definitions and concepts
define truth functional truth and falsity:
truth: a property of sentences, the sentence is truth functionally true if and only if it is true on all truth value assignments
falsity: property of sentences, the sentence is truth functionally false if and only if is false on all truth value assignments
Construct a truth table for the following sentence and determine if it is Truth Functionally True, Truth Functionally False or Truth Functionally Indeterminate
C v ~ C
Is the following sentence Truth Functionally True, Truth Functionally False, or Truth Functionally Indeterminate?
Z Horseshoe ( X v Z)
https://docs.google.com/presentation/d/1uBWqUUXwbgkDuQiz2CEpaPKHUxyuzVkGCjOsW1caRok/edit?usp=sharing
Truth Functionally True
True or false: you can construct a shortened truth table to prove inconsistency
false - you need to consider every combination of truth values
what is the corresponding material conditional for this argument:
K = (triple bar) L
L horseshoe J
Negation J
conclusion:( negation K v L )
([K =L] & [L horseshoe J] & negation J]) horseshoe (negation k v L)
define truth functional entailment
and explain in your own words how to identify this property on a truth table
relationship between a set and a sentence; set gamma truth functional entails sentence p if and only if there is no truth value assignment on which all members of the set are true and the sentence false
(similar to validity, if there is a row where all members of gamma have a true truth value, and the sentence P has a false truth value, it does not entail each other)
Construct a truth table for the following sentence and determine if it is Truth Functionally True, Truth Functionally False or Truth Functionally Indeterminate
Z ⸧ (X v Z)
Is the following sentence Truth Functionally True, Truth Functionally False, or Truth Functionally Indeterminate?
(A horseshoe B) horseshoe A
https://docs.google.com/presentation/d/1uBWqUUXwbgkDuQiz2CEpaPKHUxyuzVkGCjOsW1caRok/edit?usp=sharing
Indeterminate
construct a shortened truth table for: {F ⊃ (J ∨ K), F <--> J}
truth functionally consistent
F J K | F ⊃ ( J ∨ K) F <--> ∼ J
T F T T T F T T T T T F
write a proof for truth functionally false
Assume that P is truth-functionally false. Then, by definition, there is no truth-value assignment on which P is true. Consequently, as P is the only member of the unit set {P}, there is no truth-value assignment on which every member of that set is true. So {P} is truth-functionally inconsistent. Now assume that {P} is truth-functionally inconsistent. Then, by definition, there is no truth-value assignment on which every member of {P} is true. Since P is the only member of its unit set, there is no truth-value assignment on which P is true. Hence P is truth- functionally false.
define truth functional consistency:
and explain in your own words how to identify this property in a truth table
if your set is truth functionally consistent there will never be any truth value assignment where all members of the set is true and the sentence false
(Look for one row where all values are true)
Construct a Truth Table to determine if the following argument is valid:
A
A ⸧ B
_____
B
Is the following truth table Truth-Functionally equivalent?
Negation (C & negation C)
A horseshoe ( B horseshoe A)
https://docs.google.com/presentation/d/1uBWqUUXwbgkDuQiz2CEpaPKHUxyuzVkGCjOsW1caRok/edit?usp=sharing
Yes
construct a shortened truth table: A <--> (∼A <--> A)
A | A <--> (∼ A <--> A) ∼ (A ⊃ ∼ A)
T T F F T F T T T F F T
prove: If {∼ P} is truth-functionally inconsistent, then P is truth-functionally true.
If {∼ P} is truth-functionally inconsistent, then there is no truth-value assignment on which ∼ P is true (since ∼ P is the only member of its unit set). But then ∼ P is false on every truth-value assignment, so P is true on every truth-value assignment and is truth-functionally true.
define truth functional equivalence and validity!
and explain in your own words how to identify these properties in a truth table
equivalence: Sentences P and Q of SL are truth-functionally equivalent if and only if there is no truth-value assignment on which P and Q have different truth-values.
(truth values identical in each row)
validity: An argument of SL is truth-functionally valid if and only if there is no truth value assignment on which all the premises are true and the conclusion is false.
(there is no row where the left side has true truth values and the right side has false truth values)
Construct a truth table for the following set and determine if it is logically consistent:
[C v ~C, ~C & D, ~D]
Is the following set of sentences Truth Functionally Consistent?
L, L horseshoe J, Negation J
https://docs.google.com/presentation/d/1uBWqUUXwbgkDuQiz2CEpaPKHUxyuzVkGCjOsW1caRok/edit?usp=sharing
No
symbolize and make a shortened truth table for:
Eugene O’Neil was an alcoholic. His plays show it. But The Iceman Cometh must have been written by a teetotaler. O’Neill was an alcoholic unless he was a fake.
Truth-functionally consistent
E: Eugene O’Neill was an alcoholic
P: Eugene O’Neill’s plays show that he was an alcoholic.
I: The Iceman Cometh must have been written by a teetotaler.
F: Eugene O’Neill was a fake.
E F I P | E P I E ∨ F
T T T T |T T T T T T
prove: If Γ is truth-functionally inconsistent, then Γ truth-functionally entails every sentence of SL.
Assume that Γ is truth-functionally inconsistent. Then there is no truth-value assignment on which every member of Γ is true. Let P be an arbitrarily selected sentence of SL. Then there is no truth-value assignment on which every member of Γ is true and P false since there is no truth-value assignment on which every member of Γ is true. Hence Γ P.
A truth functional inconsistent set will _____ every sentence?
AND
Assume that the set not P is truth functionally inconsistent: what kind of sentence must P be? and explain.
ENTAIL
The set consisting of not p is truth functionally inconsistent if and only if not p is truth functionally false, but if not p is truth functionally false then p must be truth functionally true.
Construct the truth tables for each of the logical connectives
Is the following argument Truth Functionally Valid?
(A & G) v (B horseshoe G)
negation G v B
conclusion: Negation B v G
https://docs.google.com/presentation/d/1uBWqUUXwbgkDuQiz2CEpaPKHUxyuzVkGCjOsW1caRok/edit?usp=sharing
Yes
symbolize and make a shortened truth table for the following:
If the Red Sox win next Sunday, then if Joan bet $5 against them she’ll buy Ed a hamburger. The Red Sox won’t win, and Joan won’t buy Ed a hamburger.
R: The Red Sox will win next Sunday.
J: Joan bet $5.00 against the Red Sox.
E: Joan will buy Ed a hamburger.
E J R | R ⊃ ( J ⊃ E) ∼ R & ∼ E
F T F F T T F F T F T T F
prove: If Γ is truth-functionally consistent and P is truth-functionally true, then Γ ∪ {P} is truth-functionally consistent.
(this is very hard def, wont be on exam)
Since Γ is a truth-functionally consistent set there is at least one truth-value assignment on which every member of Γ is true. But P is also true on such an assignment since a truth-functionally true sentence is true on every truth-value assignment. Hence on at least one truth-value assignment every member of Γ ∪ {P} is true; so the set is truth-functionally consistent.