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29 Cards in this Set
- Front
- Back
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->
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Conditional Elimination:
from a conclusion AND its antecedant we may infer its consequent aka modus ponens |
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~E
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Negation Elimination:
From a wff of the form ~~P, we may infer P e.g. 'It is not the case that Richard Nixon was not president.' Removing just one of the eliminations would render a falsehood: 'It is not the case that Richard Nixon was the president.' So, a double negation is needed to render: 'It is the case that Richard Nixon was president.' |
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&I and
&E |
Conjunction Introduction:
from any wffs P and S, we may infer the conjunction P&S. Conjunction Elimination: from a conjunction (i.e., P&S) we may infer either of its conjuncts. |
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VI
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Disjunction Introduction:
from a wff P, we may infer the disjunction of P with any wff. (P may be either the first or second disjunct of this disjunction) If either or both of the disjuncts are true, then the whole disjunction is true. e.g., 'Today is Tuesday' is ture, then the disjunction Today is either Tuesday or Wednesday' must also be true . Indeed if today is Tuesday, then the disjunction of 'Today is Tuesday' with any statement (including itself) is true |
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VE
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Disjunction Elimination:
from wffs of the forms PVQ, P->S, and Q->S, we may infer the wff S. aka Constructive Dilemma (CD) e.g., Today is either Saturday or Sunday (a disjunction). Assume that if today is Saturday, then tonight there will be a concert and if today is Sunday, then again there will be a concert. Therefore, there will be a concert whether it is Saturday or Sunday. So, in order to apply VE, there need be three relevant assumptions. |
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<->I
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Biconditional Introduction:
from any wffs of the forms (P->S) and (S->P), we my infer P<->S. NB: NB: sort of the inverse of <-> Elimination. |
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<->E
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Biconditional Elimination:
from any wff of the form (P<->S), we may infer either (P->S) or (S->P). NB: sort of the inverse of <-> Introduction. |
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Inference Rules
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The rules of a formal system that determine which steps of reasoning are admissible in proofs.
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Hypothetical Derivations
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a derivation which begins with a hypothesis and ends when that hypothesis is discharged
NB: one may also need to embed a hypothetical argument within another. This is often the case when a conclusion is a conditional with a conditional consequent. To prove such a conclusion, one begins as awith other conditioina conclusions: hypothesize the antecedent and derive the consequent from the hypothesis for ->I. But since the consequent is itself a conditional, in order to derive it from the antecedent, we must hypothesize its antecedent for a subsidiary step of ->I. The result is a nesting of one ->I strategy inside another. |
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->I
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Conditional Introduction (hypothetical rule):
given a derivation of a wff P with the help of a hypothesis S, we may discharge the hypothesis and infer that S->P. |
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Guidelines for hypothetical reasoning
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1. Each hypothesis introduced into a proof begins a new vertical line. The hypothesis can be discharged by either ->I or ~I.
2. No occurrence of a formula to the right of a vertical line may be cited in any rule applied after that vertical line has ended. "you can bring stuff into the bubble, but stuff cannot leave the bubble." 3. If two or more hypotheses are in effect simultaneously, then the order in which they are discharged must be the reverse of the order in which they are introduced. 4.A proof is not complete until all hypotheses have been discharged. |
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~I
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Negation Introduction (hypothetical rule):
given a derivation of an absurdity from a hypothesis P, we may discharge the hypothesis and infer ~P. aka reductio ad absurdum or indirect proof To prove a negated conclusion by negation introduction, we hypothesize the conclusion WITHOUT its negation sign and derive from it an a "absurdity." This shows the hypothesis to be false, whence it follows that the negated conclusion to be true. NB: ~I can be used together with ~E to derive conclusions that are not negated. |
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If the conclusion is an atomic formula
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Then, if no other strategy is immediately apparent, hypothesize the negation of the conclusion for ~I. If this is successful, then the conclusion can be obtained after teh ~I by ~E.
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If the conclusion is a negated formula
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Then, hypothesize the conclusion without its negation sign for ~I. If a contradiction follows, the conclusion can be obtained by ~I.
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If the conclusion is a conjunction
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Then, prove each of the conjuncts separately and then conjoin them with &I.
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If the conclusion is a disjunct
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Then, sometimes (though not often) a disjunctive conclusion can be proved directly simply by proving one of its disjuncts and applying VI. Otherwise, hypothesize the negation of the conclusion and try ~I.
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If the conclusion is a conditional
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Then, hypothesize its antecedent and derive its consequent by ->I.
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If the conclusion is a biconditional
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Then, use ->I twice to prove the two conditionals needed to obtain the conclusion by <->I.
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Derived rules of inference
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a rule of inference which is not one of the basic or defining rules of a formal system but which can be proved in that system.
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Substitution Instance
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an instance of a a wff or an argument form is the result of replacing zeror or more of its sentence letters by wffs, each occurrence of the same sentence letter being replaced by the same wff.(zero instead of one to allow each form to count as a substitution instance of itself)
[4.4, pg.97] P->Q, ~Q |- ~P substitution instance of this form is: (RVS)->~C, ~~C|-~(RVS) |
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Associated Inference Rule
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from the premises of any substitution instance of the form, we may validly infer the conclusion of that substitution instance
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Modus Tollens (MT)
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a derived rule which allows you to go straight to the negation of the antecedent if you already have the negation of the consequent
4.31, pg97-8 (RVS)->~C |- ~~C->~(RVS) 1.(RVS)->~C...A 2.|~~C...Hyp 3.|~(RVS)...MT 1,2 4.~~C->~(RVS)...->I 2-3 instead of, 1.(RVS)->~C...A 2.|~~C...Hyp 3.| |RVS...Hyp 4.||~C...->E 1,3 5.||~C&~~C...&I 2,4 6.|~(RVS)...~I 3-5 7.~~C->~(RVS)->I 2-6 |
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Repeat or Reiteration (RE)
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allows us to derive in the context of a hypothetical derivation any wff that already occurs in the main derivation, provided that it is not part of a hypothetical derivation whose hypothesis has been discharged.
4.35, pg.99 P|-Q->P 1. P...A 2. |Q...Hyp 3. |P...RE 1 5. Q->P...->I 2-3 instead of, 1. P...A 2. |Q...Hyp 3. |P&P...&I 1,1 4. |P...&E 3 5. Q->P...->I 2-4 |
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Disjunctive Syllogism (DS)
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PVQ,
~P |- Q Either P is true, or Q is true. It i not the case that P is true, therefore Q. So, as long as we eliminate all the disjuncts but one, that one must be true--assuming, of course, that the disjunctive premise is true to begin with. The disjunctive syllogism proceeds by denying one of the disjuncts. |
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Constructive Dilemma (CD)
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PVQ,
P->R, Q->S |- RVS It is either raining (P) or it is snowing (Q). If it is raining, then it is wet (R). If it is snowing, then it is cold (S). Therefore, it is either wet or cold (RVS). |
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Theorems
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wffs provable without making any non-hypothetical assumptions.
The proof of a theorum typically begins with one or more hypotheses, which are later discharged by ->I or ~I. Good strategies for deriving a theorem: If there is -> or <-> present, then hypothesize the antecedent. If there is V or & present, then hypothesize the negation |
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Equivalences
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...a biconditional which i a theorem.
To prove an equivalence, follow the usual strategy for proving biconditionals: prove the two conditionals needed for <->I by two separate conditional proofs |
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Distribution (DIST)
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P V (Q&R) <-> (P&Q) V (P&R)
P V (Q&R) <-> (PVQ) & (PVR) lets you go from V to & so you can employ &E |
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DeMorgan's (DM)
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~(PVQ) <-> ((~P) & (~Q))
~(P&Q) <-> ((~P) V (~Q)) |
