Tag Archives: Sentential Logic

Sentential Logic: Rules of Inference for Deriving Proofs

EXCELLENT RESOURCE AVAILABLE HERE: https://courses.umass.edu/phil110-gmh/text/c05.pdf

Ampersand-In (&I): If one has available lines, A and B, then one is entitled to write down their conjunction, in one order A&B, or the other order B&A.

Ampersand-Out (&O): If one has available a line of the form A&B, then one is entitled to write down either conjunct A or conjunct B.

Wedge-In (∨I): If one has available a line A, then one is entitled to write down the disjunction of A with any formula B, in one order AvB, or the other order BvA.

Wedge-Out (∨O): If one has available a line of the form A∨B, and if one additionally has available a line which is the negation of the first disjunct, ~A, then one is entitled to write down the second disjunct, B. Likewise, if one has available a line of the form A∨B, and if one additionally has available a line which is the negation of the second disjunct, ~B, then one is entitled to write down the first disjunct, A.

Double-Arrow-In (↔I): If one has available a line that is a conditional A→B, and one additionally has available a line that is the converse B→A, then one is entitled to write down either the biconditional A↔B or the biconditional B↔A.

Double-Arrow-Out (↔O): If one has available a line of the form A↔B, then one is entitled to write down both the conditional A→B and its converse B→A.

Arrow-Out (→O): If one has available a line of the form A→B, and if one additionally has available a line which is the antecedent A, then one is entitled to write down the consequent B. Likewise, if one has available a line of the form A→B, and if one additionally has available a line which is the negation of the consequent, ~B, then one is entitled to write down the negation of the antecedent, ~A.

Double Negation (DN): If one has available a line A, then one is entitled to write down the double-negation ~~A. Similarly, if one has available a line of the form ~~A, then one is entitled to write down the formula A.

Sentential Logic Practice: Symbolizing More Natural Sentences

1.) Natural sentence: Either I will eat ham or I will eat turkey.
Library: H = I will eat ham, T = I will eat turkey
Symbolization: H v T

2.) Natural sentence: Yesterday, we danced, played, and ate so much!
Library:  D = we danced so much, P = we played so much, A = we ate so much
Symbolization: [D & (P&A)]

3.) Natural sentence: Harrison or John will win Prom King
Library: H = Harrison will win Prom King, J = John will win Prom King
Symbolization: H v J

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Sentential Logic Practice: Assessing Some Proofs

1.) A&B, B > (D&E), derive B > E
(A&B)
B
(D&E)
E
B > E

2.) S > (Q&R), S, derive R
S
(Q&R)
R

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Sentential Logic Practice: Symbolizing Natural Sentences

1.) Natural sentence: Either Joe Biden or Bernie Sanders will get the Democratic Presidential Nomination.
Library: B = Biden will get the Democratic Presidential Nomination
S = Sanders will get the Democratic Presidential Nomination
Symbolization: B∥S

2.) Natural sentence: If you take proper precautions, then you can help slow the spread of the novel coronavirus.
Library: T = you take proper precautions
S = you can help slow the spread of the novel coronavirus

Symbolization: T→S

3.) Natural sentence: Eat your vegetables and your meat before you have dessert.
Library: V = eat your vegetables
M = eat your meat
D = you have dessert

Symbolization: (V&M)→D; (V&M)≡D*
*material bi-conditional/material equivalence, stronger logical symbolization of the statement

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