Rules for Symbolic Logic in
Elementary Logic, by Benson Mates

BASIC SENTENTIAL CALCULUS (SC) RULES

P, Introduction of Premises {i} i. P P Any SC sentence (a "well formed formula," or WFF) may be entered on a line. The premise number is the line number {i}.
MP, Modus Ponens {n} i. P -> Q {m} j. P {n+m} k. Q i,j MP If a conditional appears on one line (i) and the antecedent of the conditional appears on another line (j), the consequent of the conditional may be entered on a new line (k). The order of lines i and j does not matter. Line k inherits the premises numbers {n+m} of line i and line j.
MT, Modus Tollens {n} i. -P -> -Q {m} j. Q {n+m} k. P i,j MT If a conditional with negations on both antecedent and consequent appears on one line (i) and the unnegated consequent of the conditional appears on another line (j), the unnegated antecedent of the conditional may be entered on a new line (k). The order of lines i and j does not matter. Line k inherits the premise numbers {n+m} of line i and line j.
C, Conditionalization {n} i. Q {n} j. P -> Q i C A conditional with any well-formed antecedent may be entered on a line (j) if the consequent appears on an earlier line (i). Line j inherits the premise numbers {n} of line i. OR: {m} i. P {n+i} j. Q {n} k. P -> Q i,j C A conditional with any well-formed antecedent may be entered on a line (k) if the consequent appears on an earlier line (j). Line k inherits the premise numbers {n+i} of line j, unless the antecedent appears in an earlier line (i), in which case the line number i may be removed if it occurs among the premise numbers of j, reducing {n+i} to {n}.
D, Definitional Interchange The following equivalent expressions may be exchanged for each other wherever they occur. The new line inherits the premise numbers of the old line: (P v Q) <-> (-P -> Q) (P & Q) <-> -(P -> -Q) (P <-> Q) <-> ((P -> Q) & (Q -> P))
STRATEGY FOR USE OF SENTENTIAL CALCULUS RULES
1. CONDITIONAL PROOF If you want to prove a conditional, assume the antecedent (P). Using Conditionalization (C) at the end of the proof then removes the extra premise and forms the conditional to be proven. If assuming the antecedent leaves a conditional still to be proven, then assume the antecedent of that conditional also. This procedure can be repeated as many times as necessary (i.e. recursively). The proof will end with as many applications of Conditionalization as the number of Premises that were added by this strategy.
ADVANCED SENTENTIAL CALCULUS (SC) RULES
TH, Theorem i. Th TH # Any well formed substitution instance of a Sentential Calculus (SC) Theorem may be entered on a line (i), with the null set () as premise number. OR: {n} i. P {n} j. Q i TH # [where Th# is of the form (P -> Q)] If a Theorem is a conditional, the consequent of the Theorem may be entered on a line (j) if the antecedent occurs on an earlier line (i). Line j inherits the premise numbers {n} of line i. OR: {n} i. P {m} j. Q {m+n} k. R i,j TH # [where Th# is of the form (P -> (Q -> R))] If a Theorem is a conditional, and its consequent is a conditional, the consequent of the consequent of the Theorem may be entered on a line (k) if the antecedent occurs on an earlier line (i) and the antecedent of the consequent occurs on another earier line (j). The order of lines i and j does not matter. Line k inherits the premise numbers {m+n} of line i and line j. OR, in general: The theorem rule may telescope any number of applications of Modus Ponens with theorems that have successive embedded conditionals in consequents and with the successive antecedents available on previous lines, citing all the lines and adding their premise numbers.
R, Replacement {n} i. P {n} j. Q i R # [where Th# is of the form (P <-> Q), a biconditional] If a line (i) is or contains a part of a line that is a substitution instance of one side of a theorem that is a biconditional, it or the part may be replaced with the corresponding substitution instance of the other side of the theorem and entered on a new line (j). Line j inherits the premise numbers {n} of line i.
STRATEGY FOR USE OF SENTENTIAL CALCULUS RULES
2. INDIRECT PROOF (Reductio ad Absurdum) Assume the negation of the conclusion (P). The task then is to derive a contradiction. If the contradiction involves the conclusion and its negation, then the proof ends with an application of Conditionalization (C), with the conclusion in the consequent, and Clavius (Th 16 or 17), to give the conclusion, free of the added premise. If the contradiction does not involve the conclusion, then the proof ends with an application of Duns Scotus (Th 8), to give the conclusion, Conditionalization (C), with the conclusion in the consequent, and Clavius (Th 16 or 17), to give the conclusion, free of the added premise. Use Indirect Proof when: 1. You don't know what else to do. 2. The conclusion is simple -- giving a simple negation to assume. 3. There is something immediate or obvious that can be done with the new assumption. Note: Indirect Proofs are not always simpler or easier than Direct Proofs. However, Indirect Proof was used quite early by the Greeks to show that 2 was an
irrational number and that there was no largest prime number.
QUANTIFIER RULES
US, Universal Specification {n} i. (x)Fx {n} j. Fc i US The universal quantifier (x) may be dropped and the variable x replaced with any constant c. Line j inherits the premise numbers {n} of line i.
EG, Existential Generalization {n} i. Fc {n} j. (x)Fx i EG The existential quantifier (x) may be added and the constant c replaced with a variable x. Line j inherits the premise numbers {n} of line i.
UG, Universal Generalization {n} i. Fc {n} j. (x)Fx i UG The universal quantifier (x) may be added and the constant c replaced with a variable x, IF the constant c does not occur in Fx OR in any of the premises {n} of line i. Line j inherits the premise numbers {n} of line i.
ES, Existential Specification {n} i. (x)Fx {j} j. Fc P {m+j} k. Q {m+n} k+1. Q i,j,k ES If the existential quantifier (x) from a line i is dropped and the variable x replaced with a constant c as a premise on line j, and Q is derived on line k with j as a premise, then Q may be entered on line k+1, IF the constant c does not occur in Q, OR in line i, OR in any premise {m} of line k. Line k+1 inherits the premise numbers {m+n} of lines i and k, except for j.
Q, Quantifier Exchange The following equivalent expressions may be exchanged for each other wherever they occur. The new line inherits the premise numbers of the old line: -(x)Fx <-> (x)-Fx (x)-Fx <-> -(x)Fx -(x)-Fx <-> (x)Fx (x)Fx <-> -(x)-Fx
PROPERTIES OF RELATIONS Symmetrical: (x)(y)(Rxy -> Ryx) next to, married to Asymmetrical: (x)(y)(Rxy -> -Ryx) north of, older than Nonsymmetrical loves, brother of Transitive: (x)(y)(z)((Rxy & Ryz) -> Rxz) north of, ancestor of Intransitive: (x)(y)(z)((Rxy & Ryz) -> -Rxz) mother of, father of Nontransitive: loves, different from Totally Reflexive: (x)Rxx identical to Reflexive: (x)((y)(Rxy v Ryx) -> Rxx) same age as Irreflexive: (x)-Rxx north of, married to Nonreflexive: loves, hates, critizes
QED At the end of a successful proof, the logician or mathematician optionally can write QED, an abbreviation for the Latin phrase quod erat demonstrandum, meaning "which was to be proven." This was a translation of the Greek phrase hóper édei deîksai, which meant "the very thing it was needed to show."

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