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Trying to better understand the concept of premise and conclusion.

In classic propositional logic, I think it usually takes on the following form: $p \to q$, where $p$ is the premise and $q$ is the conclusion.

But doesn't this depend on our semantic interpretation of the $\to$ operator? And the very concept of "premise" and "conclusion" being defined and only making sense in this particular logical framework?

Or is "premise" and "conclusion" an even stronger claim we can define on the metalogical level, e.g. $p \vdash q$ or something to this effect, that would apply to any logic system?

blub
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user525966
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  • NO, see Argument : "In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable) intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language." – Mauro ALLEGRANZA Sep 06 '18 at 14:53
  • See your previous post : $p \to q$ is a formula; a conditional with $p$ as antecedent and $q$ as consequent. – Mauro ALLEGRANZA Sep 06 '18 at 15:11
  • I know that $\Gamma \vdash \varphi$ is a "syntactic consequence" saying that $\varphi$ is "provable" from $\Gamma$ but I don't really know what this means. I also know that $\Gamma \vDash \varphi$ is a "semantic consequence" saying that $\varphi$ is true if everything in $\Gamma$ is true, but I don't know if these are concepts that apply to any system, nor what they really mean – user525966 Sep 06 '18 at 16:24
  • Is it possible to talk about multiple premises and conclusions? e.g. $a, b, c, d \vdash j, k, l$ – user525966 Sep 06 '18 at 16:27
  • premises can be many; usually, the conclusion is one. – Mauro ALLEGRANZA Sep 06 '18 at 16:47

2 Answers2

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An argument is a linguistic "object":

In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (both spellings are acceptable) intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language.

The concept of valid (deductive) argument has been defined firstly by Aristotle :

A deduction is speech (logos) in which, certain things having been supposed, something different from those supposed results of necessity because of their being so. (Prior Analytics, I.2, 24b18–20)

Each of the “things supposed” is a premise (protasis) of the argument, and what “results of necessity” is the conclusion (sumperasma).

The key discovery of Aristotle is that, in order to assess the validity of an argument, we have to consider its Logical Form.

In order to do this, is useful to "formalize" an argument using variable (i.e. reducing the linguistic argument to its "schematic" structure); see Syllogism :

Major premise: All $M$ are $P$.

Minor premise: All $S$ are $M$.

Conclusion: All $S$ are $P$.

Modern mathematical logic has improved "formalization" using the modern mathematical symbols developed for algebra.

Propositional logic is useful because in it we can have a simplified model of language: it proxy statements of natural language with propositional symbols (or variables). Thus, propositional logic provides a simple model for deductive arguments.

In propositional logic we define a formal counterpart of entailment (or: logical consequence) : $Γ⊨φ$.

The symbol reads : "formula $φ$ is a logical (or: tautological, in the case of propositional logic) consequence of the set of formulas $Γ$" and it is defined in terms of semantical concept: truth assignments (or interpretations).

The semantical concepts are related to the syntactical ones: setting up the logical calculus, we introduce rules of inference that allow us to infer a formula (the conclusion) from an initial set of formulas (the premises).

With them we define the relation of derivability, defined as follows: "$Γ ⊢ φ$ iif there is a derivation with conclusion $φ$ and with all hypotheses (or assumptions) in $Γ$."

A derivation, in turn, is a finite sequence of applications of rules of inference.

The two sides: semantical and syntactical, are linked by the property of soundness and completeness.

In propositional logic, $p \to q$ is a formula: it is a conditional with $p$ as antecedent and $q$ as consequent.

$p → q,p ⊢ q$ is the formal counterpart of a valid argument (modus ponens), where $p → q$ and $p$ are the premises and $q$ is the conclusion.

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Mauro ALLEGRANZA
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But doesn't [the meaning of p→q] depend on our semantic interpretation of the → operator? And the very concept of "premise" and "conclusion" being defined and only making sense in this particular logical framework?

The answer from mathematical logic is yes; for instance, propositional logic has a particular form of logical entailment called the material conditional. There are other forms of conditionals that have different truth mappings. From WP:

Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.

In fact, there are whole books which delve into the semantic interpretation of logical consequence which are of interest to logicians because the if-then of formal systems of mathematics comes in lots of flavors. Formal systems are language games in the Wittgensteinian sense, and are abstractions that are not free of interpretation and convention as noted by the private language argument.

J D
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