the semantics of sl defining logical notions (validity, logical equivalence, and so forth) in...

The semantics of SL

Defining logical notions (validity, logical equivalence, and so forth) in terms of truth-value assignments

A truth-value assignment: the assignment of T or F to each of the atomic sentences included in a sentence, or a set of sentences, or a group of sentences.

The semantics of SL Truth tables: an effective procedure for

establishing the logic status of individual sentences, sets of sentences, arguments, and so forth.

Each row of a truth table contains a truth value assignment.

Taken together the rows that include truth value assignments represent all the ways the world might be relevant to the sentence(s) involved.

The semantics of SL Defining logical notions in terms of truth-value

assignments: the case of sentences A sentence is truth-functionally true IFF it is true

on every TVA (or IFF there is no TVA on which it is false).

A sentence is truth-functionally false IFF it is false on every TVA (or IFF there is no TVA on which it is true.

A sentence is truth-functionally indeterminate IFF it is neither truth-functionally true nor truth functionally false (of IFF it is true on at least one TVA and false on at least one TVA).

Truth table conventions continued

GG HH ~(G ~(G H) & (G H) & (G ~H) ~H)

TT TT F F F F TT

TT FF

FF TT

FF FF

Truth table shortcuts

On any TVA: If one conjunct is false, the conjunction is

false. If one disjunct is true, the disjunction is true. If the antecedent of a material conditional is

false, the conditional is true. If the consequent of a material conditional is

true, the conditional is true. There are no shortcuts for establishing the

truth value of a biconditional.

Truth table to establish the truth functional status of individual sentences

1. Only an entire truth table can establish that a sentence is truth functionally true.

2. Only an entire truth table can establish that a sentence is truth functionally false.

3. A two row truth table can establish that a sentence is truth functionally indeterminate.

4. A one row truth table can establish that a sentence is not truth functionally true.

5. A one row truth table can establish that a sentence is not truth functionally false.

Proving that a sentence is truth-functionally indeterminate using a shortened truth table

CC DD EE C C (D (D E) E)

FF

TT



TT F F FF

TT



TT TT FF TT F F FF

TT



TT TT FF TT F F FF

FF TT TT FF T T

Truth functional validity

Defining logical notions in terms of truth-value assignments: the case of arguments.

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

An argument is truth functionally invalid IFF there is a truth value assignment on which all the premises are true and the conclusion is false.

Compare (again!)

If you studied hard, you did If you studied hard, you did well in PHIL 120.well in PHIL 120.

You studied hard.You studied hard.------------------------------------------------------------------You did well in PHIL 120.You did well in PHIL 120.

S S W WSS--------------------WW

If you studied hard, you did If you studied hard, you did well in PHIL 120.well in PHIL 120.

You did well in PHIL 120.You did well in PHIL 120.--------------------------------------------------------------------You studied hard.You studied hard.

S S W WWW----------------SS

Establishing truth functional validity: the first argument

SS WW S S W W SS WW

TT TT T T TT TT

TT FF FF TT FF

FF TT TT FF TT

FF FF TT FF FF

Establishing truth functional invalidity:the second argument

SS WW S S W W WW SS

TT TT T T TT TT

TT FF FF FF TT

FF TT TT TT FF

FF FF

Using a one row truth table to prove that an argument is Using a one row truth table to prove that an argument is truth functionally invalid:truth functionally invalid:

A v BA v BBB

__________~A~A

AA BB A v BA v B BB ~A~A

T T TT FF


A v BA v BBB

__________~A~A


TT T T TT FF


A v BA v BBB

__________~A~A


TT TT T T TT FF

Truth functional equivalence

Sentences P and Q are truth functionally equivalent IFF there is no TVA on which P and Q have different truth values.

Members of a pair of sentences are truth functionally non-equivalent IFF there is a TVA on which P and Q have different truth tables.

Only an entire truth table can prove that 2 sentences are truth functionally equivalent.

A one row truth table can prove that 2 sentences are not truth functionally equivalent.

Proving truth functional equivalenceProving truth functional equivalence

LL MM (L (L M) & (M M) & (M L) L) L L M M

TT TT T TT T TT

TT FF

FF TT

FF FF



TT TT T T TT T T TT

TT FF

FF TT

FF FF



TT TT T T TT T T TT

TT FF F F FF FF

FF TT F F FF FF

FF FF T T T T TT TT

Proving that 2 sentences are Proving that 2 sentences are notnot truth- truth-functionally equivalentfunctionally equivalent

DD OO ~(D v O)~(D v O) ~D v ~O~D v ~O

TT TT F F TT F F F F FF

TT FF F F TT F F T T T T

FF TT FF T T TT

FF FF TT TT

Truth functional consistency

A set of sentences is truth functionally consistent IFF there is at least one TVA on which all the members of the set are true.

A set of sentences is truth functionally inconsistent IFF there is no TVA on which all the members of the set are true.

A one row truth table can prove a set of sentences is truth functionally consistent.

Only an entire table can prove a set of sentences is truth functionally inconsistent

Proving a set of sentences is truth functionally consistent the short way:{C & ~D, F, ~F ~D}

CC DD FF C & ~DC & ~D FF ~F ~F ~D ~D

TT



TT FF TT



TT FF TT TT



TT FF TT TT TT



TT FF TT TT TT F F TT

Truth functional entailment

Conventions: are used to indicate sets, with individual

sentences separated by comas (gamma) is used as a meta variable for a

set of sentences ╞ (double turnstile) symbolizes the

relationship of entailment that can obtain between a set of sentences of SL and an individual sentence of SL


╞ P (a formula in the meta language)Is read as “A set of sentences truth

functionally entails a sentence P” A set of sentences truth functionally

entails a sentence P IFF there is no truth value assignment on which all the members of are true and P is false.

In SL:M (A v B), ~A v ~B, ~A M} ╞ ~M


╞ with a line drawn through it from top right to bottom left symbolizes that the relationship of truth functional entailment does not hold.

A set of sentences does not truth functionally entail a sentence P IFF there is one truth value assignment on which all the members of are true and P is false.

Using a truth table to prove entailment:Using a truth table to prove entailment:{A v C, ~C{A v C, ~C ╞ ╞ AA

AA CC A v CA v C ~C~C AATT TTTT FFFF TTFF FF


AA CC A v CA v C ~C~C AATT TT TTTT FF TTFF TT FFFF FF FF


AA CC A v CA v C ~C~C AATT TT TTTT FF TTFF TT TT FFFF FF FF


AA CC A v CA v C ~C~C AATT TT TTTT FF TTFF TT TT FF FFFF FF FF FF

Using a truth table to prove a set does Using a truth table to prove a set does notnot entail entail a sentence (that the following is false:a sentence (that the following is false:

{A v C, ~C{A v C, ~C ╞ ╞ ~A~A

AA CC A v CA v C ~C~C ~A~ATT TT TT FF FFTT FF FFFF TT TTFF FF TT

Using a truth table to prove a set does Using a truth table to prove a set does notnot entail entail a sentence (that the following is false):a sentence (that the following is false):

{A v C, ~C{A v C, ~C ╞ ╞ ~A~A

AA CC A v CA v C ~C~C ~A~ATT TT TT FF FFTT FF TT TT FFFF TT TTFF FF TT

the semantics of sl defining logical notions (validity, logical equivalence, and so forth) in...

Documents

row truth table

truth table shortcuts

entire truth table

truth table conventions

truth value assignments

truth functional validity

semantics of sl truth

true iff