Introductory Logic PHI 120 Presentation: "Truth Tables – Validity vs. Soundness" This PowerPoint Presentation contains a large number of slides, a good many of which are nearly identical. If you print this Presentation, I recommend six or nine slides per page.

Homework 1.Study Allen/Hand Logic Primer – Sec. 1.1, p. 2: “soundness” – Sec. 2.2, p. 45, “incompatible premises” 2.Ex. 2.2: i-xii

VALIDITY vs. SOUNDNESS P & Q, ~P ⊢ R Validity: An argument is valid if and only if: if all of its premises are true its conclusion is true. Validity: An argument is valid if and only if: if all of its premises are true its conclusion is true. Corollary: It is impossible for a valid argument to have: all true premises false conclusion Corollary: It is impossible for a valid argument to have: all true premises false conclusion

Validity vs. Soundness Valid Argument No invalidating assignment Criteria of a Sound Argument 1.argument is valid and 2.all premises are True. Valid but Unsound  no invalidating assignment  not all premises true Invalidating Assignment (1) conclusion is False (2) all premises are True Invalidating Assignment (1) conclusion is False (2) all premises are True

Incompatible Premises PQRP&R,~P ⊢ Q

PQRP&R,~P ⊢ Q Atomic statements MUST be written in alphabetical order

Incompatible Premises PQRP&R,~P ⊢ Q

PQRP&R,~P ⊢ Q T T T T F F F F

PQRP&R,~P ⊢ Q TT TT TF TF FT FT FF FF

PQRP&R,~P ⊢ Q TTT TTF TFT TFF FTT FTF FFT FFF

PQRP&R,~P ⊢ Q TTT TTF TFT TFF FTT FTF FFT FFF

PQRP&R,~P ⊢ Q TTT TTF TFT TFF FTT FTF FFT FFF

PQRP&R,~P ⊢ Q TTTF TTFF TFTF TFFF FTTT FTFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFF TFTF TFFF FTTT FTFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFF FTTT FTFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFFF FTTT FTFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFFF FTTFT FTFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFFF FTTFT FTFFT FFTT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFFF FTTFT FTFFT FFTFT FFFT

PQRP&R,~P ⊢ Q TTTF TTFFF TFTF TFFFF FTTFT FTFFT FFTFT FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFT Testing for Validity: Find the Invalidating Assignment

Incompatible Premises PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTFF TFFFF FTTFT FTFFT FFTFT FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFFF FTTFT FTFFT FFTFT FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFTF FFFFT

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFTF

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFT No Invalidating Assignment So the argument is valid Incompatible Premises

PQRP&R,~P ⊢ Q TTTTF TTFFF TFTTF TFFFF FTTFT FTFFT FFTFT FFFFT

Validity vs. Soundness Valid Argument Impossible for conclusion to be False and all premises True Sound Argument An argument is sound if and only if it is valid and all its premises are true. Valid but Unsound  No invalidating assignment  Not all premises true

SEQUENTS Truth Tables Determine truth-values of: 1.atomic statements 2.negations of atomics 3.inside parentheses 4.negation of the parentheses 5.any remaining connectives Determine truth-values of: 1.atomic statements 2.negations of atomics 3.inside parentheses 4.negation of the parentheses 5.any remaining connectives

Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency. ~P, R v ~P P v Q ⊢ ~Q First, identify the governing connectives.

~P, R v ~P P v Q ⊢ ~Q First, identify the governing connectives. Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q First, identify the governing connectives. Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, (R v ~P) (P v Q) ⊢ ~Q The second premise is a complex binary: Φ Ψ Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q The conclusion is a negation. Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

CONSTRUCT THE TRUTH TABLE Truth Tables – Sequents

~P, R v ~P P v Q ⊢ ~Q Determine the number of rows for the sequent 2 3 simple statements = 8 rows

~P, R v ~P P v Q ⊢ ~Q ~P,Rv~P PvQ ⊢ ~Q _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q _ __ Valid ____ Invalid Alphabetical Sequence! Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q T T T T F F F F _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TT TT TF TF FT FT FF FF _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTT TTF TFT TFF FTT FTF FFT FFF _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFF TTFFF TFTFF TFFFF FTTTT FTFTT FFTTT FFFTT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFFF TTFFFF TFTFFT TFFFFT FTTTTF FTFTTF FFTTTT FFFTTT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFFF TTFFFF TFTFFT TFFFFT FTTTTF FTFTTF FFTTTT FFFTTT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency. (R v ~P) (P v Q)

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFFF TTFFFFF TFTFFT TFFFFFT FTTTTF FTFTTF FFTTTT FFFTTT _ __ Valid ____ Invalid (R v ~P) (P v Q) Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFFF TTFFFFF TFTFFT TFFFFFT FTTTTF FTFTTF FFTTTT FFFTTT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFF TTFFFFF TFTFTFT TFFFFFT FTTTTTF FTFTTTF FFTTTTT FFFTTTT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFF TTFFFFF TFTFTFT TFFFFFT FTTTTTF FTFTTTF FFTTTTFT FFFTTTFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFF TTFFFFF TFTFTFT TFFFFFT FTTTTTF FTFTTTF FFTTTTFT FFFTTTFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTF TTFFFFTF TFTFTFTT TFFFFFTT FTTTTTTF FTFTTTTF FFTTTTFT FFFTTTFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTF TTFFFFFTF TFTFTFTT TFFFFFFTT FTTTTTTF FTFTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTF TTFFFFFTF TFTFTFTT TFFFFFFTT FTTTTTTF FTFTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Testing for Validity: Find the Invalidating Assignment Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid ____ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTT T TT T T F FTFTTTTTF FFTTTTFFT FFFTTTFFT _ __ Valid _ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT TautologyInconsistencyContingency _ __ Valid _ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

~P, R v ~P P v Q ⊢ ~Q PQR~P,Rv~P PvQ ⊢ ~Q TTTFTFTTF TTFFFFFTF TFTFTFTTT TFFFFFFTT FTTTTTTTF FTFTTTTTF FFTTTTFFT FFFTTTFFT TautologyInconsistencyContingency _ __ Valid _ Invalid Truth Tables Directions: (i) Construct a Truth Table in the grids provided. (ii) Circle the governing connective in each sentence. (iii) If the sequent is invalid, circle the invalidating assignment and check the line which reads INVALID. If the sequent is valid, check the line which reads VALID. (iv) In the space provided, identify what kind of truth value the conclusion has: Tautology, Inconsistency, or Contingency.

Homework 1.Study Allen/Hand Logic Primer – Sec. 1.1, p. 2: “soundness” – Sec. 2.2, p. 45, “incompatible premises” 2.Ex. 2.2: i-xii