ostap

Propositions and Arguments

• A proposition is either true or false.
• An argument is either valid or invalid.
• Consistency is applied to multiple propositions.

A set of propositions is jointly consistent if and only if there exists a possible situation in which all of these propositions are true.

Example of a consistent set of propositions:

{ All humans are mammals, Lorenz is a human, Brussels is the capital of Belgium }

This is a set of propositions, not an argument (there’s no conclusion).

• These propositions are consistent because they can be true together.

Another consistent set:

{ All humans are mammals, Lorenz is a crocodile, Leuven is the capital of Belgium }

• We are not concerned with whether these propositions are true, but whether they could be true together.

Argument Transformation

You can transform an argument of the type:

All Y are Z, X is Y, so X is Z

into a set of premises by negating the conclusion:

{ All Y are Z, X is Y, X is not Z }

• If the argument is valid, this will result in an inconsistent set of premises.

Example: Invalid Argument

All logicians are smart, Einstein is smart, so Einstein is a logician.

• This is an invalid argument.

The corresponding set of premises:

{ All logicians are smart, Einstein is smart, Einstein is not a logician }

• This is a consistent set of premises.

Equivalence

Two propositions are equivalent if and only if they are true in exactly the same possible situations.

Examples:

• Equivalent Propositions:
  • “Pete kisses Mary.”
  • “Mary is kissed by Pete.”
• Non-equivalent Propositions:
  • “Lorenz is a human.”
  • “Lorenz is a mammal.”

Formal Definitions

• The propositions A and B are equivalent if and only if the arguments “A, therefore B” and “B, therefore A” are both valid.
• Validity is independent of the actual truth (de facto truth) of the propositions.

Example Exam Question

Suppose you are given an argument with three premises. Two premises are true, one premise is false, and the conclusion is false. We cannot determine whether the argument is valid unless it is said that the premises are true and the conclusion is false — in that case, the argument would be invalid.

Example Arguments:

• True premise, true conclusion:

  Brussels is the capital of Belgium, so Paris is the capital of France.

  This is an invalid argument.

• True premise, false conclusion:

  Brussels is the capital of Belgium, so Toulouse is the capital of France.

  This is an invalid argument.

• False premise, true conclusion:

  Leuven is the capital of Belgium, so Paris is the capital of France.

  This is an invalid argument.

• False premise, false conclusion:

  Leuven is the capital of Belgium, so Toulouse is the capital of France.

  This is an invalid argument.

Premise-Conclusion Validity Table

Premise(s)	Conclusion	Validity
false	false	unknown
false	true	unknown
true	true	unknown
true	false	invalid

This follows the logic of a NOT P OR C operation.

Soundness

An argument is sound if and only if:

• The argument is valid.
• The premises are true.