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(You'll also sometimes see that symbol used in sentences of the form "PREMISES |= conclusion"; this is shorthand for the relation that "all models that make PREMISES true also make conclusion true")

The power of deductive systems and the simplicity of the notion of ontology of this kind of framework is what makes it so widespread in the methodology of analytical philosophy, mathematics and science. It is by no means the only, or even necessarily the most appropriate, logic for use in formulating languages about particular subjects, and it is known to have several limitations and unusual consequences. Nonethteless, it is certainly a system that has become well established in a great many areas of study in the mathematical sciences.

(You'll also sometimes see that symbol used in the form "PREMISES |= conclusion"; this is shorthand for the Semantic Entailment relation: "all models that make PREMISES true also make conclusion true")

The power of deductive systems and the simplicity of the notion of ontology of this kind of framework is what makes it so widespread in the methodology of analytical philosophy, mathematics and science. It is by no means the only, or even necessarily the most appropriate, logic for use in formulating languages about particular subjects, and it is known to have several limitations and unusual consequences. Nonetheless, it is certainly a system that has become well established in a great many areas of study in the mathematical sciences.

What makes Logic interesting is that deductive systems and semantics can be given rigorous treatments that tell us about what we can do with formulae. For instance, in Classical Propositional Logic, True or False are understood as forming a two-valued Boolean Algebra - a mathematical structure described in, for example, set theory or category theory. This means we can talk about particular kinds of mathematical function that we know are described in that algebra - if you have a true sentence A and a false sentence B, then according to the algebra, we can evaluate A ^ B as a false sentence (interpreting ^ as a function that takes T and F to F), and A v B as a true sentence (interpreting v to take T and F to T).

So in Classical Propositional Logic, rather than having to go through and manually specify values for every single formula, you can instead require that the interpretations assign values to all of the Basic formulae (like the atomic propositions) and build up values for more complex sentences inductively. This structure means you can have a theory of deductive inference that "gets things right" on the level of interpreting sentences in line with their mathematical relationships.

What makes Logic interesting is that deductive systems and semantics can be given rigorous treatments that tell us about what we can do with formulae. For instance, in Classical Propositional Logic, True and False are understood as forming a two-valued Boolean Algebra - a mathematical structure described in set theory and category theory. This means we can talk about particular kinds of mathematical function that we know are described in that algebra - if you have a true sentence A and a false sentence B, then according to the algebra, we can evaluate A ^ B as a false sentence (interpreting ^ as a function that takes T and F to F), and A v B as a true sentence (interpreting v to take T and F to T).

So in Classical Propositional Logic, rather than having to go through and manually specify values for every single formula, you can instead require that the interpretations assign values to all of the basic formulae (the Atomic propositions) and build up values for more complex sentences inductively. This structure means you can have a theory of deductive inference that "gets things right" on the level of interpreting sentences in line with their mathematical relationships.

What is generally now called Classical logic is the framework that is thought to encapsulate both the Predicate calculus formalism and the compositional aspect that a structured, two-valued algebra of logic gave us in the propositional case. Predicate logic works with the idea that our domain of objects is something that we refer to in the logical structure of making assertions - when I say that "That chair is green", the logical form of what I'm saying actually identifies something in the domain to be "that chair", and to attribute to it the property of "being green". The logical structure of language is such that what I say is true when it turns out that "being green" is, according to the interpretation we make, something that the object that is "that chair" satisfies. We can introduce first-order Variables and Quantifiers to make generalizations about whether properties apply to some object, every object, no object etc. And in the Classical framework, every property is understood to either determinately apply to a given object or to determinately fail to apply to a given object; the payoff being that we can use another (more complex) algebraic analysis to let us build up expressive and complete deductive systems for our logic.

The power of deductive systems and the simplicity of the notion of ontology of this kind of framework is what makes it so widespread in the methodology of analytical philosophy, mathematics and science. It is by no means the only, or even necessarily the most appropriate, logic for use in formulating languages about particular subjects, but it is certainly one that has become well established in a great many areas of study.

What is generally now called Classical logic is the framework that is thought to encapsulate both the Predicate calculus formalism and the compositional aspect that a structured, two-valued algebra of logic gave us in the propositional case. Predicate logic works with the idea that our domain of objects is something that we refer to in the logical structure of making assertions - when I say that "That chair is green", the logical form of what I'm saying actually identifies something in the domain to be "that chair", and to attribute to it the property of "being green". The logical structure of language is such that what I say is true when it turns out that "being green" is, according to the interpretation we make, something that the object that is "that chair" satisfies. We can introduce first-order Variables and Quantifiers to make generalizations about whether properties apply to some object, every object, no object etc. And in the Classical framework, every property is understood to either determinately apply to a given object or to determinately fail to apply to that given object; the payoff being that we can use another (more complex) algebraic analysis to let us build up expressive and complete deductive systems for our logic.

The power of deductive systems and the simplicity of the notion of ontology of this kind of framework is what makes it so widespread in the methodology of analytical philosophy, mathematics and science. It is by no means the only, or even necessarily the most appropriate, logic for use in formulating languages about particular subjects, and it is known to have several limitations and unusual consequences. Nonethteless, it is certainly a system that has become well established in a great many areas of study in the mathematical sciences.