Material conditional

Logical connectives
Tautology ${displaystyle top }$ Opposite conjunction ${displaystyle uparrow }$ Opposite implication ${displaystyle leftarrow }$ Conditional material ${displaystyle rightarrow }$ Logical disjunction ${displaystyle lor }$ Logical denial ${displaystyle lnot }$ Exclusive disjunction ${displaystyle nleftrightarrow }$ Biconditional ${displaystyle leftrightarrow }$ Logical assertion Opposition ${displaystyle downarrow }$ Logical attachment ${displaystyle nrightarrow }$ Opposite attachment ${displaystyle nleftarrow }$ Logical conjunction ${displaystyle land }$ Contradiction ${displaystyle bot }$
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The conditional material, known as conditional, really functional conditional, or imprecisely confused with material involvement, it is a logical connective that connects two propositions, usually represented as ${displaystyle Ato B}$. In propositional logic, the material conditional is a binary truth function, which becomes false when B is fake being A true, and it becomes true in any other case. In the logic of preaching, it can be seen as a relation of subsets between the extension of preached (possibly complex).

Unlike the colloquial Spanish where expressions "yes..., then..." imply cause and effect, the conditional material ${displaystyle Ato B}$ No. conventionally establishes a causal relationship between its propositions. It's just an expression that assumes a true value. false when, simultaneously, ${displaystyle A}$ is true and ${displaystyle B}$ It's fake.

The material conditional can be denoted in several ways,

${displaystyle Ato B}$

${displaystyle Asupset B}$ (although this symbol can be used as the symbol of a superset in set theory)

${displaystyle ARightarrow B}$ (although not recommended, as it is used for logical implication)

${displaystyle CAB}$ (in Polish notation)

With respect to variables A and B,

${displaystyle A}$ It's him. background conditional

${displaystyle B}$ the consecutive conditional

It is important not to confuse the concept of conditional material with the logical implication. The confusion is exacerbated because the symbols ${displaystyle Ato B}$ and ${displaystyle ARightarrow B}$ are imprecisely used as equivalent expressions by many, when they really are not. Although in day-to-day conversations the difference has no greater impact, the subtle difference between both concepts is significant in the correct understanding of the propositional logic.

Definition

The material conditional is a truth function that can take two truth values (usually the values of propositions):

• return false when the first value is true and the second false,
• and return true in any other case.

In other words, the truth table of the material conditional is the following:

${displaystyle {begin{array}{intexcluding}{hline A fakeB}{to B\\hline v fakevv}{v exposef}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}{f}}}}}}}}}}}}}{f}}$

As seen, the material conditional returns 0 (false) only when the antecedent is true and the consequent is false. In all other cases, it returns 1 (true).

Formal properties

Some of the formal properties of the material conditional are:

• Distribution: ${displaystyle (Ato (Bto C))iff (Ato B)to (Ato C),}$

• Transitivity: ${displaystyle (Ato B)to (Bto C)Rightarrow (Ato C)},$

• Background: ${displaystyle Ato (Bto C)}iff (Bto (Ato C),}$

• Idempotence: ${displaystyle Ato A,}$ It's always true.

• Preservation of the Truth: The interpretation by which all variables are assigned a true value of "true" produces a true value of "true" as a result of the material conditional.

• Right-to-left partnership: ${displaystyle Ato Bto Ciff Ato (Bto C)}$

Classical logic ${displaystyle prightarrow q}$ equivalent to ${displaystyle neg (pland neg q)}$and by the laws of De Morgan equivalent to ${displaystyle neg plor q}$. However, in minimal logic (and therefore also in intuiistic logic) ${displaystyle prightarrow q}$ It just implies ${displaystyle neg (pland neg q)}$; and in intuiistic logic (but not in minimal logic) ${displaystyle neg plor q}$ implies ${displaystyle prightarrow q}$.

Correlation with set theory

In group theory, the equivalent notion of conditional material ${displaystyle Ato B}$ is:

${displaystyle Ato Biff A^{C}cup B=(Asetminus B)^{C}}$

that is, it is the union of the complement of A and the set B, or equivalently, the complement of A minus B.

Difference between material conditional and logical implication

The material conditional should not be confused with the relation of logical implication. The difference is subtle but very important in propositional logic.

• The conditional material is a hypothetical claim does not speak of reality ; that is, it is not possible to know the true value of A or B simply by observing the expression "Yeah. A, then. B", without any additional information. The conditional establishes a relationship between A and B, but does not clarify its true value.

• On the other hand, the logical implication «A, therefore B" is a non-hypothetical affirmation but with truth content that expresses something of reality; that is, it clearly states that A is true, and that therefore B is true. It is possible to establish the value of A, and B, without any additional entry.

The difference between the two also depends on the field in which you are working. In mathematical logic, the fundamental difference between the two is that the material conditional is a truth function that can be either true or false, while the implication is always true—it is therefore a tautology—that is, there is a logical impossibility for the statement "If A, then B" to be false. For this reason, the precise way to express implication is "A implies B" or "A is a sufficient condition of B." This is analogous to the statement "B is a necessary condition of A."

Common properties

There is, however, a close relationship between the two in most logical systems, including classical logic. For example, the following principles are held:

• Yeah. ${displaystyle Gamma vdash A}$, then ${displaystyle (vdash Gamma)to A}$where A is any formula and ${displaystyle Gamma }$ It's a set of any formulas. This is a particular case of the theorem of deduction.

• Yeah. ${displaystyle (vdash Gamma)to A}$, then ${displaystyle Gamma vdash A}$. This is a particular case of the reverse theorem of deduction.

• Both the material conditional and the logical consequence are monotonous. I mean, yeah. ${displaystyle Gamma vdash A}$, then ${displaystyle Delta cup Gamma vdash A}$ And yes. ${displaystyle Ato B,}$, then ${displaystyle (Aland C)to B}$.

In these examples the trinquet symbol ()) has been used as a substitute for ${displaystyle to }$ or ${displaystyle Rightarrow }$. These principles, however, do not apply in all logical systems. For example, they are not held in non-monotonical logics.

Philosophical problems around the material conditional

Outside of mathematics, there is some controversy about whether the truth function of material implication provides an adequate treatment of conditional sentences in a natural language such as Spanish, that is, indicative and counterfactual conditionals. An indicative conditional is a statement in the indicative mood with an attached conditional clause. A counterfactual conditional is a statement from false to fact in the subjunctive mood. That is, critics maintain that in some non-mathematical cases, the truth value of a compound statement, "if p then q", is not adequately determined by the truth values of p and q. Examples of non-truth statements- Functional include: "q because p", "p before q" and "it is possible that p".

"[Of] the sixteen possible truth functions of A and B, material implication is the only serious candidate. First of all, it is indisputable that when A is true and B is false, "If A, B i>" it's false. A basic rule of inference is the modus dondendoPONs: from "If A, B" and A, B can be inferred. If it were possible to have A true, B false and "If A, B" true, this inference would be invalid. Secondly, it is indisputable that "If A, B" is sometimes true sometimes when A and B are, respectively, (true, true), or (false, true), or (false, false)... Non-true functional accounts agree that "If A, B" is false when A is true and B is false; and they agree that the conditional is sometimes true for the other three combinations of truth values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with truth functionalism that when A and B are both true, "If A, B " must be true. Some do not, demanding an additional relationship between the facts that A and that B.

The functional-truth theory of the conditional was an integral part of Frege's new logic (1879). It was enthusiastically accepted by Russell (who called it "material application"), Wittgenstein in the Tractatus, and the logical positivists, and is now found in all text of logic. It's the first conditional theory students find. Normally, students don't get the impression that it's obviously right. It's the first surprise of logic. However, as the textbooks testify, it does a creditworthy job in many circumstances. And he has many defenders. It is a surprisingly simple theory: "If A, B" is false when A is true and B is false. In all other cases, "Yes A, B" is true. Therefore, it is equivalent to "~(A fake~B##A or B". "A ▪ B"has, by stipulation, these conditions of truth.

Dorothy Edgington, Stanford Philosophy Encyclopedia, "Conditional"

The meaning of the material conditional can sometimes be used in the Spanish construction "If condition then consequence" (a type of conditional statement), where the condition and consequence must be filled in with sentences in Spanish. However, this construction also implies a "reasonable" between the condition (protasis) and the consequence (apodosis) (see Connective Logic).^{[citation required]}

Conditional material can produce some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see empty truth). So the statement "if 2 is odd then 2 is even" is true. Likewise, any conditional material with a true consequent is true. So the statement "if I have a penny in my pocket then Paris is in France" It's always true, regardless of whether or not there's a penny in my pocket. These problems are known as the paradoxes of material implication, although they are not really paradoxes in the strict sense; that is, they do not cause logical contradictions. These unexpected truths arise because speakers of Spanish (and other natural languages) are tempted to mistake between the material conditional and the indicative conditional or other conditional statements, such as the counterfactual conditional and the material biconditional.

It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by the statements 'If...then...' in natural languages. For an overview of some of the various analyses, formal and informal, of conditionals, see the "References" next. The relevant logic attempts to capture these alternative concepts of implication that material implication leaves out.