The Truth Value of Valid and Invalid Inferences?

Aletheia_LogoTruth Omnipresent

Given that the examination of an argument’s validity or invalidity is expressible in the propositions “P is a valid syllogism” or “P is not a valid syllogism,” the examination of an argument’s validity or invalidity is implicitly also an examination of the veracity or falsity of the propositions “P is a valid syllogism” or “P is not a valid syllogism.” More explicitly, the process of identifying whether or not a syllogism is valid or invalid takes the form of a syllogism.

MP: All valid syllogisms have three terms.

mp: P does not have three terms.

C: Therefore, P is not a valid syllogism.

Conversely:

MP: No valid syllogisms has four terms.

mp: P has four terms.

C: Therefore, P is not a valid syllogism.

And so on. The validity or falsity of P, therefore, is a matter of the truth. It either is or it is not the case that P is true.

Strictly speaking, therefore, considerations of validity and invalidity are not divorced from considerations of veracity and falsity. As mentioned elsewhere, prior to induction is deduction, and prior to deduction are the axioms of thought. Without an axiomatic foundation, there can be neither induction (which utilizes the axiom of induction and deduction in the process of amassing particulars into a set) or deduction (which draws particular inferences from propositions). The structure takes the following form:

1. Identity:  A is A.

a.’ Property sharing entities are set members.

a.” MP: All entities bearing a and b properties belong to set S.

    mp: x and y bear a and b properties.

    C: Therefore, x and y belong to set S.

Antecedent to either process of reasoning, therefore, are the axioms of inference/laws of logic, and the axiom of induction.

Thus, logic is always concerned with truth, albeit at different levels. Validity has respect to the veracity or falsity of the proposition “Some argument p necessarily implies some conclusion q.” Soundness has respect to the veracity or falsity of the individual premises and their attendant conclusions.

-h.

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