Substructural Logics
Created: 2026-01-31
A structural rule is a logical rule that applies indiscriminately to all propositions without regard to their form.
See in context at Substructural Logics
Created: 2026-01-31
Structural rules can also be motivated by characterising logical validity by way of models.
See in context at Substructural Logics
Created: 2026-01-31
In this reasoning, we did not have to appeal to the truth conditions for any formulas in models: the rule is not beholden to any particular scheme concerning how formulas are evaluated.
See in context at Substructural Logics
Created: 2026-01-31
regardless of whether we think of validity in terms of proofs or in terms of models, the structural rules are facts about validity that hold irrespective of the logical form of the components of the arguments. They arise from the general definition of validity and its structural features, rather than the rules governing particular connectives, quantifiers or other items of vocabulary.
See in context at Substructural Logics
Created: 2026-01-31
and also describes what one can infer from a sentence involving that connective. The residuation condition for p→qp→qp\to q only answers this question implicitly, by way of an appeal to the Cut rule, and the assumption that p→q⊢p→qp→q⊢p→qp\to q\vdash p\to q.
即蕴涵规则没有给定蕴涵式在premise的情况，并需要依靠切规则来隐式推导。
See in context at Substructural Logics
Created: 2026-01-31
The behaviour of premise combination may vary, and as a result, the behaviour of the conditional and other logical connectives vary as a result.
See in context at Substructural Logics
Created: 2026-01-31
Here, we are to deny that ppp follows from p,qp,qp, q. In general, we are to reject the inference rule that has this form: X⊢AX,Y⊢AX⊢AX,Y⊢A \frac{X \vdash A} {X, Y \vdash A}
This is the rule of weakening: it steps from a stronger statement, that AAA follows from XXX to a possibly weaker one, that AAA follows from XXX together with YYY. In this rule, the members of XXX, and YYY and the conclusion AAA are arbitrary, so it is a purely structural rule.
See in context at Substructural Logics
Created: 2026-01-31
One of the early motivating examples comes from a concern for relevance. If the logic is relevant (if to say that ppp entails qqq is true is to say, at least, that qqq truly depends on p)p)p) then weakening need not hold.
See in context at Substructural Logics
Created: 2026-01-31
So, in relevant logics, the inference from an arbitrary premise to a logical truth such as q→qq→qq \rightarrow q may well fail.
See in context at Substructural Logics
Created: 2026-01-31
The associativity of premise combination is another structural rule which has an impact on what inferences are valid.
See in context at Substructural Logics
Created: 2026-01-31
If premise combination is not commutative, then residuation can go in two ways. In addition to the residuation condition giving the behaviour of →→\rightarrow, we may wish to define a new arrow ←←\leftarrow as follows:
See in context at Substructural Logics
Created: 2026-01-31
A final important example is the rule of contraction which dictates how premises may be reused
See in context at Substructural Logics
Created: 2026-02-01
Cut and Identity as structural rules involve the relationship between occurrences of formulas in the left and right of sequents, while the other structural rules govern the ways that formulas may be manipulated inside the left or the right of sequents.