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?%tFPart Four:Epistemic Cognition, FocussingFirst on Deductive Reasoning 3Epistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningGEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningHEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoninguEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningvEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningxEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningyEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningzEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoning}Epistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoning_OSCAR as a Deductive ReasonerOSCAR s greatest virtue as an automated reasoner is that it is capable of performing defeasible reasoning. However, the defeasible reasoner is built on top of a deductive reasoner, and is best understood by looking at the deductive reasoner first.
aNatural DeductionOSCAR s reasoning is in the style of natural deduction .
I take this to mean that it reasons about what follows from the premises given suppositions.
This is implemented by having OSCAR reason about sequents. These are pairs <supposition,formula> where supposition is a set of formulas. Abbreviated formula / supposition.
The most characteristic rule of suppositional reasoning is CONDITIONALIZATION:
Given an interest in (P Q)/X suppose {P} and adopt interest in inferring Q/{P}X.
Another example, DILEMMA:
Given (P v Q) and an interest in R/X, adopt interest in R/X{P} and R/X{Q}. :[U!O!:I3%=;> `Bidirectional ReasoningDPerhaps the most novel feature of OSCAR s deductive reasoning is that reasonschemas are segregated into backward and forward schemas.
forward schemas lead from conclusions to conclusions
From (P & Q), infer P.
backward schemas lead from interests to interests
From P, Q infer (P & Q).
`5!2!bSome Inference Rulesz adjunction simplification
p/X q/Y (p&q)/X
(p&q)/XY p/X q/X
negation introduction negation elimination
p/X ~~p/X
~~p/X p/X
addition disjunctive syllogism
p/X ~p/X, (pvq)/Y ~q/X, (pvq)/Y
(pvq)/X (qvp)/X q/Y p/X
conditionalization modus ponens modus tollens
q/X{p} p/X, (p q)/Y ~q/X, (p q)/Y
(p q)/X q/XY ~p/XY
reductio1 reductio2
p/X{~p} (q & ~q)/X{~p}
p/X p/X!
7
e
9(8C
u*cDirectionalityMost of these inference rules have natural directions, and are combinatorially explosive when applied in the opposite direction.
A plausible classification:
Forwards reasons Backwards reasons
simplification adjunction
negation elimination negation introduction
disjunctive syllogism addition
modus ponens conditionalization
modus tollens reductio1
Note that reductio2 fits neither category. I will return to this.
l%DD>?RdInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests..NTNS,  beInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bfInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bgInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bhInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  biInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bjInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  b{InterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  blInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bmInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bnInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  boInterestDriven Reasoning{
reasonforwards
If a set of sequents X is a forwards reason for a sequent S, some member of X is newly concluded, and the other members of X have already been concluded, then conclude S.
reasonbackwards
If interest is adopted in a sequent S, and a set X of sequents is a backwards reason for S, then adopt interest in any members of X that have not already been concluded. If every member of X has been concluded, conclude S.
dischargeinterest
If interest was adopted in the members of X as a way of getting the sequent S, and some member of X is concluded and the other members of X have already been concluded, then conclude S.~ , ppGeneralized Backwards Reasons:This is inadequate for reductio2:
reductio2
(q & ~q)/X{~p}
p/X
The proper interpretation of this rule should be:
Given an interest in p/X, suppose {~p}. Then for each conclusion q/X{~p} drawn relative to the reductio supposition, adopt interest in ~q/X{~p}. When such a contradiction is concluded, conclude p/X.
Generalized backwards reasons have both forwards and backwards premises:
example: ("x)(Fx Gx)/X (forwards premise)
Fa/X (backwards premise)
Ga/XB"!"
2FG>J
t]yGeneralized Forward ReasonsWe can also allow forwards reasons to have backwards premises as well as forwards premises. The intent is that once the forwards premises have been instantiated by conclusions, we adopt interest in the backwards premises.
reasonforwards
If a triple X,Y,S instantiates a forward reasonschema, some member of X is newly concluded, and the other members of X have already been concluded, then adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.
q
reasonbackwards
Given a new interest in a sequent S such that for some X,Y, the triple X,Y,S instantiates a backward reasonschema and all members of X have already been concluded, then adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S. If some members of X have not been concluded, then simply record X,Y as a potential reason for S, for use by dischargeinterest.
dischargeinterest
If X,Y,S instantiates a backward reasonschema, interest has been adopted in S, some member of X is newly concluded and all other members of X have already been concluded, adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.
If X,Y,S instantiates a forward reasonschema, all members of X have already been concluded, and some member of Y is newly concluded, adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.<Gb*>`t%rInterestDriven Reasoning
zsDefining ReasonSchemasd(defforwardsreason symbol
:forwardspremises list of formulas
:backwardspremises list of formulas
:conclusions list of formulas
:variables list of symbols)
(defbackwardsreason symbol
:conclusions list of formulas
:forwardspremises list of formulas
:backwardspremises list of formulas
:variables list of symbols)
!!32,tDefining ReasonSchemas(defforwardsreason MODUSPONENS
:conclusions Q
:forwardspremises
P
(P Q)
:variables P Q)
(defbackwardsreason ADDITION
:conclusions (P&Q)
:backwardspremises
P
Q
:variables P Q)
&!P}6e`vBQuantifiers Instantiation RulesForwards reasons:
quantifier negation eliminations:
infer ($x) P from ~("x)P
infer ("x) P from ~($x)P
universal instantiation:
infer Sb(c,x)P/X from ("x)P/X where c is a term already occurring in some conclusion Q/Y such that Y X and Sb(c,x)P results from substituting c for all free occurrences of x in P. If there are no such terms, infer Sb(@,x)P/X from ("x)P/X.
existential instantiation:
infer Sb(@x,x)P/X from ($x)P/X where @x is a constant that has not previously occurred in any conclusions.
Auxiliary rule for forwards reasoning:
If Q/Y is a newly adopted conclusion, then for each conclusion of the form ("x)P/X such that Y X, infer Sb(c,x)P/X from ("x)P/X where c is a term occurring in Q/Y but not occurring in any previous conclusions.
!
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Udi8iw!Quantifiers  instantiation rules(Backwards reasons:
quantifier negation introductions:
adopt interest in ($x) P to infer ~("x)P
adopt interest in ("x) P to infer ~($x)P
universal generalization:
adopt interest in Sb(^x,x)P/X to infer ("x)P/X, where ^x is a free variable that has not previously occurred in any conclusions.
existential generalization:
adopt interest in Sb(c,x)P/X to infer ($x)P/X where c is a term already occurring in some conclusion Q/Y such that Y X. If there are no such terms, adopt interest in Sb(@,x)P/X to infer ($x)P/X.
Auxiliary rule for backwards reasoning:
If Q/Y is a newly adopted conclusion, then for each interest of the form ($x)P/X such that Y X, adopt interest in Sb(c,x)P/X to infer ($x)P/X where c is a term occurring in Q/Y but not occurring in any previous conclusions. !!I04&D
0PlxVQuantifiers Skolemizationand Unification\In forwardsreasoning, universally bound variables are instantiated by free variables (this is the rule UI), and existentially bound variables are instantiated by skolemfunctions whose arguments are all the free variables already occurring in the formula (this is EI).
In backwardsreasoning, existentially bound variables are instantiated by free variables (this is the rule EG), and universally bound variables are instantiated by skolemfunctions whose arguments are all the free variables already occurring in the formula (this is UG).
Forwards reasoning and interestdischarge then use unification.,uDeductive Reasoning in OSCAR,OSCAR is surprisingly efficient as a deductive reasoner.
In a recent comparison with the the highly respected OTTER resolutionrefutation theorem prover on a set of 163 problems chosen by Geoff Sutcliffe from the TPTP theorem proving library:
OTTER failed to get 16
OSCAR failed to get 3
On problems solved by both theorem provers, OSCAR (written in LISP) was on the average 40 times faster than OTTER (written in C)
OSCAR s advantage lies in its startling efficiency in proofsearch.
Completeness and Soundness of Natural DeductionH9v9v
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'US Letter Paper):/TimesArial PalatinoSymbolMyfontMicrosoft Office 98GPart Four: Epistemic Cognition, Focussing First on Deductive ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningEpistemic ReasoningOSCAR as a Deductive ReasonerNatural DeductionBidirectional ReasoningSome Inference RulesDirectionalityInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningInterestDriven ReasoningGeneralized Backwards ReasonsGeneralized Forward ReasonsNo Slide TitleInterestDriven ReasoningNo Slide TitleDefining ReasonSchemasDefining ReasonSchemas"Quantifiers Instantiation Rules"Quantifiers  instantiation rules,Quantifiers Skolemization and UnificationDeductive Reasoning in OSCARNo Slide TitleFonts UsedDesign Template
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"$A'+()*#&,O$~r$CHpqը¯å$i~r$+QTbM#"\Dii~r$VnfqƠ>i~`1? ܃+@g4~d~d@m40ppp@<4BdBdg ?%MtFPart Four:Epistemic Cognition, FocussingFirst on Deductive Reasoning 3Epistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningGEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningHEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoninguEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningvEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningxEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningyEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningzEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoningEpistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoning}Epistemic ReasoningEpistemic reasoning is driven by both input from perception and queries passed from practical cognition.
The way in which epistemic interests effect the course of cognition is by initiating backward reasoning.
Example of bidirectional reasoning_OSCAR as a Deductive ReasonerOSCAR s greatest virtue as an automated reasoner is that it is capable of performing defeasible reasoning. However, the defeasible reasoner is built on top of a deductive reasoner, and is best understood by looking at the deductive reasoner first.
aNatural DeductionOSCAR s reasoning is in the style of natural deduction .
I take this to mean that it reasons about what follows from the premises given suppositions.
This is implemented by having OSCAR reason about sequents. These are pairs <supposition,formula> where supposition is a set of formulas. Abbreviated formula / supposition.
The most characteristic rule of suppositional reasoning is CONDITIONALIZATION:
Given an interest in (P Q)/X suppose {P} and adopt interest in inferring Q/{P}X.
Another example, DILEMMA:
Given (P v Q) and an interest in R/X, adopt interest in R/X{P} and R/X{Q}. :[U!O!:I3%=;> `Bidirectional ReasoningDPerhaps the most novel feature of OSCAR s deductive reasoning is that reasonschemas are segregated into backward and forward schemas.
forward schemas lead from conclusions to conclusions
From (P & Q), infer P.
backward schemas lead from interests to interests
From P, Q infer (P & Q).
`5!2!bSome Inference Rules adjunction simplification
p/X q/Y (p&q)/X
(p&q)/XY p/X q/X
negation introduction negation elimination
p/X ~~p/X
~~p/X p/X
addition disjunctive syllogism
p/X ~p/X, (pvq)/Y ~q/X, (pvq)/Y
(pvq)/X (qvp)/X q/XY p/XY
conditionalization modus ponens modus tollens
q/X{p} p/X, (p q)/Y ~q/X, (p q)/Y
(p q)/X q/XY ~p/XY
reductio1 reductio2
p/X{~p} (q & ~q)/X{~p}
p/X p/XL!
7
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u*cDirectionalityMost of these inference rules have natural directions, and are combinatorially explosive when applied in the opposite direction.
A plausible classification:
Forwards reasons Backwards reasons
simplification adjunction
negation elimination negation introduction
disjunctive syllogism addition
modus ponens conditionalization
modus tollens reductio1
Note that reductio2 fits neither category. I will return to this.
l%DD>?RdInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests..NTNS,  beInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bfInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bgInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bhInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  biInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bjInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  b{InterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  blInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bmInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  bnInterestDriven ReasoningWe can think of interestdriven reasoning as consisting of three operations:
(1) we reason forwards from previously drawn conclusions to new conclusions;
(2) we reason backwards from interests to interests;
(3) when we have reasoned backwards to a set of sequents as interests and forwards to the same set of sequents as conclusions, then we discharge interest and conclude the sequent that led to those interests.$NTNT,  boInterestDriven Reasoning{
reasonforwards
If a set of sequents X is a forwards reason for a sequent S, some member of X is newly concluded, and the other members of X have already been concluded, then conclude S.
reasonbackwards
If interest is adopted in a sequent S, and a set X of sequents is a backwards reason for S, then adopt interest in any members of X that have not already been concluded. If every member of X has been concluded, conclude S.
dischargeinterest
If interest was adopted in the members of X as a way of getting the sequent S, and some member of X is concluded and the other members of X have already been concluded, then conclude S.~ , ppGeneralized Backwards Reasons:This is inadequate for reductio2:
reductio2
(q & ~q)/X{~p}
p/X
The proper interpretation of this rule should be:
Given an interest in p/X, suppose {~p}. Then for each conclusion q/X{~p} drawn relative to the reductio supposition, adopt interest in ~q/X{~p}. When such a contradiction is concluded, conclude p/X.
Generalized backwards reasons have both forwards and backwards premises:
example: ("x)(Fx Gx)/X (forwards premise)
Fa/X (backwards premise)
Ga/XB"!"
2FG>J
t]yGeneralized Forward ReasonsWe can also allow forwards reasons to have backwards premises as well as forwards premises. The intent is that once the forwards premises have been instantiated by conclusions, we adopt interest in the backwards premises.
reasonforwards
If a triple X,Y,S instantiates a forward reasonschema, some member of X is newly concluded, and the other members of X have already been concluded, then adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.
q
reasonbackwards
Given a new interest in a sequent S such that for some X,Y, the triple X,Y,S instantiates a backward reasonschema and all members of X have already been concluded, then adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S. If some members of X have not been concluded, then simply record X,Y as a potential reason for S, for use by dischargeinterest.
dischargeinterest
If X,Y,S instantiates a backward reasonschema, interest has been adopted in S, some member of X is newly concluded and all other members of X have already been concluded, adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.
If X,Y,S instantiates a forward reasonschema, all members of X have already been concluded, and some member of Y is newly concluded, adopt interest in the first member of Y that has not already been concluded. If every member of Y has been concluded, conclude S.<Gb*>`t%rInterestDriven Reasoning
zsDefining ReasonSchemasd(defforwardsreason symbol
:forwardspremises list of formulas
:backwardspremises list of formulas
:conclusions list of formulas
:variables list of symbols)
(defbackwardsreason symbol
:conclusions list of formulas
:forwardspremises list of formulas
:backwardspremises list of formulas
:variables list of symbols)
!!32,tDefining ReasonSchemas(defforwardsreason MODUSPONENS
:conclusions Q
:forwardspremises
P
(P Q)
:variables P Q)
(defbackwardsreason ADDITION
:conclusions (P&Q)
:backwardspremises
P
Q
:variables P Q)
&!P}6e`vBQuantifiers Instantiation RulesForwards reasons:
quantifier negation eliminations:
infer ($x) P from ~("x)P
infer ("x) P from ~($x)P
universal instantiation:
infer Sb(c,x)P/X from ("x)P/X where c is a term already occurring in some conclusion Q/Y such that Y X and Sb(c,x)P results from substituting c for all free occurrences of x in P. If there are no such terms, infer Sb(@,x)P/X from ("x)P/X.
existential instantiation:
infer Sb(@x,x)P/X from ($x)P/X where @x is a constant that has not previously occurred in any conclusions.
Auxiliary rule for forwards reasoning:
If Q/Y is a newly adopted conclusion, then for each conclusion of the form ("x)P/X such that Y X, infer Sb(c,x)P/X from ("x)P/X where c is a term occurring in Q/Y but not occurring in any previous conclusions.
!
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Udi8iw!Quantifiers  instantiation rules(Backwards reasons:
quantifier negation introductions:
adopt interest in ($x) P to infer ~("x)P
adopt interest in ("x) P to infer ~($x)P
universal generalization:
adopt interest in Sb(^x,x)P/X to infer ("x)P/X, where ^x is a free variable that has not previously occurred in any conclusions.
existential generalization:
adopt interest in Sb(c,x)P/X to infer ($x)P/X where c is a term already occurring in some conclusion Q/Y such that Y X. If there are no such terms, adopt interest in Sb(@,x)P/X to infer ($x)P/X.
Auxiliary rule for backwards reasoning:
If Q/Y is a newly adopted conclusion, then for each interest of the form ($x)P/X such that Y X, adopt interest in Sb(c,x)P/X to infer ($x)P/X where c is a term occurring in Q/Y but not occurring in any previous conclusions. !!I04&D
0PlxVQuantifiers Skolemizationand Unification\In forwardsreasoning, universally bound variables are instantiated by free variables (this is the rule UI), and existentially bound variables are instantiated by skolemfunctions whose arguments are all the free variables already occurring in the formula (this is EI).
In backwardsreasoning, existentially bound variables are instantiated by free variables (this is the rule EG), and universally bound variables are instantiated by skolemfunctions whose arguments are all the free variables already occurring in the formula (this is UG).
Forwards reasoning and interestdischarge then use unification.,uDeductive Reasoning in OSCAR,OSCAR is surprisingly efficient as a deductive reasoner.
In a recent comparison with the the highly respected OTTER resolutionrefutation theorem prover on a set of 163 problems chosen by Geoff Sutcliffe from the TPTP theorem proving library:
OTTER failed to get 16
OSCAR failed to get 3
On problems solved by both theorem provers, OSCAR (written in LISP) was on the average 40 times faster than OTTER (written in C)
OSCAR s advantage lies in its startling efficiency in proofsearch.
Completeness and Soundness of Natural DeductionH9v9v
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