15 – 16. Tautological Entailment

Question 1

Turn into a Mathematical Exercise (4)

Answer 1

A) formalize the argument into a PL language
B) make truth tables for the premises and the conclusion
C) consider those rows where all premises are true
D) check that the conclusion is true in all those rows

Question 2

Tautological Validity (3)

Answer 2

A) all valuations which make all its PREM=1, and CON=1
B) No valuation which make all its PREM=1, and CON=0
C) Truth-preser. in virtue of its propositional connectives
D) Remplace ∴ for ⊨

Question 3

Tautologically Invalid

Answer 3

There is a valuation which make all its PREM=1, and CON=0

Question 4

Intuitive Mismatch

Answer 4

Intuitive assessment for deductive validity, can be at odds with tautological connectives, since there are valid arguments that do posses truth in base of their connectives

Question 5

NO PREMISES (Steps and consequences)

Answer 5

Tautology viewed as the conclusion of a tautologically valid argument with no premises.
Tautology can be understood negatively, “there is not a possible valuation.”
EX: ⊨ (P ∧ Q) ∨ ¬P ∨ ¬Q
1) Make truth table
2) Delete rows without true premises
3) All rows remain since there is no row with premises
4) All conclusions are true, hence it is a tautology

Question 6

Tautological Equivalence

Answer 6

A) α ≈ β if and only if both α ⊨ β and β ⊨ α
a) Same truth tables!
B) Replacing PREM/CON with equivalent PREM/CON does not impact validity of argument. (SAME TRUTH TABLES)

Question 7

Tautological Consistency

Answer 7

A) WFF tautologically consistent if and only if a valuation makes all WFF true together.
B) Premises and the negation of its conclusion are tautologically inconsistent.

Question 8

Tautological Inconsistency

Answer 8

A set is tautologically inconsistent, if its conjunction is a contradiction!

Question 9

Indirect Method

Answer 9

Look for a row in which all PREM=1, yet CON=0
Work back from conclusion, to see if it is possible