Is second-order logic logic?

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

What is wrong with second-order logic?

Problems with second-order logic

For example, the property of being a cube is not itself a cube; the property of being large is not large, etc. Facts such as these seem to be expressible in a second-order language as follows: ¬Cube(Cube) ¬Large(Large) ¬Tet(Tet) …

What is the difference between first-order logic and second-order logic?

Wikipedia describes the first-order vs. second-order logic as follows: First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals.

Is first-order logic truth functional?

First-order validities (or consequences, or equivalences) are truths (or consequences, or equivalences) solely in virtue of the truth- functional connectives, the quantifiers, and the identity symbol.

Why is second-order logic incomplete?

Theorem: 2nd order logic is incomplete: 1) The set T of theorems of 2nd order logic is effectively enumerable. 2) The set V of valid sentences of 2nd order logic is not effectively enumerable. 3) Thus, by Lemma One, V is not a subset of T.

What is the meaning of second order?

Adjective. second-order (not comparable) (mathematics, logic) describing the second in a numerical sequence of models, languages, relationships, forms of logical discourse etc. Of secondary importance.

Are all tautologies first-order Validities?

In the context of first-order logic, a distinction is maintained between logical validities, sentences that are true in every model, and tautologies (or, tautological validities), which are a proper subset of the first-order logical validities. In the context of propositional logic, these two terms coincide.

What is the truth-functional conditional?

Assuming truth-functionality — that the truth value of the conditional is determined by the truth values of its parts — it follows that a conditional is always true when its components have these combinations of truth values.

How many types of truth functions are there?

Table of binary truth functions. In two-valued logic, there are sixteen possible truth functions, also called Boolean functions, of two inputs P and Q.

What is the difference between tautology and pleonasm?

Difference between pleonasm and tautology

Pleonasm has a sense of using an unnecessary overabundance of redundant words in one description. Tautology has a sense of saying the exact same in different words, using multiple words with the same meaning.

Are all tautologies logically equivalent?

Furthermore, by definition, two sentences (or propositions) are logically equivalent if and only if they have the same truth values (no matter what truth values their atomic constituents, if any, have). So, because tautologies always have the same truth value (namely, true), they are always logically equivalent.

Are tautologies valid?

It is not originally defined in the context of premise-conclusion as you said. However, it can be proven that tautological sentences as defined previously is always the ‘true conclusion’ of any argument regardless of truth of the premises. Therefore, tautology is always valid.

Are tautologies informative?

A Tautology is a statement that is always true because of its structure—it requires no assumptions or evidence to determine its truth. A tautology gives us no genuine information because it only repeats what we already know.

Can you have a valid argument with contradictory premises?

But on a classical conception of validity, any argument with contradictory premises counts as valid, since it is impossible for all the premises of an argument with contradictory premises to be true, and so a fortiori impossible for the argument to have true premises and false conclusion.

Can a valid argument be a contradiction?

No propositionally valid argument can have a contradiction as a conclusion.

Can a valid argument have a false conclusion?

TRUE: A valid argument cannot have all true premises and a false conclusion. So if a valid argument does have a false conclusion, it cannot have all true premises. Thus at least one premise must be false.

Are all fallacies invalid?

@Curious Yes informal fallacies may be valid, but formal fallacies would be invalid, that is, the form is what is invalid about the argument.

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