A\to B\vdash B$
$A,B\vdash A\land B$
$A\to B\vdash (C\to A)\to (C\to B)$
$A\to B\vdash (B\to C)\to (A\to C)$
$A\to B\vdash \lnot B\to \lnot A$

Stronger logics can be obtained by adding any of the following axioms.

$(A\to B)\to (\lnot B\to \lnot A)$
$(A\to B)\land (B\to C)\to (A\to C)$
$(A\to B)\to ((B\to C)\to (A\to C))$
$(A\to B)\to ((C\to A)\to (C\to B))$
$(A\to (A\to B))\to (A\to B)$
$(A\land (A\to B))\to B$
$(A\to \lnot A)\to \lnot A$
$(A\to (B\to C))\to (B\to (A\to C))$
$A\to ((A\to B)\to B)$
$((A\to A)\to B)\to B$
$A\lor \lnot A$
$A\to (A\to A)$

There are some notable logics stronger than B that can be obtained by adding axioms to B as follows.

For DW, add axiom 1.
For DJ, add axioms 1, 2.
For TW, add axioms 1, 2, 3, 4.
For RW, add axioms 1, 2, 3, 4, 8, 9.
For T, add axioms 1, 2, 3, 4, 5, 6, 7, 11.
For R, add axioms 1-11.
For E, add axioms 1-7, 10, 11, $((A\to A)\land (B\to B)\to C)\to C$ , and $\Box A\land \Box B\to \Box (A\land B)$ , where $\Box A$ is defined as $(A\to A)\to A$ .
For RM, add all the additional axioms.

Semantics

The standard model theory for relevance logics is the Routley-Meyer ternary-relational semantics developed by Richard Routley and Robert Meyer. A Routley-Meyer frame F for a propositional language is a quadruple (W,R,*,0), where W is a non-empty set, R is a ternary relation on W, and * is a function from W to W, and $0\in W$ . A Routley-Meyer model M is a Routley-Meyer frame F together with a valuation, $\Vdash$ , that assigns a truth value to each atomic proposition relative to each point $a\in W$ . There are some conditions placed on Routley-Meyer frames. Define $a\leq b$ as $R0ab$ .

$a\leq a$ .
If $a\leq b$ and $b\leq c$ , then $a\leq c$ .
If $d\leq a$ and $Rabc$ , then $Rdbc$ .
$a^{**}=a$ .
If $a\leq b$ , then $b^{*}\leq a^{*}$ .

Write $M,a\Vdash A$ and $M,a\nVdash A$ to indicate that the formula $A$ is true, or not true, respectively, at point $a$ in $M$ . One final condition on Routley-Meyer models is the hereditariness condition.

If $M,a\Vdash p$ and $a\leq b$ , then $M,b\Vdash p$ , for all atomic propositions $p$ .

By an inductive argument, hereditariness can be shown to extend to complex formulas, using the truth conditions below.

If $M,a\Vdash A$ and $a\leq b$ , then $M,b\Vdash A$ , for all formulas $A$ .

The truth conditions for complex formulas are as follows.

$M,a\Vdash A\land B\iff M,a\Vdash A$ and $M,a\Vdash B$
$M,a\Vdash A\lor B\iff M,a\Vdash A$ or $M,a\Vdash B$
$M,a\Vdash A\to B\iff \forall b,c((Rabc\land M,b\Vdash A)\Rightarrow M,c\Vdash B)$
$M,a\Vdash \lnot A\iff M,a^{*}\nVdash A$

A formula $A$ holds in a model $M$ just in case $M,0\Vdash A$ . A formula $A$ holds on a frame $F$ iff A holds in every model $(F,\Vdash )$ . A formula $A$ is valid in a class of frames iff A holds on every frame in that class. The class of all Routley-Meyer frames satisfying the above conditions validates that relevance logic B. One can obtain Routley-Meyer frames for other relevance logics by placing appropriate restrictions on R and on *. These conditions are easier to state using some standard definitions. Let $Rabcd$ be defined as $\exists x(Rabx\land Rxcd)$ , and let $Ra(bc)d$ be defined as $\exists x(Rbcx\land Raxd)$ . Some of the frame conditions and the axioms they validate are the following.

Name	Frame condition	Axiom
Pseudo-modus ponens	$Raaa$	$(A\land (A\to B))\to B$
Prefixing	$Rabcd\Rightarrow Ra(bc)d$	$(A\to B)\to ((C\to A)\to (C\to B))$
Suffixing	$Rabcd\Rightarrow Rb(ac)d$	$(A\to B)\to ((B\to C)\to (A\to C))$
Contraction	$Rabc\Rightarrow Rabbc$	$(A\to (A\to B))\to (A\to B)$
Conjunctive syllogism	$Rabc\Rightarrow Ra(ab)c$	$(A\to B)\land (B\to C)\to (A\to C)$
Assertion	$Rabc\Rightarrow Rbac$	$A\to ((A\to B)\to B)$
E axiom	$Ra0a$	$((A\to A)\to B)\to B$
Mingle axiom	$Rabc\Rightarrow a\leq c$ or $b\leq c$	$A\to (A\to A)$
Reductio	$Raa^{*}a$	$(A\to \lnot A)\to \lnot A$
Contraposition	$Rabc\Rightarrow Rac^{}b^{}$	$(A\to B)\to (\lnot B\to \lnot A)$
Excluded middle	$0^{*}\leq 0$	$A\lor \lnot A$
Strict implication weakening	$0\leq a$	$A\to (B\to B)$
Weakening	$Rabc\Rightarrow b\leq c$	$A\to (B\to A)$

The last two conditions validate forms of weakening that relevance logics were originally developed to avoid. They are included to show the flexibility of the Routley-Meyer models.

References

^ Ackermann, W. (1956), "Begründung einer strengen Implikation", Journal of Symbolic Logic, 21 (2): 113–128, JSTOR 2268750
^ Moh, Shaw-kwei (1950), "The Deduction Theorems and Two New Logical Systems", Methodos, 2: 56–75 Moh Shaw-Kwei, 1950, "," Methodos 2 56-75.
^ Church, A. (1951), The Weak Theory of Implication in Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften, Kommissions-Verlag Karl Alber, edited by A. Menne, A. Wilhelmy and H. Angsil, pp.22-37.

Bibliography

Alan Ross Anderson and Nuel Belnap, 1975. Entailment: the logic of relevance and necessity, vol. I. Princeton University Press. ISBN 0-691-07192-6
------- and J. M. Dunn, 1992. Entailment: the logic of relevance and necessity, vol. II, Princeton University Press.
Mares, Edwin, and Meyer, R. K., 2001, "Relevant Logics," in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic. Blackwell.

External links

Stanford Encyclopaedia of Philosophy: "Relevance logic" -- by Edwin Mares.

[1] Ackermann, W. (1956), "Begründung einer strengen Implikation", Journal of Symbolic Logic, 21 (2): 113–128, JSTOR 2268750

[2] Moh, Shaw-kwei (1950), "The Deduction Theorems and Two New Logical Systems", Methodos, 2: 56–75 Moh Shaw-Kwei, 1950, "," Methodos 2 56-75.

[3] Church, A. (1951), The Weak Theory of Implication in Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften, Kommissions-Verlag Karl Alber, edited by A. Menne, A. Wilhelmy and H. Angsil, pp.22-37.

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