TF truth, falsity, and indeterminacy
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TF truth, falsity, and indeterminacy. P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff it has the value F for any truth-value assignment. P is tf-false iff ~P is tf-true

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TF truth, falsity, and indeterminacy

P is truth-functionally true iff it has the value T for any truth-value assignment.

P is truth-functionally false iff it has the value F for any truth-value assignment.

P is tf-false iff ~P is tf-true

P is truth-functionally indeterminate iff it has the value T for some truth-value assignments, and the value F for some other truth-value assignments.

P is tf-indeterminate iff it is neither tf-true nor th-false.


TF equivalence and consistency

P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment.

A set of sentences is truth-functionally consistent iff there is a truth-value assignment that on which all the members of the set have the value T.

A set of sentences is truth-functionally inconsistent iff it is not tf-consistent.


TF entailment and validity

A set  of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of  is true and P false.


TF entailment and validity

A set  of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of  is true and P false.

An argument of SL is truth-functionally valid iff there is no truth-value assignment on which all the premises are true and the conclusion false.


TF entailment and validity

A set  of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of  is true and P false.

An argument of SL is truth-functionally valid iff there is no truth-value assignment on which all the premises are true and the conclusion false.

An argument of SL is truth-functionally invalid iff it is not tf-valid.


TF entailment and validity

A set  of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of  is true and P false.

An argument of SL is truth-functionally valid iff there is no truth-value assignment on which all the premises are true and the conclusion false.

An argument of SL is truth-functionally invalid iff it is not tf-valid.

An argument is tf-valid iff the premises tf-entail the conclusion.


TF properties

P is truth-functionally true iff it has the value T for any truth-value assignment.

P is truth-functionally false iff ~P is tf-true.

P is truth-functionally indeterminate iff P is neither tf-true nor tf-false.

P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment.

A set of sentences is truth-functionally consistent iff there is a truth-value assignment that on which all the members of the set have the value T.

A set  of SL sentences truth-functionally entails a sentence P iff there is no truth-value assignment on which every member of  is true and P false.

An argument of SL is truth-functionally valid iff there is no truth-value assignment on which all the premises are true and the conclusion false.












































Shortened truth tables
shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that ~B(B&~B) is not tf-true


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (~B~A)&C is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false

COTRADICTION!


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false

CONTRADICTION!


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-false


shortened truth tables

Show that (AB)  (BA) is not tf-true


shortened truth tables

Show that (AB)  (BA) is not tf-true


shortened truth tables

Show that (AB)  (BA) is not tf-true


shortened truth tables

Show that (AB)  (BA) is not tf-true

CONTRADICTION!

Therefore, the sentence is tf-true


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


shortened truth tables

Practice

Show that:

{A(B&C), B  (AC), C~C} is tf-consistent

{B(A&~C), (CA)  B, ~BA}  ~(AC)


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