In a liquid, the pressure is exerted by the weight of the liquid. As the force due to it's weight acts downwards, why is there an upward force at any point in a liquid?

Pressure ("hydrostatic") forces on a submerged body get evenly distributed around it because the liquid in which it is submerged cannot transmit shear forces.

The buoyant force acting upwards on a submerged body arises when the submerged body weighs less than the liquid it displaces. This means it behaves like a bubble, and experiences a net force which acts to move it in the direction opposite to the direction in which gravity points.

The defining characteristic of a fluid is that the shear stress is proportional to the shear rate. For a fluid in hydrostatic equilibrium this means that the shear stress must be zero everywhere.

When you decompose the strain tensor into the principal strains then the maximum shear strain is given by half the difference between the maximum and minim eigenvalues. Since that quantity must be zero it follows that all of the principal strains must be the same. This means that a plane in any direction, including upwards, will receive the same force.

If it were otherwise then there would be some direction where there would be shear and therefore the fluid would flow.

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