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It is easier to list the differences than similarities between the two kinds of logic.

Mathematical logic is the branch of mathematics that studies mathematical activity. It has all the usual properties of a mathematical branch. It uses the standard mathematical methods, such as the axiomatic method, informal set theory, and symbolic notation. It idealizes away from the actual situation by ignoring many aspects of actual mathematical activity, for example, it focuses mostly on how mathematical statements are proved and what they mean, but says little about conjectures, analogies, elegance, or about mathematics as a human activity. It phrases its results in terms of mathematical theorems (as opposed to, say, critical essays or historical studies).

Philosophical logic on the other hand attempts to attack its object of interest, which we could broadly characterize as reasoning, as a whole and from many different angles, as is customary in philosophy. Thus, apart from using the deductive method, we might consider linguistic aspects of logic, or logic as it relates to religion, we might learn something about logic by looking at its historical development, we may put it in the sociological context, etc. Consequently, no single treatment of logic will be accepted as a comprehensive one by (good) philosophers. In fact, it will be hard to get philosophers to agree on what precisely philosophical logic is.

It is naive to think of mathematical logic as being superior to philosophical logic. Certainly, mathematical logic is superior to philosophical logic in certain aspects, but one should never forget that the scope of mathematical logic is very narrow and it is therefore not surprising that mathematical logic boasts with deeper and technically more complicated insights than philosophical logic.

2 added 50 characters in body

It is easier to list the differences than similarities between the two kinds of logic.

Mathematical logic is a the branch of mathematics which that studies mathematical activity. It has all the usual properties of a mathematical branch. It uses the standard mathematical methods, such as the axiomatic method, informal set theory, and symbolic notation. It idealizes away from the actual situation by ignoring many aspects of actual mathematical activity, for example, it only considers focuses mostly on how mathematical statements are proved and proofswhat they mean, but says little about conjectures, analogies, elegance, or about mathematics as a human activity. It phrases its results in terms of mathematical theorems (as opposed to, say, critical essays or historical studies).

Philosophical logic on the other hand attempts to attack its object of interest as a whole and from many different angles, as is customary in philosophy. Thus, apart from using the deductive method, we might consider linguistic aspects of logic, or logic as it relates to religion, we might learn something about logic by looking at its historical development, we may put it in the sociological context, etc. Consequently, no single treatment of logic will be accepted as a comprehensive one by (good) philosophers. In fact, it will be hard to get philosophers to agree on what precisely philosophical logic is.

It is naive to think of mathematical logic as being superior to philosophical logic. Certainly, mathematical logic is superior to philosophical logic in certain aspects, but one should never forget that the scope of mathematical logic is very narrow and it is therefore not surprising that mathematical logic boasts with deeper and technically more complicated insights than philosophical logic.

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It is easier to list the differences than similarities between the two kinds of logic.

Mathematical logic is a branch of mathematics which studies mathematical activity. It has all the usual properties of a mathematical branch. It uses the standard mathematical methods, such as the axiomatic method, informal set theory, and symbolic notation. It idealizes away from the actual situation by ignoring many aspects of actual mathematical activity, for example, it only considers statements and proofs, but says little about conjectures, analogies, or about mathematics as a human activity. It phrases its results in terms of mathematical theorems (as opposed to, say, critical essays or historical studies).

Philosophical logic on the other hand attempts to attack its object of interest as a whole and from many different angles, as is customary in philosophy. Thus, apart from using the deductive method, we might consider linguistic aspects of logic, or logic as it relates to religion, we might learn something about logic by looking at its historical development, we may put it in the sociological context, etc. Consequently, no single treatment of logic will be accepted as a comprehensive one by (good) philosophers. In fact, it will be hard to get philosophers to agree on what precisely philosophical logic is.

It is naive to think of mathematical logic as being superior to philosophical logic. Certainly, mathematical logic is superior to philosophical logic in certain aspects, but one should never forget that the scope of mathematical logic is very narrow and it is therefore not surprising that mathematical logic boasts with deeper and technically more complicated insights than philosophical logic.